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Tigray Region

The Tigray Region (or simply Tigray; officially the Tigray National Regional State) is the northernmost regional state in Ethiopia. The Tigray Region is the homeland of the Tigrayan, Irob and Kunama people. Its capital and largest city is Mekelle. Tigray is the fifth-largest by area, the fourth-most populous, and the fifth-most densely populated of the 11 regional states.

Tigray is bordered by Eritrea to the north, the Amhara Region to the south, the Afar Region to the east, and Sudan to the west. Towns in Tigrai include: Mekelle, Adigrat, Axum, Shire, Adwa, Humera, Dansha, Mai Kadra, Enticho, Wukro, Sheraro, Abiy Addi, Atsbi, Hawzen, Adi Gudom, Korem, Maychew, Alamata, Mekoni, Rama, May Tsebri, Addi Remets, Hagere Selam, and Zalambessa.

Tigray's official language is Tigrinya, similar to that of southern Eritrea. The estimated population as of 2019 is approximately 5,443,000. The majority of the population (c. 80%) are farmers, contributing 46% to the regional gross domestic product (2009). The highlands have the highest population density, especially in eastern and central Tigray. The much less densely populated lowlands comprise 48% of Tigray's area. Although the percentage of Muslims in Tigray is less than 5%, it has supposedly been historically Islam’s doorway to the region and to Africa at large. 96% of Tigrayans are Orthodox Christian. After Armenians, ethnic Tigrayans have the highest percentage of Orthodox Christians in the world.

The government of Tigray consists of the executive branch, led by the president, Getachew Reda; the legislative branch, which comprises the state council; and the judicial branch, which is led by the state supreme court. In early November 2020, a conflict between the Tigray People's Liberation Front (TPLF) and the Ethiopian federal government (with support from Eritrea) rapidly escalating into the Tigray War, destabilizing the region, and exposing a well-organized campaign to wipe out the region of ethnic Tigrayans. As many as 600,000 people were killed as a result of the war. As of 2023, the region is run by the Interim Regional Administration of Tigray.

3rd millennium to 1st century BC
Tigray is often regarded as the cradle of Ethiopian civilization. Its landscape has many historic monuments. Three major monotheistic religions, Judaism, Christianity and Islam arrived in Ethiopia through the Red Sea and then Tigray.

Given the presence of a large temple complex and fertile surroundings, the capital of the 3,000-year-old kingdom of Dʿmt may have been near present-day Yeha. Dʿmt developed irrigation schemes, used the plough, grew millet, and made iron tools and weapons. Some modern historians, including Stuart Munro-Hay, Rodolfo Fattovich, Ayele Bekerie, Cain Felder, and Ephraim Isaac consider this civilization to be indigenous, although Sabaean-influenced due to the latter's dominance of the Red Sea. Others, including Joseph Michels, Henri de Contenson, Tekletsadik Mekuria, and Stanley Burstein, have viewed Dʿmt as the result of a mixture of Sabaean and indigenous peoples. The most recent research, however, shows that Ge'ez, the ancient Semitic language spoken in Tigray, Eritrea and northern Ethiopia in ancient times, is not likely to have been derived from Sabaean. There is evidence of a Semitic-speaking presence in Tigray, Eritrea and northern Ethiopia at least as early as 2000 BC. It is now believed that Sabaean influence was minor, limited to a few localities and disappearing after a few decades or a century, It may have represented a trading or military colony, in some sort of symbiosis or military alliance with the civilization of Dʿmt or some other proto-Aksumite state.

After the fall of Dʿmt in the 5th century BC, the plateau came to be dominated by smaller, unknown successor kingdoms. This lasted until the rise of one of these polities during the first century BC, the Aksumite Kingdom, which succeeded in reunifying the area and is, in effect, the ancestor of medieval and modern states in Eritrea and Ethiopia using the name "Ethiopia" as early as the 4th century.

1st to 10th century AD

The Kingdom of Aksum was a trading empire rooted in northern Ethiopia. It existed from approximately 100–940 AD, growing from the proto-Aksumite Iron Age period c. 4th century BC to achieve prominence by the 1st century AD.

According to the Book of Axum, Axum's first capital, Mazaber, was built by Itiyopis, son of Cush. The capital was later moved to Aksum in northern Ethiopia.

The Empire of Aksum, at its height, at times extended across most of present-day Eritrea, Ethiopia, Djibouti, Sudan, Yemen and Saudi Arabia. The capital city of the empire was Axum, now in northern Ethiopia. Today a smaller community, the city of Axum was once a bustling metropolis and a cultural and economic hub. Two hills and two streams lie on the east and west expanses of the city; perhaps providing the initial impetus for settling this area. Along the hills and plain outside the city, the Aksumites had cemeteries with elaborate grave stones, which are called stelae, or obelisks. Other important cities included Yeha, Hawulti-Melazo, Matara, Adulis, and Qohaito, the last three of which are now in Eritrea. By the reign of Endubis in the late 3rd century, Aksum had begun minting its own currency and was named by Mani as one of the four great powers of his time, along with China and the Sassanid and Roman empires. It converted to Christianity in 325 or 328 under King Ezana and was the first state to use the image of the cross on its coins.

11th to 19th century AD

In the 11th century the Tigrinya-speaking lands (Tigray-Mareb Melash) were divided into two provinces, separated by the Mereb River, by the newly enthroned Agaw emperors. The governor of the northern province received the title Bahre Negash (Ruler of the sea), whereas the governor of the southern province was given the title of Tigray Mekonen (Lord of Tigray). The Portuguese Jesuit Emanuele Baradas's work titled "Do reino de Tigr", written in 1633–34, states that the "Reino de Tigr" (Kingdom of Tigray) extended from Hamasien to Temben, from the borders of Dankel to the Adwa mountain. He also stated that Tigray-Mereb Melash was divided into 24 smaller political units (principalities), twelve of which were located south of the Mereb and governed by the Tigray Mekonen, based in Enderta. The other twelve were located north of the Mereb, under the authority of the Bahre Negash, based in the district of Serae.

The Book of Aksum, written and compiled minly in the period from the sixteenth to seventeenth centuries, shows a traditional schematic map of Tigray with the city of Aksum at its center, surrounded by the 13 principal provinces: "Tembien, Shire, Serae, Hamasien, Bur, Sam’a, Agame, Amba Senayt, Garalta, Enderta, Sahart and Abergele."

During the Middle Ages, the position of Tigray Mekonnen ("Governor of Tigray") was established to rule over the area. Other districts included Akele Guzay (now part of Eritrea), and the kingdom of the Bahr negus, who ruled much of what is now Eritrea and Shire district and town in Western Tigray. At the time when Tigray Mekonnen existed simultaneously with that of Bahr negus, their frontier seems to have been the Mareb River, which is currently constitutes the border between the Ethiopian province of Tigray and Eritrea.

After the loss of power of the Bahr negus in the aftermath of Bahr negus Yeshaq's rebellions, By the unsettled Zemene Mesafint period ("Era of the Princes"), both designations had declined to little more than empty titles, and the lord who succeeded them used (and received from the Emperor) the title of either Ras or Dejazmach, beginning with Ras Mikael Sehul. Rulers of Tigray such as Ras Wolde Selassie alternated with others, chiefly those of Begemder or Yejju, as warlords to maintain the Ethiopian monarchy during the Zemene Mesafint.

In the mid-19th century, the lords of Tembien and Enderta managed to establish an overlordship of Tigray. One of its members, Dejazmach Kahsay Mercha, ascended the imperial throne in 1872 under the name Yohannes IV. Following his 1889 death in the Battle of Metemma, the Ethiopian throne came under the control of the king of Shewa, and the center of power shifted south and away from Tigray.

20th century
In 1943, a rebellion broke out all over southern and eastern Tigray under the slogan, "there is no government; let's organize and govern ourselves". Throughout Enderta Awraja, including Mekelle, Didibadergiajen, Hintalo, Saharti, Samre and Wajirat, Raya Awraja, Kilte-Awlaelo Awraja and Tembien Awraja, local assemblies, called gerreb, were formed. The gerreb sent representatives to a central congress, called the shengo, which elected leaders and established a military command system. Although the first Woyane rebellion of 1943 had shortcomings as a prototype revolution, historians agree that it involved a fairly high level of spontaneity and peasant initiative. It demonstrated considerable popular participation and reflected widely shared grievances. The uprising was specifically directed against the central Shoan Amhara regime of Haile Selassie I, rather than the Tigrayan imperial elite.

Ethiopian Civil War

Following the outbreak of the Ethiopian Revolution in February 1974, the first signal of any mass uprising was the actions of the soldiers of the 4th Brigade of the 4th Army Division in Nagelle in southern Ethiopia. The Coordinating Committee of the Armed Forces, Police, and Territorial Army, or the Derg (Ge'ez "Committee"), was officially announced 28 June 1974 by a group of military officers. The committee elected Major Mengistu Haile Mariam as its chairman and Major Atnafu Abate as its vice-chairman. In July 1974, the Derg obtained key concessions from the emperor, Haile Selassie, which included the power to arrest not only military officers but government officials at every level. Soon both former Prime Ministers Tsehafi Taezaz Aklilu Habte-Wold and Endalkachew Makonnen, along with most of their cabinets, most regional governors, many senior military officers and officials of the Imperial court were imprisoned. In August 1974, after a proposed constitution creating a constitutional monarchy was presented to the emperor, the Derg began a program of dismantling the imperial government in order to forestall further developments in that direction. The Derg deposed and imprisoned the emperor on 12 September 1974.

In addition, the Derg in 1975 nationalized most industries and private and somewhat secure urban real-estate holdings. But mismanagement, corruption, and general hostility to the Derg's violent rule, coupled with the draining effects of constant warfare with the separatist guerrilla movements in Tigray, led to a drastic fall in general productivity of food and cash crops. In October 1978, the Derg announced the National Revolutionary Development Campaign to mobilize human and material resources to transform the economy, which led to a Ten-Year Plan (1984/1985-1993/1994) to expand agricultural and industrial output, forecasting a 6.5% growth in GDP and a 3.6% rise in per capita income. Instead per capita income declined 0.8% over this period. Famine scholar Alex de Waal observes that while the famine that struck the country in the mid-1980s is usually ascribed to drought, "closer investigation shows that widespread drought occurred only some months after the famine was already under way". Hundreds of thousands fled economic misery, conscription, and political repression, and went to live in neighboring countries and all over the Western world, creating an Ethiopian diaspora.

Toward the end of January 1991, a coalition of rebel forces, the Ethiopian People's Revolutionary Democratic Front (EPRDF) captured Gondar, the ancient capital city, Bahar Dar, and Dessie.

Postwar
John Young, who visited the area several times in the early 1990s, attributes this delay in part to "central budget restraint, structural readjustment, and lack of awareness by government bureaucrats in Addis Ababa of conditions in the province", but notes "an equally significant obstacle was posed by an entrenched, and largely Amhara-dominated, central bureaucracy which used its power to block government-authorised funds from reaching Tigray". At the same time, a growing urban middle class of traders, businessmen and government officials emerged that was suspicious of and distant from the victorious EPRDF. From 1991 to 2001, the president of Tigray was Gebru Asrat. In 1998, war erupted between Eritrea and Ethiopia over a portion of territory that had been administered as part of Tigray, which included the town of Badme. A 2002 United Nations decision awarded much of this land to Eritrea, but Ethiopia did not accept the ruling until 2018, when a bilateral agreement ended the border conflict. The text of this agreement has not been publicly availed.

21st century
From 2001 to 2010 the president was Tsegay Berhe.

2020 administrative reorganisation
Between 2018 and 2020, as part of a reform aimed to deepen and strengthen decentralisation, woredas were reorganised, and new boundaries established. As smaller towns had been growing, they had started providing a larger range of services, such as markets and even banks, that encouraged locals to travel there rather than to their formal woreda centre. However, these locals still had to travel to their local woreda centre for most local government services - often in a different direction. In 2018 and 2019, after multiple village discussions that were often vigorous in the more remote areas, 21 independent urban administrations were added and other boundaries re-drawn, resulting in an increase from 35 to 88 woredas in January 2020.

Tigray War
Following the 2020 Tigray regional election, on 4 November, the Ethiopian military launched attacks on the government headquarters in Mekelle, initiating the Tigray War after months of preparation with the Eritrean army. Ethiopian and Amhara forces advanced through southern Tigray, while Eritrean troops occupied northern border towns. Amhara militias continued to control Western Tigray as of 2023.

Warfare, the COVID-19 pandemic in Ethiopia, and a locust outbreak contributed to an emergency food situation in the region by January 2021. Approximately two million people faced food shortages, with a critical situation in Shire Inda Selassie, hosting 100,000 refugees. The Famine Early Warning Systems Network indicated that parts of central and eastern Tigray were likely in emergency phase 4, a step below famine.

Location and size
Tigray is situated between 12° – 15°N and 36° 30' – 40° 30'E.

A 2006 national statistics report stated the land area as 50079 km2. The 2011 National Statistics gave an area of 41410 km2, but the sum of the figures it gave for the Tigray zones was substantially different, rendering the 2011 report internally inconsistent. The figure of 50,079 km2 is supported by the Google Maps area calculator.

Overview
The East African Orogeny led to the growth of a mountain chain in the Precambrian (up to 800 Ma [million years ago]), which was largely eroded afterwards. Around 600 Ma, the Gondwana break-up led to the presence of tectonic structures and a Palaeozoic planation surface, that extents to the north and west of the Dogu'a Tembien massif.

Subsequently, there was the deposition of sedimentary and volcanic formations, from older (at the foot of the massif) to younger, near the summits. From Palaeozoic to Triassic, Tigray was located near the South Pole. The (reactivate) Precambrian extensional faults guided the deposition of glacial sediments (Edaga Arbi Glacials and Enticho Sandstone). Later alluvial plain sediments were deposited (Adigrat Sandstone). The break-up of Gondwana (Late Palaeozoic to Early Triassic) led to an extensional tectonic phase, what caused the lowering of large parts of the Horn of Africa. As a consequence a marine transgression occurred, leading to the deposition of marine sediments (Antalo Limestone and Agula Shale). The region has an estimated 3.89 billion tons of mostly "excellent" quality oil shale. At the end of the Mesozoic tectonic phase, a new (Cretaceous) planation took place. After that, the deposition of continental sediments (Amba Aradam Formation) indicates the presence of less shallow seas, probably caused by a regional uplift. At the beginning of the Caenozoic, there was a relative tectonic quiescence, during which the Amba Aradam Sandstones were partially eroded, which led to the formation of a new planation surface.

In the Eocene, the Afar plume, a broad regional uplift, deformed the lithosphere, leading to the eruption of flood basalts. Three major formations may be distinguished: lower basalts, interbedded lacustrine deposits and upper basalts. Almost at the same time, the Mekelle Dolerite intruded into the Mesozoic sediments, following joints and faults.

A new magma intrusion occurred in the Early Miocene, which gave rise to phonolite plugs, mainly in the Adwa area and also in Dogu’a Tembien. The present geomorphology is marked by deep valleys, eroded as a result of the regional uplift. Throughout the Quaternary, deposition of alluvium and freshwater tufa occurred in the valley bottoms.

Fossils
In Tigray, there are two main fossil-bearing geological units. The Antalo Limestone (upper Jurassic) is the largest. Its marine deposits comprise mainly benthic marine invertebrates. Also, the Tertiary lacustrine deposits, interbedded in the basalt formations, contain a range of silicified mollusc fossils.

In the Antalo Limestone: large Paracenoceratidae cephalopods (nautilus); Nerineidae indet.; sea urchins; Rhynchonellid brachiopod; crustaceans; coral colonies; crinoid stems.

In the Tertiary silicified lacustrine deposits: Pila (gastropod); Lanistes sp.; Pirenella conica; and land snails (Achatinidae indet.).

All snail shells, both fossil and recent, are called t’uyo in Tigrinya language, which means ‘helicoidal’.

Traditional uses of rock
As Tigray holds a wide variety of rock types, there is expectedly a varied use of rock.
 * Natural stone masonry. Preferentially, the easier shaped limestone and sandstone are used to build homesteads and churches, but particularly in the upland areas, basalt is also used. Traditionally, fermented mud will be used as mortar
 * Fencing of homesteads, generally in dry stones
 * Church bells, generally three elongated plates in phonolite or clinkstone, with different tonalities
 * Milling stone: for this purpose plucked-bedrock pits, small rock-cut basins that naturally occur in rivers with kolks, are excavated from the river bed and further shaped. Milling is done at home using an elongated small boulder
 * Door and window lintels, prepared from rock types that frequently have an elongated shape (sandstone, phonolite, limestone), or that are easily shaped (tufa)
 * Troughs for livestock watering and feeding, generally hewn from tufa
 * Footpath paving, generally done as community work. Some very ancient paved footpaths occur on major communication lines dating back to the period before the introduction of the automobile
 * foot travellers stop, pray and put an additional stone
 * Stones collected from farmlands in order to free space for the crop, and heaped in typical rounded metres-high heaps, called zala
 * Contour bunding or gedeba: terrace walls in dry stone, typically laid out along the contour for sake of soil conservation
 * Check dams or qetri in gullies for sake of gully erosion control
 * Cobble stones, used for paving secondary streets in the towns. Generally limestone is used.

Major mountains

 * Ferrah Imba, 3954 metres, summit of the Tsibet massif in Endamekoni woreda (12.84278°N, 39.5175°W), and highest peak of Tigray
 * Imba Alaje, 3438 metres, in Alaje woreda (12.99417°N, 39.50361°W)
 * Mugulat, 3263 metres, in Ganta Afeshum woreda (14.26667°N, 39.41667°W); one of its spurs is crossed by the Siqurto foot tunnel
 * Asimba, 3199 metres, in Irob woreda (14.43333°N, 39.6°W)
 * Upper plateaus of the Atsbi Horst at 3057 metres in Atsbi Wenberta woreda (14.05°N, 39.71667°W)
 * Maebino, 3031 metres, in Irob woreda (14.45°N, 39.55°W)
 * Imba Tsion, 2917 metres, in Hawzen woreda (14.1°N, 39.43333°W)
 * Ekli Imba, 2799 metres, summit of the Arebay massif in Degua Tembien woreda (13.71667°N, 39.26667°W)
 * Imba Aradom – sometimes transliterated as Amba Aradam, 2756 metres, in Hintalo Wajirat woreda (13.33333°N, 39.51667°W)
 * Soloda, 2436 metres, part of the Adwa plugs in Adwa woreda (14.18333°N, 38.9°W)
 * Imba Neway, 2388 metres, in Abergele (woreda)(13.26667°N, 38.95°W)

Water challenge
Overall, the region is semi-arid. The wet season lasts only for a couple of months. The farmers are adapted to this, but the problem arises when rains are less than normal. Another major challenge is providing water to urban areas. Smaller towns, but particularly Mekelle, face endemic water shortages. Reservoirs have been built, but their management is sub-optimal.

Large mammals
Besides elephants in Western Tigray and the endemic gelada baboon on the highest mountains, large mammals in the region, with scientific (italics), English and Tigrinya language names, are:

 * Cercopithecus aethiops; grivet monkey, ወዓግ
 * Crocuta crocuta, spotted hyena, ዝብኢ
 * Caracal caracal, caracal, ጭክ ኣንበሳ
 * Panthera pardus, leopard, ነብሪ
 * Xerus rutilus, unstriped ground squirrel, ምጹጽላይ or ጨጨራ
 * Canis mesomelas, black-backed jackal, ቡኳርያ
 * Canis anthus, golden jackal, ቡኳርያ
 * Papio hamadryas, hamadryas baboon, ጋውና
 * Procavia capensis, rock hyrax, ጊሐ
 * Felis silvestris, African wildcat, ሓክሊ ድሙ
 * Civettictis civetta, African civet, ዝባድ
 * Papio anubis, olive baboon, ህበይ
 * Ichneumia albicauda, white-tailed mongoose, ፂሒራ
 * Herpestes ichneumon, large grey mongoose, ፂሒራ
 * Hystrix cristata, crested porcupine, ቅንፈዝ
 * Oreotragus oreotragus; klipspringer, ሰስሓ
 * Orycteropus afer, aardvark, ፍሒራ
 * Genetta genetta, common genet, ስልሕልሖት
 * Lepus capensis, cape hare, ማንቲለ
 * Mellivora capensis, honey badger, ትትጊ

Small rodents
The most common pest rodents with widespread distribution in agricultural fields and storage areas are three Ethiopian endemic species: the Dembea grass rat (Arvicanthis dembeensis, sometimes considered a subspecies of Arvicanthis niloticus), Ethiopian white-footed rat (Stenocephalemys albipes), and Awash multimammate mouse (Mastomys awashensis).

Bats
Bats occur in natural caves, church buildings and abandoned homesteads. The large colony of bats that roosts in Zeyi cave comprises Hipposideros megalotis (Ethiopian large-eared roundleaf bat), Hipposideros tephrus, and Rhinolophus blasii (Blasius's horseshoe bat).

Birds
With its numerous exclosures, forest fragments and church forests, Tigray is a birdwatcher's paradise. Detailed inventories list at least 170 bird species, including numerous endemic species. Species belonging to the Afrotropical Highland Biome occur in the dry evergreen montane forests of the highland plateau but can also occupy other habitats. Wattled Ibis can be found feeding in wet grassland and open woodland. Black-winged Lovebird, Banded Barbet, Golden-mantled or Abyssinian Woodpecker, Montane White-eye, Rüppell's Robin-chat, Abyssinian Slaty Flycatcher and Tacazze Sunbird are found in evergreen forest, mountain woodlands and areas with scattered trees including fig trees, Euphorbia abyssinica and Juniperus procera. Erckel's spurfowl, Dusky Turtle Dove, Swainson's or Grey-headed Sparrow, Baglafecht Weaver, African Citril, Brown-rumped Seedeater and Streaky Seedeater are common Afrotropical breeding residents of woodland edges, scrubland and forest edges. White-billed Starling and Little Rock Thrush can be found on steep cliffs; Speckled or African rock pigeon and White-collared Pigeon in gorges and rocky places but also in towns and villages.

Species belonging to the Somali-Masai Biome. Hemprich's Hornbill and White-rumped Babbler are found in bushland, scrubland and dense secondary forest, often near cliffs, gorges or water. Chestnut-Winged or Somali Starling and Rüppell's Weaver are found in bushy and shrubby areas. Black-billed wood hoopoes have some red at the base of the bill or an entirely red bill in this area.

Species belonging to the Sudan-Guinea Savanna Biome: Green-backed eremomela and Chestnut-crowned Sparrow-Weaver.

Species that are neither endemic nor biome-restricted but that have restricted ranges or that can be more easily seen in Ethiopia than elsewhere in their range: Abyssinian Roller is an Ethiopian relative of Lilac-breasted Roller, which is an intra-tropical breeding migrant of south and east Africa, and of European Roller, an uncommon Palearctic passage migrant. Black-billed Barbet, Yellow-breasted Barbet and Grey-headed Batis are species from the Sahel and Northern Africa but also occur in Acacia woodlands in the area.

The most regularly observed raptor birds in crop fields in Tigray are Augur buzzard (Buteo augur), Common Buzzard (Buteo buteo), Steppe Eagle (Aquila nipalensis), Lanner falcon (Falco biarmicus), Black kite (Milvus migrans), Yellow-billed kite (Milvus aegyptius) and Barn owl (Tyto alba).

Birdwatching can be done particularly in exclosures and forests. Eighteen bird-watching sites have been inventoried in Enderta and Degua Tembien and mapped.

Administrative zones and districts
Like other Regions in Ethiopia, Tigray is subdivided into administrative zones, and further into woredas or districts. Up to January 2020, these were the woredas of Tigray:

 * Central Tigray
 * Abergele
 * Abiy Addi Town
 * Adwa
 * Adwa Town
 * Aksum Town
 * Dogu'a Tembien
 * Enticho
 * Kola Tembien
 * La'ilay Maychew
 * Mereb Lehe
 * Naeder Adet
 * Tahtay Maychew
 * Werie Lehe
 * East Tigray
 * Adigrat Town
 * Atsbi Wenberta
 * Ganta Afeshum
 * Gulomahda
 * Hawzen
 * Irob
 * Kilte Awulaelo
 * Saesi Tsaedaemba
 * Wukro Town
 * North West Tigray
 * Asigede Tsimbela
 * La'ilay Adiyabo
 * Medebay Zana
 * Tahtay Adiyabo
 * Tahtay Koraro
 * Tselemti
 * Shiraro Town
 * Shire Town
 * South Tigray
 * Alaje
 * Alamata
 * Alamata Town
 * Endamehoni
 * Korem Town
 * Maychew Town
 * Ofla
 * Raya Azebo
 * South East Tigray
 * Enderta
 * Hintalo Wajirat
 * Samre
 * West Tigray
 * Kafta Humera
 * Humera Town
 * Wolqayt
 * Tsegede
 * Mekelle (special zone)

In 2018 and 2019, after multiple village discussions that were often vigorous in the more remote areas, 21 independent urban administrations were added and other boundaries re-drawn, resulting in an increase from 35 to 94 woredas in January 2020:

 * Central Tigray
 * Abergele
 * Adet
 * Abiy Addi Town
 * Adwa
 * Adwa Town
 * Ahferom
 * Ahse'a
 * Aksum Town
 * Chila
 * Edaga Arbi
 * Egela
 * Emba Seneyti
 * Enda Felasi
 * Enticho town
 * Hahayle
 * Keyh Tekli
 * Kola Temben
 * Laelay Maychew
 * Naeder
 * Rama
 * Tahtay Maychew
 * Tanqua Melash
 * East Tigray
 * Adigrat Town
 * Agulae
 * Atsbi
 * Atsbi Town
 * Bizet
 * Edaga Hamus Town
 * Erob
 * Freweyni Town
 * Ganta Afeshum
 * Geralta
 * Hawzen
 * Hawzen Town
 * Irob
 * Kelete Awelallo
 * Saesie
 * Tsaeda Emba
 * Wukro Town
 * Fatsi town
 * Zalambessa town
 * North West Tigray
 * Adi Daero town
 * Adi Hageray
 * Asgede
 * Laelay Tselemti
 * Inda Aba Guna town
 * Shire Inda Slasse city
 * Laelay Adiabo
 * May Tsebri Town
 * Selekleka
 * Seyemti Adiyabo
 * Sheraro Town
 * Tahtay Adiyabo
 * Tahtay Koraro
 * Tselemti
 * Tsimbia
 * Zena
 * South Tigray
 * Alamata Town
 * Bora
 * Raya Chercher
 * Emba Alaje
 * Endamehoni
 * Korem Town
 * Maichew Town
 * Mekoni Town
 * Neqsage
 * Ofla
 * Raya Alamata
 * Raya Azebo
 * Selewa
 * Zata
 * South East Tigray
 * Adi Gudem town
 * Dogu'a Temben
 * Enderta
 * Hagere Selam town
 * Hintalo
 * Saharti
 * Samre
 * Wajirat
 * West Tigray
 * Awura
 * Dansha town
 * Kafta Humera
 * Qorarit town
 * May Kadra town
 * May Gaba town
 * Humera town
 * Tsegede
 * Welkalt
 * Mekelle (special zone)

Major cities
Mekelle, home to Mekelle University, Mekelle Institute of Technology, Tigray Institute of Policy Studies, Admas University, Microlink College, Nile College, and Mekelle College of Teacher Education is the capital of Tigray, near the geographic center of the state.

Other Tigray cities functioning as centers of Ethiopian metropolitan areas include:
 * Adigrat (home of Adigrat University, Debre Damo monastery and Addis Pharmaceutical Factory)
 * Adwa (home of Adwa Pan African University,)
 * Axum (home of Aksum University,)
 * Maychew (home of Raya University)

Of the 10 largest cities in Tigray, Maychew has the highest elevation at 2479 meter above sea level. Plenty of smaller towns, like Atsbi and Edaga Hamus are located at even higher elevations. Of the large cities, Humera is located at the lowest altitude (585 m).

Executive branch
The executive branch is headed by the Chief Administrator of Tigray. The current president is Getachew Reda Kahsay, a TPLF member, elected in 2023. A Vice President of Tigray succeeds the president in the event of any removal from office, and performs any duties assigned by the president. The other elected constitutional offices in the executive branch are the Regional Health Bureau (Ato Hagos Godefy), Educational Bureau (Ato Gebre'egziabher), Auditor General (Ato Alemseged Kebedew), and 12 other officials.

Judicial branch
There are three levels of the Tigray state judiciary. The lowest level is the court of common pleas: each woreda maintains its own constitutionally mandated court of common pleas, which maintain jurisdiction over all justiciable matters. The intermediate-level court system is the district court system. Four courts of appeals exist, each retaining jurisdiction over appeals from common pleas, municipal, and county courts in an administrative zone. A case heard in this system is decided by a three-judge panel, and each judge is elected.

The highest-ranking court, the Tigray Supreme Court, is Tigray's "court of last resort". A seven-justice panel composes the court, which, by its own discretion, hears appeals from the courts of appeals, and retains original jurisdiction over limited matters. The chief judge is called the President of Tigray Supreme Court (W/ro Hirity Miheretab).

Legislative branch
The State Council, which is the highest administrative body of the state, is made up of 152 members.

National politics
Tigray is represented by 38 representatives in the Federal Democratic Republic of Ethiopia House of Peoples' Representatives. But currently after the illegitimate postponement of the national election of Ethiopia Tigray has pulled it representative from the House of House of Peoples' Representatives and has no representation in the Federal parliament.

Demographics

Based on the 2007 census conducted by the Central Statistical Agency of Ethiopia (CSA), the Tigray Region has a population of 4,316,988, of whom 2,126,465 are men and 2,190,523 women; urban inhabitants number 844,040 or 19.55% of the population. With an estimated area of 84,722 km2, the region had an estimated density of 51 people per km2. In the entire region 992,635 households were counted, for an average of 4.4 people per household, with urban households having on average 3.4 and rural households 4.6.

In the previous census, conducted in 1994, the region's population was 3,136,267, of whom 1,542,165 were men and 1,594,102 women; urban inhabitants numbered 621,210, or 14% of the population.

According to the CSA,, 53.99% of the total population had access to safe drinking water, of whom 42.68% were rural inhabitants and 97.28% were urban. Values for other reported common indicators of the standard of living for Tigray include: 31.6% of the inhabitants fall into the lowest wealth quintile; adult literacy for men is 67.5% and for women 33.7%; and the infant mortality rate is 67 infant deaths per 1,000 live births, less than the national average of 77; at least half of these deaths occurred in the infants' first month of life.

The predominant religion in Tigray is Orthodox Christianity

Ethnicity

With 96.55% of the local population, the region is predominantly inhabited by the Tigrinya-speaking Tigrayan people. The Tigrinya language belongs to the Semitic branch of the Afro-Asiatic family of languages. Most other residents hail from other Afro-Asiatic-speaking communities, including the Amhara, Irob, Afar, Agaw and Oromo. Partly assimilated Oromo live in remoter villages in Raya Azebo and Alamata (woreda), whereas there are Agaw in Abergele (woreda). There are also Nilo-Saharan-speaking Kunama as well.

Languages
The working language is Tigrinya. Saho and Kunama are also spoken, and people in urban areas are also able to speak Amharic.

Notable people

 * Mehari Taddele Maru, author, professor at European university Institute
 * Tedros Adhanom Ghebreyesus, Director General, World Health Organization WHO
 * Meles Zenawi Asres, Prime Minister, Ethiopia 1995–2012
 * Zeresenay Alemseged, Anthropologist, known for his discovery of Selam, also referred to as "Lucy’s child", the almost-complete fossilized remains of a 3.3 million-year-old child of the species Australopithecus afarensis.
 * Araya Zerihun
 * Sebhat Gebre-Egziabher well known fiction writer and phylosopher
 * Tewolde Berhan Gebre Egziabher, well known environmentalist and brother of Sibhat Gebregziabhier
 * Miruts Yifter, Olympic Runner
 * Gebrehiwot Baykedagn, was an Ethiopian doctor, economist, and intellectual.
 * Yohannes IV, the king of kings of Ethiopia
 * Ras Alula, the first full African general
 * Bashai Awalom, The first Ethiopia spy
 * Saint Yared, the first church singer, composer and writer of three books
 * Deqiqe Estefanos

Terracing and dam construction
An important aspect of the agricultural work in Tigray after the end of the 1991 civil war was to minimize the problems of drought. In the past, Tigray was covered with forests and had a micro-climate that favoured the rains. Subsequently, the forests were cut down, usually to impoverish the population during the wars. Consequently, Tigray achieved a fair amount of rainfall during the rainy season, from August to September, but quickly lost these waters downstream. In the process the fertile soil of the fields eroded. After a few weeks of rain, the country again dried up.

The government undertook two projects in Tigray. The first was the construction of terraces which, with the agreement and help of local communities, go up to the tops of the mountains at 2,500 metres. The goal was to prevent the rainfall flowing away immediately so that it could be conserved for the agricultural season. On the highest terraces were planted trees, mainly eucalyptus, the dominant tree in Ethiopia and native to Australia. These plants created a new microclimate. The terracing method was very simple but required good organization. Long stretches of the fields were terraced by the villagers using stone walls from stones that erosion had exposed. The rains eroding the still non-terraced ground formed mudslides that were held by the topmost walls, which permitted construction of a new terrace field and another wall with uncovered stones, creating new ground terraced farmland every year.

Another endeavour involved the construction of small reservoirs for local irrigation. As rains last only for a couple of months per year, reservoirs of different sizes allow harvesting runoff from the rainy season for further use in the dry season. The dams needed to create these basins are typically an embankment of a few hundreds of meters, closing off one part of a valley, with a maximum height of 20 metres. Each took months of work, in which people carried earth on their back, and with assistance of donkeys. Generally 2,000–3,000 people — men, women and children — carried the earth in simple baskets.

The small reservoirs in Tigray include:

 * Addi Abagiè
 * Addi Akhor
 * Addi Amharay
 * May Leiba
 * Hiza'iti Wedi Cheber
 * Addi Asme'e
 * Chini
 * Addi Gela
 * Addi Hilo
 * Addi Qenafiz
 * Addi Shihu
 * Aqushela
 * Arato
 * Belesat
 * Betqua
 * Chichat
 * Dibdibo
 * Dur Anbesa
 * Imbagedo
 * Inda Zib'i
 * Era (reservoir)
 * Era Quhila
 * Gereb Mihiz
 * Filiglig
 * Gereb May Zib'i
 * Gereb Bi'ati
 * Gereb Awso
 * Felaga
 * Gereb Segen (Hintalo)
 * Gereb Segen (May Gabat)
 * Gereb Shegal
 * Ginda'i
 * Godew

Overall, these reservoirs suffer from rapid siltation. Part of the water that could be used for irrigation is lost through seepage; the positive side-effect is that this contributes to groundwater recharge.

Vegetation and enclosures
Tigray holds numerous exclosures, areas that are set aside for regreening. Logging and livestock grazing are not allowed there. Besides effects on biodiversity, water infiltration, protection from flooding, sediment deposition, carbon sequestration, people commonly have economic benefits from these exclosures through grass harvesting, beekeeping and other non-timber forest products. The local inhabitants also consider it as "land set aside for future generations". In Dogu'a Tembien, several exclosures are managed by the EthioTrees project. They have as an additional benefit that the villagers receive carbon credits for the sequestered CO2, as part of a carbon offset programme. The revenues are then reinvested in the villages, according to the priorities of the communities; it may be for an additional class in the village school, a water pond, conservation in the exclosures, or a store for incense.

Livestock

The CSA estimated in 2005 that farmers in Tigray had a total of 2,713,750 cattle (representing 7.0% of Ethiopia's total cattle), 72,640 sheep (0.42%), 208,970 goats (1.61%), 1,200 horses (less than 0.1%), 9,190 mules (6.24%), 386,600 asses (15.43%), 32,650 camels (7.15%), 3,180,240 poultry of all species (10.3%), and 20,480 beehives (0.47%). Cattle are an essential component in the dominant grain-plough agricultural system. In the rainy season, a large part of the cattle herds are in transhumance.

Mainly used for draught, there are several cattle landraces in Tigray:
 * Arado cattle, the dominant variety
 * Raya cattle, long horned, especially raised in Southern Tigray and traded widely as plough oxen
 * Irob cattle, particularly in the Irob woreda
 * Abergele cattle, particularly in Abergele (woreda) and on the southwestern slopes of Dogu’a Tembien
 * Begayt cattle, in western Tigray. They are known for better milk production
 * In small towns: Cross-bred Arado x Begayt, and Arado x Holstein-Friesian milk cows

Landmarks

A distinctive feature of Tigray are its rock-hewn churches. Similar in design to those of Lalibela in the Amhara Region, these churches are found in four or five clusters – Gheralta, Teka-Tesfay, Atsbi and Tembien – with Wukro sometimes included. Some of the churches are considered earlier than those of Lalibela, perhaps dating from the eighth century. Mostly monolithic, with designs partly inspired by classical architecture, they are often located at the top of cliffs or steep hills, for security. For example, Tigray's ancient Debre Damo monastery is accessible only by climbing a rope 25 metres up a sheer cliff.

Looting has become a major issue in the Tigray Region, as archaeological sites have become sources for construction materials and ancient artifacts used for everyday purposes by local populations.

The area is famous for a single rock sculptured 23 meter long obelisk in Axum as well as for other fallen obelisks. The Axum treasure site of ancient Tigrayan history is a major landmark. Yeha is another important local landmark that is little-known outside the region.

Ground travel
A major north–south road corridor goes through Tigray. This is facilitated by Highway 2 which goes from Adigrat to Addis Ababa and Highway 3 which goes from Shire to Addis Ababa.

Air travel
Tigray has one international airport and four commercial airports. The international airport is Alula Aba Nega Airport (MQX) near Mekelle. The region's four other commercial airports are Shire Airport (SHC), Humera Airport (HUE), Dansha Airport, and Emperor Yohannes IV Airport (AXU), which serves Axum.

Sports
Mekelle 70 Enderta F.C. (Tigrinya: ጋንታ መቐለ 70 እንደርታ) is an Ethiopian football club based in the capital, Mekelle. They are a member of the Ethiopian Football Federation and currently play in the top division of Ethiopian football, the Ethiopian Premier League. They are known by the nickname the Lion's Den (ምዓም ኣንበሳ /ምዓም አናብስት/ኣናብስቶቹ). The club won its first Ethiopian Premier League title in the 2018–2019 Ethiopian Premier League Season.

Shire Indasillasie F.C. (Tigrinya: ጋንታ ስሑል ሽረ, also known as Sihul Shire FC) is an Ethiopian football club based in Shire. They are a member of the Ethiopian Football Federation and play in the Ethiopian Premier League, the first division of football in Ethiopia.

Mekele City, Suhul Shire, and Adigrat University football clubs were Tigray-based clubs among the 14 clubs to participate in the Ethiopian Premier League in 2020/2021. However, due to the war, they were replaced by other clubs from the League one rank below the Ethiopian Premier League.

Tigrayans are known for their good performance in road cycling. For many years cyclists from this region have been dominant in the Ethiopian national cycling championships. Tsgabu Grmay is one of the best Ethiopian cyclists and the first Ethiopian to participate in the Tour de France.

Education
At the regional level, the Tigray Education Bureau governs primary and secondary educational institutions. At the municipal level, there are approximately 300 school districts region-wide.

Colleges and universities

 * Adigrat University
 * Axum University
 * Adwa Pan-African University
 * Raya University
 * Ethio-lmage College
 * Greenwich College
 * Hashenge College
 * Mars Engineering College
 * Mekelle University
 * Mekelle Institute of Technology
 * New Millennium College
 * Nile College
 * Sheba University College
 * Signal College
 * St. Mary's University College
 * Winner college Axum

Libraries
Tigray is home to Ethiopia's most extensive church libraries that are found in the eastern and central zones of the region. There are several ongoing digitization projects to preserve previous historical texts.

 * Axum Heritage Foundation
 * Romanat Qeddus Mika'el Church
 * Gunda Gunde Monastery
 * Agwaza Monastery
 * Debre Damo Monastery

Non-governmental organisations
Major NGOs, involved in development activities are:
 * Relief Society of Tigray
 * Tigrai Development Association
 * Tegaru Disaster Relief Fund (TDRF)

title: Tigray Region


text:
Large Hadron Collider

The Large Hadron Collider (LHC) is the world's largest and highest-energy particle collider. It was built by the European Organization for Nuclear Research (CERN) between 1998 and 2008 in collaboration with over 10,000 scientists and hundreds of universities and laboratories across more than 100 countries. It lies in a tunnel 27 km in circumference and as deep as 175 m beneath the France–Switzerland border near Geneva.

The first collisions were achieved in 2010 at an energy of 3.5 teraelectronvolts (TeV) per beam, about four times the previous world record. The discovery of the Higgs boson at the LHC was announced in 2012. Between 2013 and 2015, the LHC was shut down and upgraded; after those upgrades it reached 6.5 TeV per beam (13.0 TeV total collision energy). At the end of 2018, it was shut down for maintenance and further upgrades, reopening over three years later in April 2022.

The collider has four crossing points where the accelerated particles collide. Nine detectors, each designed to detect different phenomena, are positioned around the crossing points. The LHC primarily collides proton beams, but it can also accelerate beams of heavy ions, such as in lead–lead collisions and proton–lead collisions.

The LHC's goal is to allow physicists to test the predictions of different theories of particle physics, including measuring the properties of the Higgs boson, searching for the large family of new particles predicted by supersymmetric theories, and studying other unresolved questions in particle physics.

Background
The term hadron refers to subatomic composite particles composed of quarks held together by the strong force (analogous to the way that atoms and molecules are held together by the electromagnetic force). The best-known hadrons are the baryons such as protons and neutrons; hadrons also include mesons such as the pion and kaon, which were discovered during cosmic ray experiments in the late 1940s and early 1950s.

A collider is a type of a particle accelerator that brings two opposing particle beams together such that the particles collide. In particle physics, colliders, though harder to construct, are a powerful research tool because they reach a much higher center of mass energy than fixed target setups. Analysis of the byproducts of these collisions gives scientists good evidence of the structure of the subatomic world and the laws of nature governing it. Many of these byproducts are produced only by high-energy collisions, and they decay after very short periods of time. Thus many of them are hard or nearly impossible to study in other ways.

Purpose
Many physicists hope that the Large Hadron Collider will help answer some of the fundamental open questions in physics, which concern the basic laws governing the interactions and forces among elementary particles and the deep structure of space and time, particularly the interrelation between quantum mechanics and general relativity.

These high-energy particle experiments can provide data to support different scientific models. For example, the Standard Model and Higgsless model required high-energy particle experiment data to validate their predictions and allow further theoretical development. The Standard Model was completed by detection of the Higgs boson by the LHC in 2012.

LHC collisions have explored other questions, including:
 * Do all known particles have supersymmetric partners, as part of supersymmetry in an extension of the Standard Model and Poincaré symmetry?
 * Are there extra dimensions, as predicted by various models based on string theory, and can we detect them?
 * What is the nature of the dark matter, a hypothetical form of matter which appears to account for 27% of the mass-energy of the universe?

Other open questions that may be explored using high-energy particle collisions include:
 * It is already known that electromagnetism and the weak nuclear force are different manifestations of a single force called the electroweak force. The LHC may clarify whether the electroweak force and the strong nuclear force are similarly just different manifestations of one universal unified force, as predicted by various Grand Unification Theories.
 * Why is the fourth fundamental force (gravity) so many orders of magnitude weaker than the other three fundamental forces? See also Hierarchy problem.
 * Are there additional sources of quark flavour mixing beyond those already present within the Standard Model?
 * Why are there apparent violations of the symmetry between matter and antimatter? See also CP violation.
 * What are the nature and properties of quark–gluon plasma, thought to have existed in the early universe and in certain compact and strange astronomical objects today? This will be investigated by heavy ion collisions, mainly in ALICE, but also in CMS, ATLAS and LHCb. First observed in 2010, findings published in 2012 confirmed the phenomenon of jet quenching in heavy-ion collisions.

Design
The collider is contained in a circular tunnel, with a circumference of 26.7 km, at a depth ranging from 50 to 175 m underground. The variation in depth was deliberate, to reduce the amount of tunnel that lies under the Jura Mountains to avoid having to excavate a vertical access shaft there. A tunnel was chosen to avoid having to purchase expensive land on the surface and to take advantage of the shielding against background radiation that the Earth's crust provides.

The 3.8 m wide concrete-lined tunnel, constructed between 1983 and 1988, was formerly used to house the Large Electron–Positron Collider. The tunnel crosses the border between Switzerland and France at four points, with most of it in France. Surface buildings hold ancillary equipment such as compressors, ventilation equipment, control electronics and refrigeration plants. The collider tunnel contains two adjacent parallel beamlines (or beam pipes) each containing a beam, which travel in opposite directions around the ring. The beams intersect at four points around the ring, which is where the particle collisions take place. Some 1,232 dipole magnets keep the beams on their circular path (see image ), while an additional 392 quadrupole magnets are used to keep the beams focused, with stronger quadrupole magnets close to the intersection points in order to maximize the chances of interaction where the two beams cross. Magnets of higher multipole orders are used to correct smaller imperfections in the field geometry. In total, about 10,000 superconducting magnets are installed, with the dipole magnets having a mass of over 27 tonnes. About 96 tonnes of superfluid helium-4 is needed to keep the magnets, made of copper-clad niobium-titanium, at their operating temperature of 1.9 K, making the LHC the largest cryogenic facility in the world at liquid helium temperature. LHC uses 470 tonnes of Nb–Ti superconductor.

During LHC operations, the CERN site draws roughly 200 MW of electrical power from the French electrical grid, which, for comparison, is about one-third the energy consumption of the city of Geneva; the LHC accelerator and detectors draw about 120 MW thereof. Each day of its operation generates 140 terabytes of data.

When running an energy of 6.5 TeV per proton, once or twice a day, as the protons are accelerated from 450 GeV to 6.5 TeV, the field of the superconducting dipole magnets is increased from 0.54 to 7.7 teslas (T). The protons each have an energy of 6.5 TeV, giving a total collision energy of 13 TeV. At this energy, the protons have a Lorentz factor of about 6,930 and move at about 0.999 999  990, or about 3.1 m/s slower than the speed of light (c). It takes less than 90 microseconds (μs) for a proton to travel 26.7 km around the main ring. This results in 11,245 revolutions per second for protons whether the particles are at low or high energy in the main ring, since the speed difference between these energies is beyond the fifth decimal.

Rather than having continuous beams, the protons are bunched together, into up to 2,808 bunches, with 115 billion protons in each bunch so that interactions between the two beams take place at discrete intervals, mainly 25 nanoseconds (ns) apart, providing a bunch collision rate of 40 MHz. It was operated with fewer bunches in the first years. The design luminosity of the LHC is 1034 cm−2s−1, which was first reached in June 2016. By 2017, twice this value was achieved.

Before being injected into the main accelerator, the particles are prepared by a series of systems that successively increase their energy. The first system is the linear particle accelerator Linac4 generating 160 MeV negative hydrogen ions (H− ions), which feeds the Proton Synchrotron Booster (PSB). There, both electrons are stripped from the hydrogen ions leaving only the nucleus containing one proton. Protons are then accelerated to 2 GeV and injected into the Proton Synchrotron (PS), where they are accelerated to 26 GeV. Finally, the Super Proton Synchrotron (SPS) is used to increase their energy further to 450 GeV before they are at last injected (over a period of several minutes) into the main ring. Here, the proton bunches are accumulated, accelerated (over a period of 20 minutes) to their peak energy, and finally circulated for 5 to 24 hours while collisions occur at the four intersection points.

The LHC physics programme is mainly based on proton–proton collisions. However, during shorter running periods, typically one month per year, heavy-ion collisions are included in the programme. While lighter ions are considered as well, the baseline scheme deals with lead ions (see A Large Ion Collider Experiment). The lead ions are first accelerated by the linear accelerator LINAC 3, and the Low Energy Ion Ring (LEIR) is used as an ion storage and cooler unit. The ions are then further accelerated by the PS and SPS before being injected into LHC ring, where they reach an energy of 2.3 TeV per nucleon (or 522 TeV per ion), higher than the energies reached by the Relativistic Heavy Ion Collider. The aim of the heavy-ion programme is to investigate quark–gluon plasma, which existed in the early universe.

Detectors
Nine detectors have been built in large caverns excavated at the LHC's intersection points. Two of them, the ATLAS experiment and the Compact Muon Solenoid (CMS), are large general-purpose particle detectors. ALICE and LHCb have more specialized roles, while the other five—TOTEM, MoEDAL, LHCf, SND and FASER—are much smaller and are for very specialized research. The ATLAS and CMS experiments discovered the Higgs boson, which is strong evidence that the Standard Model has the correct mechanism of giving mass to elementary particles.

Computing and analysis facilities
Data produced by LHC, as well as LHC-related simulation, were estimated at 200 petabytes per year.

The LHC Computing Grid was constructed as part of the LHC design, to handle the massive amounts of data expected for its collisions. It is an international collaborative project that consists of a grid-based computer network infrastructure initially connecting 140 computing centres in 35 countries (over 170 in more than 40 countries ). It was designed by CERN to handle the significant volume of data produced by LHC experiments, incorporating both private fibre optic cable links and existing high-speed portions of the public Internet to enable data transfer from CERN to academic institutions around the world. The LHC Computing Grid consists of global federations across Europe, Asia Pacific and the Americas.

The distributed computing project LHC@home was started to support the construction and calibration of the LHC. The project uses the BOINC platform, enabling anybody with an Internet connection and a computer running Mac OS X, Windows or Linux to use their computer's idle time to simulate how particles will travel in the beam pipes. With this information, the scientists are able to determine how the magnets should be calibrated to gain the most stable "orbit" of the beams in the ring. In August 2011, a second application (Test4Theory) went live which performs simulations against which to compare actual test data, to determine confidence levels of the results.

By 2012, data from over 6 quadrillion ($10,000,000,000,000,000,000,000,000,000,000,000$) LHC proton–proton collisions had been analysed. The LHC Computing Grid had become the world's largest computing grid in 2012, comprising over 170 computing facilities in a worldwide network across more than 40 countries.

Operational history
The LHC first went operational on 10 September 2008, but initial testing was delayed for 14 months from 19 September 2008 to 20 November 2009, following a magnet quench incident that caused extensive damage to over 50 superconducting magnets, their mountings, and the vacuum pipe.

During its first run (2010–2013), the LHC collided two opposing particle beams of either protons at up to 4 teraelectronvolts (4 TeV or 0.64 microjoules), or lead nuclei (574 TeV per nucleus, or 2.76 TeV per nucleon). Its first run discoveries included the long-sought Higgs boson, several composite particles (hadrons) like the χb (3P) bottomonium state, the first creation of a quark–gluon plasma, and the first observations of the very rare decay of the Bs meson into two muons (Bs0 → μ+μ−), which challenged the validity of existing models of supersymmetry.

Operational challenges
The size of the LHC constitutes an exceptional engineering challenge with unique operational issues on account of the amount of energy stored in the magnets and the beams. While operating, the total energy stored in the magnets is 10 GJ and the total energy carried by the two beams reaches 724 MJ.

Loss of only one ten-millionth part (10−7) of the beam is sufficient to quench a superconducting magnet, while each of the two beam dumps must absorb 362 MJ. These energies are carried by very little matter: under nominal operating conditions (2,808 bunches per beam, 1.15×1011 protons per bunch), the beam pipes contain 1.0×10−9 gram of hydrogen, which, in standard conditions for temperature and pressure, would fill the volume of one grain of fine sand.

Cost
With a budget of €7.5 billion (about $9bn or £6.19bn ), the LHC is one of the most expensive scientific instruments ever built. The total cost of the project is expected to be of the order of 4.6bn Swiss francs (SFr) (about $4.4bn, €3.1bn, or £2.8bn ) for the accelerator and 1.16bn (SFr) (about $1.1bn, €0.8bn, or £0.7bn ) for the CERN contribution to the experiments.

The construction of LHC was approved in 1995 with a budget of SFr 2.6bn, with another SFr 210M toward the experiments. However, cost overruns, estimated in a major review in 2001 at around SFr 480M for the accelerator, and SFr 50M for the experiments, along with a reduction in CERN's budget, pushed the completion date from 2005 to April 2007. The superconducting magnets were responsible for SFr 180M of the cost increase. There were also further costs and delays owing to engineering difficulties encountered while building the cavern for the Compact Muon Solenoid, and also due to magnet supports which were insufficiently strongly designed and failed their initial testing (2007) and damage from a magnet quench and liquid helium escape (inaugural testing, 2008). Because electricity costs are lower during the summer, the LHC normally does not operate over the winter months, although exceptions over the 2009/10 and 2012/2013 winters were made to make up for the 2008 start-up delays and to improve precision of measurements of the new particle discovered in 2012, respectively.

Construction accidents and delays

 * On 25 October 2005, José Pereira Lages, a technician, was killed in the LHC when a switchgear that was being transported fell on top of him.
 * On 27 March 2007, a cryogenic magnet support designed and provided by Fermilab and KEK broke during an initial pressure test involving one of the LHC's inner triplet (focusing quadrupole) magnet assemblies. No one was injured. Fermilab director Pier Oddone stated "In this case we are dumbfounded that we missed some very simple balance of forces". The fault had been present in the original design, and remained during four engineering reviews over the following years. Analysis revealed that its design, made as thin as possible for better insulation, was not strong enough to withstand the forces generated during pressure testing. Details are available in a statement from Fermilab, with which CERN is in agreement. Repairing the broken magnet and reinforcing the eight identical assemblies used by LHC delayed the start-up date, then planned for November 2007.
 * On 19 September 2008, during initial testing, a faulty electrical connection led to a magnet quench (the sudden loss of a superconducting magnet's superconducting ability owing to warming or electric field effects). Six tonnes of supercooled liquid helium—used to cool the magnets—escaped, with sufficient force to break 10-ton magnets nearby from their mountings, and caused considerable damage and contamination of the vacuum tube. Repairs and safety checks caused a delay of around 14 months.
 * Two vacuum leaks were found in July 2009, and the start of operations was further postponed to mid-November 2009.

Exclusion of Russia
With the 2022 Russian invasion of Ukraine, the participation of Russians with CERN was called into question. About 8% of the workforce are of Russian nationality. In June 2022, CERN said the governing council "intends to terminate" CERN's cooperation agreements with Belarus and Russia when they expire, respectively in June and December 2024. CERN said it would monitor developments in Ukraine and remains prepared to take additional steps as warranted. CERN further said that it would reduce the Ukrainian contribution to CERN for 2022 to the amount already remitted to the Organization, thereby waiving the second installment of the contribution.

Initial lower magnet currents
In both of its runs (2010 to 2012 and 2015), the LHC was initially run at energies below its planned operating energy, and ramped up to just 2 x 4 TeV energy on its first run and 2 x 6.5 TeV on its second run, below the design energy of 2 x 7 TeV. This is because massive superconducting magnets require considerable magnet training to handle the high currents involved without losing their superconducting ability, and the high currents are necessary to allow a high proton energy. The "training" process involves repeatedly running the magnets with lower currents to provoke any quenches or minute movements that may result. It also takes time to cool down magnets to their operating temperature of around 1.9 K (close to absolute zero). Over time the magnet "beds in" and ceases to quench at these lesser currents and can handle the full design current without quenching; CERN media describe the magnets as "shaking out" the unavoidable tiny manufacturing imperfections in their crystals and positions that had initially impaired their ability to handle their planned currents. The magnets, over time and with training, gradually become able to handle their full planned currents without quenching.

Inaugural tests (2008)
The first beam was circulated through the collider on the morning of 10 September 2008. CERN successfully fired the protons around the tunnel in stages, three kilometres at a time. The particles were fired in a clockwise direction into the accelerator and successfully steered around it at 10:28 local time. The LHC successfully completed its major test: after a series of trial runs, two white dots flashed on a computer screen showing the protons travelled the full length of the collider. It took less than one hour to guide the stream of particles around its inaugural circuit. CERN next successfully sent a beam of protons in an anticlockwise direction, taking slightly longer at one and a half hours owing to a problem with the cryogenics, with the full circuit being completed at 14:59.

Quench incident
On 19 September 2008, a magnet quench occurred in about 100 bending magnets in sectors 3 and 4, where an electrical fault vented about six tonnes of liquid helium (the magnets' cryogenic coolant) into the tunnel. The escaping vapour expanded with explosive force, damaging 53 superconducting magnets and their mountings, and contaminating the vacuum pipe, which also lost vacuum conditions.

Shortly after the incident, CERN reported that the most likely cause of the problem was a faulty electrical connection between two magnets. It estimated that repairs would take at least two months, owing to the time needed to warm up the affected sectors and then cool them back down to operating temperature. CERN released an interim technical report and preliminary analysis of the incident on 15 and 16 October 2008 respectively, and a more detailed report on 5 December 2008. The analysis of the incident by CERN confirmed that an electrical fault had indeed been the cause. The faulty electrical connection had led (correctly) to a failsafe power abort of the electrical systems powering the superconducting magnets, but had also caused an electric arc (or discharge) which damaged the integrity of the supercooled helium's enclosure and vacuum insulation, causing the coolant's temperature and pressure to rapidly rise beyond the ability of the safety systems to contain it, and leading to a temperature rise of about 100 degrees Celsius in some of the affected magnets. Energy stored in the superconducting magnets and electrical noise induced in other quench detectors also played a role in the rapid heating. Around two tonnes of liquid helium escaped explosively before detectors triggered an emergency stop, and a further four tonnes leaked at lower pressure in the aftermath. A total of 53 magnets were damaged in the incident and were repaired or replaced during the winter shutdown. This accident was thoroughly discussed in a 22 February 2010 Superconductor Science and Technology article by CERN physicist Lucio Rossi.

In the original schedule for LHC commissioning, the first "modest" high-energy collisions at a centre-of-mass energy of 900 GeV were expected to take place before the end of September 2008, and the LHC was expected to be operating at 10 TeV by the end of 2008. However, owing to the delay caused by the incident, the collider was not operational until November 2009. Despite the delay, LHC was officially inaugurated on 21 October 2008, in the presence of political leaders, science ministers from CERN's 20 Member States, CERN officials, and members of the worldwide scientific community.

Most of 2009 was spent on repairs and reviews from the damage caused by the quench incident, along with two further vacuum leaks identified in July 2009; this pushed the start of operations to November of that year.

Run 1: first operational run (2009–2013)

On 20 November 2009, low-energy beams circulated in the tunnel for the first time since the incident, and shortly after, on 30 November, the LHC achieved 1.18 TeV per beam to become the world's highest-energy particle accelerator, beating the Tevatron's previous record of 0.98 TeV per beam held for eight years.

The early part of 2010 saw the continued ramp-up of beam in energies and early physics experiments towards 3.5 TeV per beam and on 30 March 2010, LHC set a new record for high-energy collisions by colliding proton beams at a combined energy level of 7 TeV. The attempt was the third that day, after two unsuccessful attempts in which the protons had to be "dumped" from the collider and new beams had to be injected. This also marked the start of the main research programme.

The first proton run ended on 4 November 2010. A run with lead ions started on 8 November 2010, and ended on 6 December 2010, allowing the ALICE experiment to study matter under extreme conditions similar to those shortly after the Big Bang.

CERN originally planned that the LHC would run through to the end of 2012, with a short break at the end of 2011 to allow for an increase in beam energy from 3.5 to 4 TeV per beam. At the end of 2012, the LHC was planned to be temporarily shut down until around 2015 to allow upgrade to a planned beam energy of 7 TeV per beam. In late 2012, in light of the July 2012 discovery of the Higgs boson, the shutdown was postponed for some weeks into early 2013, to allow additional data to be obtained before shutdown.

Long Shutdown 1 (2013–2015)
The LHC was shut down on 13 February 2013 for its two-year upgrade called Long Shutdown 1 (LS1), which was to touch on many aspects of the LHC: enabling collisions at 14 TeV, enhancing its detectors and pre-accelerators (the Proton Synchrotron and Super Proton Synchrotron), as well as replacing its ventilation system and 100 km of cabling impaired by high-energy collisions from its first run. The upgraded collider began its long start-up and testing process in June 2014, with the Proton Synchrotron Booster starting on 2 June 2014, the final interconnection between magnets completing and the Proton Synchrotron circulating particles on 18 June 2014, and the first section of the main LHC supermagnet system reaching operating temperature of 1.9 K, a few days later. Due to the slow progress with "training" the superconducting magnets, it was decided to start the second run with a lower energy of 6.5 TeV per beam, corresponding to a current in the magnet of 11,000 amperes. The first of the main LHC magnets were reported to have been successfully trained by 9 December 2014, while training the other magnet sectors was finished in March 2015.

Run 2: second operational run (2015–2018)
On 5 April 2015, the LHC restarted after a two-year break, during which the electrical connectors between the bending magnets were upgraded to safely handle the current required for 7 TeV per beam (14 TeV collision energy). However, the bending magnets were only trained to handle up to 6.5 TeV per beam (13 TeV collision energy), which became the operating energy for 2015 to 2018. The energy was first reached on 10 April 2015. The upgrades culminated in colliding protons together with a combined energy of 13 TeV. On 3 June 2015, the LHC started delivering physics data after almost two years offline. In the following months, it was used for proton–proton collisions, while in November, the machine switched to collisions of lead ions and in December, the usual winter shutdown started.

In 2016, the machine operators focused on increasing the luminosity for proton–proton collisions. The design value was first reached 29 June, and further improvements increased the collision rate to 40% above the design value. The total number of collisions in 2016 exceeded the number from Run 1 – at a higher energy per collision. The proton–proton run was followed by four weeks of proton–lead collisions.

In 2017, the luminosity was increased further and reached twice the design value. The total number of collisions was higher than in 2016 as well.

The 2018 physics run began on 17 April and stopped on 3 December, including four weeks of lead–lead collisions.

Long Shutdown 2 (2018–2022)
Long Shutdown 2 (LS2) started on 10 December 2018. The LHC and the whole CERN accelerator complex was maintained and upgraded. The goal of the upgrades was to implement the High Luminosity Large Hadron Collider (HL-LHC) project that will increase the luminosity by a factor of 10. LS2 ended in April 2022. The Long Shutdown 3 (LS3) in the 2020s will take place before the HL-LHC project is done.

Run 3: third operational run (2022)
LHC became operational again on 22 April 2022 with a new maximum beam energy of 6.8 TeV (13.6 TeV collision energy), which was first achieved on 25 April. It officially commenced its run 3 physics season on 5 July 2022. This round is expected to continue until 2026. In addition to a higher energy the LHC is expected to reach a higher luminosity, which is expected to increase even further with the upgrade to the HL-LHC after Run 3.

Findings and discoveries
An initial focus of research was to investigate the possible existence of the Higgs boson, a key part of the Standard Model of physics which was predicted by theory, but had not yet been observed before due to its high mass and elusive nature. CERN scientists estimated that, if the Standard Model was correct, the LHC would produce several Higgs bosons every minute, allowing physicists to finally confirm or disprove the Higgs boson's existence. In addition, the LHC allowed the search for supersymmetric particles and other hypothetical particles as possible unknown areas of physics. Some extensions of the Standard Model predict additional particles, such as the heavy W' and Z' gauge bosons, which are also estimated to be within reach of the LHC to discover.

First run (data taken 2009–2013)
The first physics results from the LHC, involving 284 collisions which took place in the ALICE detector, were reported on 15 December 2009. The results of the first proton–proton collisions at energies higher than Fermilab's Tevatron proton–antiproton collisions were published by the CMS collaboration in early February 2010, yielding greater-than-predicted charged-hadron production.

After the first year of data collection, the LHC experimental collaborations started to release their preliminary results concerning searches for new physics beyond the Standard Model in proton–proton collisions. No evidence of new particles was detected in the 2010 data. As a result, bounds were set on the allowed parameter space of various extensions of the Standard Model, such as models with large extra dimensions, constrained versions of the Minimal Supersymmetric Standard Model, and others.

On 24 May 2011, it was reported that quark–gluon plasma (the densest matter thought to exist besides black holes) had been created in the LHC.

Between July and August 2011, results of searches for the Higgs boson and for exotic particles, based on the data collected during the first half of the 2011 run, were presented in conferences in Grenoble and Mumbai. In the latter conference, it was reported that, despite hints of a Higgs signal in earlier data, ATLAS and CMS exclude with 95% confidence level (using the CLs method) the existence of a Higgs boson with the properties predicted by the Standard Model over most of the mass region between 145 and 466 GeV. The searches for new particles did not yield signals either, allowing to further constrain the parameter space of various extensions of the Standard Model, including its supersymmetric extensions.

On 13 December 2011, CERN reported that the Standard Model Higgs boson, if it exists, is most likely to have a mass constrained to the range 115–130 GeV. Both the CMS and ATLAS detectors have also shown intensity peaks in the 124–125 GeV range, consistent with either background noise or the observation of the Higgs boson.

On 22 December 2011, it was reported that a new composite particle had been observed, the χb (3P) bottomonium state.

On 4 July 2012, both the CMS and ATLAS teams announced the discovery of a boson in the mass region around 125–126 GeV, with a statistical significance at the level of 5 sigma each. This meets the formal level required to announce a new particle. The observed properties were consistent with the Higgs boson, but scientists were cautious as to whether it is formally identified as actually being the Higgs boson, pending further analysis. On 14 March 2013, CERN announced confirmation that the observed particle was indeed the predicted Higgs boson.

On 8 November 2012, the LHCb team reported on an experiment seen as a "golden" test of supersymmetry theories in physics, by measuring the very rare decay of the $$B_s$$ meson into two muons ($$B_s^0\rightarrow\mu^+\mu^-$$). The results, which match those predicted by the non-supersymmetrical Standard Model rather than the predictions of many branches of supersymmetry, show the decays are less common than some forms of supersymmetry predict, though could still match the predictions of other versions of supersymmetry theory. The results as initially drafted are stated to be short of proof but at a relatively high 3.5 sigma level of significance. The result was later confirmed by the CMS collaboration.

In August 2013, the LHCb team revealed an anomaly in the angular distribution of B meson decay products which could not be predicted by the Standard Model; this anomaly had a statistical certainty of 4.5 sigma, just short of the 5 sigma needed to be officially recognized as a discovery. It is unknown what the cause of this anomaly would be, although the Z' boson has been suggested as a possible candidate.

On 19 November 2014, the LHCb experiment announced the discovery of two new heavy subatomic particles, bottom xi'- and bottom xi*-. Both of them are baryons that are composed of one bottom, one down, and one strange quark. They are excited states of the bottom Xi baryon.

The LHCb collaboration has observed multiple exotic hadrons, possibly pentaquarks or tetraquarks, in the Run 1 data. On 4 April 2014, the collaboration confirmed the existence of the tetraquark candidate Z(4430) with a significance of over 13.9 sigma. On 13 July 2015, results consistent with pentaquark states in the decay of bottom Lambda baryons (Λ$6$) were reported.

On 28 June 2016, the collaboration announced four tetraquark-like particles decaying into a J/ψ and a φ meson, only one of which was well established before (X(4274), X(4500) and X(4700) and X(4140)).

In December 2016, ATLAS presented a measurement of the W boson mass, researching the precision of analyses done at the Tevatron.

Second run (2015–2018)
At the conference EPS-HEP 2015 in July, the collaborations presented first cross-section measurements of several particles at the higher collision energy.

On 15 December 2015, the ATLAS and CMS experiments both reported a number of preliminary results for Higgs physics, supersymmetry (SUSY) searches and exotics searches using 13 TeV proton collision data. Both experiments saw a moderate excess around 750 GeV in the two-photon invariant mass spectrum,  but the experiments did not confirm the existence of the hypothetical particle in an August 2016 report.

In July 2017, many analyses based on the large dataset collected in 2016 were shown. The properties of the Higgs boson were studied in more detail and the precision of many other results was improved.

As of March 2021, the LHC experiments have discovered 59 new hadrons in the data collected during the first two runs. On 5 July 2022 LHCb reported the discovery of a new type of pentaquark made up of a charm quark and a charm antiquark and an up, a down and a strange quark, observed in an analysis of decays of charged B mesons.

"High-luminosity" upgrade
After some years of running, any particle physics experiment typically begins to suffer from diminishing returns: as the key results reachable by the device begin to be completed, later years of operation discover proportionately less than earlier years. A common response is to upgrade the devices involved, typically in collision energy, luminosity, or improved detectors. In addition to a possible increase to 14 TeV collision energy, a luminosity upgrade of the LHC, called the High Luminosity Large Hadron Collider, started in June 2018 that will boost the accelerator's potential for new discoveries in physics, starting in 2027. The upgrade aims at increasing the luminosity of the machine by a factor of 10, up to 1035 cm−2s−1, providing a better chance to see rare processes and improving statistically marginal measurements.

Proposed Future Circular Collider
CERN has several preliminary designs for a Future Circular Collider (FCC)—which would be the most powerful particle accelerator ever built—with different types of collider ranging in cost from around €9 billion (US$10.2 billion) to €21 billion. It would use the LHC ring as preaccelerator, similar to how the LHC uses the smaller Super Proton Synchrotron. It is CERN's opening bid in a priority-setting process called the European Strategy for Particle Physics Update, and will affect the field's future well into the second half of the century. As of 2023, no fixed plan exists and it is unknown if the construction will be funded.

Safety of particle collisions
The experiments at the Large Hadron Collider sparked fears that the particle collisions might produce doomsday phenomena, involving the production of stable microscopic black holes or the creation of hypothetical particles called strangelets. Two CERN-commissioned safety reviews examined these concerns and concluded that the experiments at the LHC present no danger and that there is no reason for concern,  a conclusion endorsed by the American Physical Society.

The reports also noted that the physical conditions and collision events that exist in the LHC and similar experiments occur naturally and routinely in the universe without hazardous consequences, including ultra-high-energy cosmic rays observed to impact Earth with energies far higher than those in any human-made collider, like the Oh-My-God particle which had 320 million TeV of energy, and a collision energy tens of times more than the most energetic collisions produced in the LHC.

Popular culture
The Large Hadron Collider gained a considerable amount of attention from outside the scientific community and its progress is followed by most popular science media. The LHC has also inspired works of fiction including novels, TV series, video games and films.

CERN employee Katherine McAlpine's "Large Hadron Rap" surpassed 8 million YouTube views as of 2022.

The band Les Horribles Cernettes was founded by women from CERN. The name was chosen so to have the same initials as the LHC.

National Geographic Channel's World's Toughest Fixes, Season 2 (2010), Episode 6 "Atom Smasher" features the replacement of the last superconducting magnet section in the repair of the collider after the 2008 quench incident. The episode includes actual footage from the repair facility to the inside of the collider, and explanations of the function, engineering, and purpose of the LHC.

The song "Munich" off of the 2012 studio album Scars & Stories by The Fray is inspired by the Large Hadron Collider. Lead singer Isaac Slade said in an interview with The Huffington Post, "There's this large particle collider out in Switzerland that is kind of helping scientists peel back the curtain on what creates gravity and mass. Some very big questions are being raised, even some things that Einstein proposed, that have just been accepted for decades are starting to be challenged. They're looking for the God Particle, basically, the particle that holds it all together. That song is really just about the mystery of why we're all here and what's holding it all together, you know?"

The Large Hadron Collider was the focus of the 2012 student film Decay, with the movie being filmed on location in CERN's maintenance tunnels.

Fiction
The novel Angels & Demons, by Dan Brown, involves antimatter created at the LHC to be used in a weapon against the Vatican. In response, CERN published a "Fact or Fiction?" page discussing the accuracy of the book's portrayal of the LHC, CERN, and particle physics in general. The movie version of the book has footage filmed on-site at one of the experiments at the LHC; the director, Ron Howard, met with CERN experts in an effort to make the science in the story more accurate.

The novel FlashForward, by Robert J. Sawyer, involves the search for the Higgs boson at the LHC. CERN published a "Science and Fiction" page interviewing Sawyer and physicists about the book and the TV series based on it.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Large Hadron Collider


text:
Cauchy–Binet formula

In mathematics, specifically linear algebra, the Cauchy–Binet formula, named after Augustin-Louis Cauchy and Jacques Philippe Marie Binet, is an identity for the determinant of the product of two rectangular matrices of transpose shapes (so that the product is well-defined and square). It generalizes the statement that the determinant of a product of square matrices is equal to the product of their determinants. The formula is valid for matrices with the entries from any commutative ring.

Statement
Let A be an m×n matrix and B an n×m matrix. Write [n] for the set {1, ..., n}, and $$\tbinom{[n]}m$$ for the set of m-combinations of [n] (i.e., subsets of [n] of size m; there are $\tbinom nm$ of them). For $$S\in\tbinom{[n]}m$$, write A[m],S for the m×m matrix whose columns are the columns of A at indices from S, and BS,[m] for the m×m matrix whose rows are the rows of B at indices from S. The Cauchy–Binet formula then states

 * $$\det(AB) = \sum_{S\in\tbinom{[n]}m} \det(A_{[m],S})\det(B_{S,[m]}).$$

Example: Taking m = 2 and n = 3, and matrices $$A = \begin{pmatrix}1&1&2\\ 3& 1& -1\\ \end{pmatrix}$$ and $$B = \begin{pmatrix}1&1\\3&1\\0&2\end{pmatrix}$$, the Cauchy–Binet formula gives the determinant

\det(AB)= \left|\begin{matrix}1&1\\3&1\end{matrix}\right| \cdot \left|\begin{matrix}1&1\\3&1\end{matrix}\right| + \left|\begin{matrix}1&2\\1&-1\end{matrix}\right| \cdot \left|\begin{matrix}3&1\\0&2\end{matrix}\right| + \left|\begin{matrix}1&2\\3&-1\end{matrix}\right| \cdot \left|\begin{matrix}1&1\\0&2\end{matrix}\right|. $$

Indeed $$AB =\begin{pmatrix}4&6\\6&2\end{pmatrix}$$, and its determinant is $$-28$$ which equals $$-2 \times -2 + -3 \times 6 + -7 \times 2$$ from the right hand side of the formula.

Special cases
If n < m then $$\tbinom{[n]}m$$ is the empty set, and the formula says that det(AB) = 0 (its right hand side is an empty sum); indeed in this case the rank of the m×m matrix AB is at most n, which implies that its determinant is zero. If n = m, the case where A and B are square matrices, $$\tbinom{[n]}m=\{[n]\} $$ (a singleton set), so the sum only involves S = [n], and the formula states that det(AB) = det(A)det(B).

For m = 0, A and B are empty matrices (but of different shapes if n > 0), as is their product AB; the summation involves a single term S = Ø, and the formula states 1 = 1, with both sides given by the determinant of the 0×0 matrix. For m = 1, the summation ranges over the collection $$\tbinom{[n]}1$$ of the n different singletons taken from [n], and both sides of the formula give $$\textstyle\sum_{j=1}^nA_{1,j}B_{j,1}$$, the dot product of the pair of vectors represented by the matrices. The smallest value of m for which the formula states a non-trivial equality is m = 2; it is discussed in the article on the Binet–Cauchy identity.

In the case n = 3
Let $$ \boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}, \boldsymbol{d}, \boldsymbol{x}, \boldsymbol{y},\boldsymbol{z},\boldsymbol{w} $$ be three-dimensional vectors.

\begin{align} & 1 = 1 & (m = 0)\\[10pt] & \boldsymbol{a}\cdot \boldsymbol{x} = a_1  x_1 + a_2 x_2 + a_3 x_3 & (m = 1)\\[10pt] & \begin{vmatrix} \boldsymbol{a}\cdot \boldsymbol{x} & \boldsymbol{a}\cdot \boldsymbol{y}\\ \boldsymbol{b}\cdot \boldsymbol{x} & \boldsymbol{b}\cdot \boldsymbol{y} \end{vmatrix} \\[4pt] = {} & \begin{vmatrix} a_2 & a_3\\ b_2 & b_3 \end{vmatrix} \begin{vmatrix} x_2 & y_2\\ x_3 & y_3 \end{vmatrix} + \begin{vmatrix} a_3 & a_1\\ b_3 & b_1 \end{vmatrix} \begin{vmatrix} x_3 & y_3\\ x_1 & y_1 \end{vmatrix} + \begin{vmatrix} a_1 & a_2\\ b_1 & b_2 \end{vmatrix} \begin{vmatrix} x_1 & y_1\\ x_2 & y_2 \end{vmatrix}\\[4pt] = {} & (\boldsymbol{a}\times\boldsymbol{b})\cdot(\boldsymbol{x}\times\boldsymbol{y}) & (m = 2)\\[10pt] & \begin{vmatrix} \boldsymbol{a}\cdot \boldsymbol{x} & \boldsymbol{a}\cdot \boldsymbol{y} & \boldsymbol{a}\cdot \boldsymbol{z}\\ \boldsymbol{b}\cdot \boldsymbol{x} & \boldsymbol{b}\cdot \boldsymbol{y} & \boldsymbol{b}\cdot \boldsymbol{z}\\ \boldsymbol{c}\cdot \boldsymbol{x} & \boldsymbol{c}\cdot \boldsymbol{y} & \boldsymbol{c}\cdot \boldsymbol{z} \end{vmatrix} = \begin{vmatrix} a_1 & a_2 & a_3\\ b_1 & b_2 & b_3\\ c_1 & c_2 & c_3 \end{vmatrix} \begin{vmatrix} x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \\ x_3 & y_3 & z_3 \end{vmatrix}\\[4pt] = {} & [\boldsymbol{a}\cdot (\boldsymbol{b} \times \boldsymbol{c})] [\boldsymbol{x}\cdot (\boldsymbol{y} \times \boldsymbol{z})] & (m = 3)\\[10pt] & \begin{vmatrix} \boldsymbol{a}\cdot \boldsymbol{x} & \boldsymbol{a}\cdot \boldsymbol{y} & \boldsymbol{a}\cdot \boldsymbol{z} & \boldsymbol{a}\cdot \boldsymbol{w} \\ \boldsymbol{b}\cdot \boldsymbol{x} & \boldsymbol{b}\cdot \boldsymbol{y} & \boldsymbol{b}\cdot \boldsymbol{z} & \boldsymbol{b}\cdot \boldsymbol{w} \\ \boldsymbol{c}\cdot \boldsymbol{x} & \boldsymbol{c}\cdot \boldsymbol{y} & \boldsymbol{c}\cdot \boldsymbol{z} & \boldsymbol{c}\cdot \boldsymbol{w} \\ \boldsymbol{d}\cdot \boldsymbol{x} & \boldsymbol{d}\cdot \boldsymbol{y} & \boldsymbol{d}\cdot \boldsymbol{z} & \boldsymbol{d}\cdot \boldsymbol{w} \end{vmatrix} = 0 & (m = 4) \end{align} $$

In the case m > 3, the right-hand side always equals 0.

A simple proof
The following simple proof relies on two facts that can be proven in several different ways:

 * 1) For any $$1 \leq k\leq n$$ the coefficient of $$z^{n-k}$$ in the polynomial $$\det(zI_n+X)$$ is the sum of the $$k\times k$$ principal minors of $$X$$.
 * 2) If $$m \leq n$$ and $$A$$ is an $$m\times n$$ matrix and $$B$$ an $$n\times m$$ matrix, then
 * $$\det(zI_n+BA)=z^{n-m}\det(zI_m+AB)$$.

Now, if we compare the coefficient of $$z^{n-m}$$ in the equation $$\det(zI_n+BA)=z^{n-m}\det(zI_m+AB)$$, the left hand side will give the sum of the principal minors of $$BA$$ while the right hand side will give the constant term of $$\det(zI_m+AB)$$, which is simply $$\det(AB)$$, which is what the Cauchy–Binet formula states, i.e.

\begin{align} &\det(AB)= \sum_{S\in\tbinom{[n]}m} \det((BA)_{S,S})=\sum_{S\in\tbinom{[n]}m} \det(B_{S,[m]}) \det(A_{[m],S}) \\[5pt] = {} & \sum_{S\in\tbinom{[n]}m}\det(A_{[m],S}) \det(B_{S,[m]}). \end{align} $$

Proof
There are various kinds of proofs that can be given for the Cauchy−Binet formula. The proof below is based on formal manipulations only, and avoids using any particular interpretation of determinants, which may be taken to be defined by the Leibniz formula. Only their multilinearity with respect to rows and columns, and their alternating property (vanishing in the presence of equal rows or columns) are used; in particular the multiplicative property of determinants for square matrices is not used, but is rather established (the case n = m). The proof is valid for arbitrary commutative coefficient rings.

The formula can be proved in two steps:

 * 1) use the fact that both sides are multilinear (more precisely 2m-linear) in the rows of A and the columns of B, to reduce to the case that each row of A and each column of B has only one non-zero entry, which is 1.
 * 2) handle that case using the functions [m] → [n] that map respectively the row numbers of A to the column number of their nonzero entry, and the column numbers of B to the row number of their nonzero entry.

For step 1, observe that for each row of A or column of B, and for each m-combination S, the values of det(AB) and det(A[m],S)det(BS,[m]) indeed depend linearly on the row or column. For the latter this is immediate from the multilinear property of the determinant; for the former one must in addition check that taking a linear combination for the row of A or column of B while leaving the rest unchanged only affects the corresponding row or column of the product AB, and by the same linear combination. Thus one can work out both sides of the Cauchy−Binet formula by linearity for every row of A and then also every column of B, writing each of the rows and columns as a linear combination of standard basis vectors. The resulting multiple summations are huge, but they have the same form for both sides: corresponding terms involve the same scalar factor (each is a product of entries of A and of B), and these terms only differ by involving two different expressions in terms of constant matrices of the kind described above, which expressions should be equal according to the Cauchy−Binet formula. This achieves the reduction of the first step.

Concretely, the multiple summations can be grouped into two summations, one over all functions f:[m] → [n] that for each row index of A gives a corresponding column index, and one over all functions g:[m] → [n] that for each column index of B gives a corresponding row index. The matrices associated to f and g are

 * $$L_f=\bigl((\delta_{f(i),j})_{i\in[m],j\in[n]}\bigr) \quad\text{and}

\quad R_g=\bigl((\delta_{j,g(k)})_{j\in[n],k\in[m]}\bigr)$$

where "$$\delta$$" is the Kronecker delta, and the Cauchy−Binet formula to prove has been rewritten as

\begin{align} & \sum_{f:[m]\to[n]}\sum_{g:[m]\to[n]}p(f,g)\det(L_fR_g) \\[5pt] = {} & \sum_{f:[m]\to[n]}\sum_{g:[m]\to[n]} p(f,g) \sum_{S\in\tbinom{[n]}m} \det((L_f)_{[m],S}) \det((R_g)_{S,[m]}), \end{align} $$

where p(f,g) denotes the scalar factor $$\textstyle(\prod_{i=1}^mA_{i,f(i)})(\prod_{k=1}^mB_{g(k),k})$$. It remains to prove the Cauchy−Binet formula for A = Lf and B = Rg, for all f,g:[m] → [n].

For this step 2, if f fails to be injective then Lf and LfRg both have two identical rows, and if g fails to be injective then Rg and LfRg both have two identical columns; in either case both sides of the identity are zero. Supposing now that both f and g are injective maps [m] → [n], the factor $$\det((L_f)_{[m],S})$$ on the right is zero unless S = f([m]), while the factor $$\det((R_g)_{S,[m]})$$ is zero unless S = g([m]). So if the images of f and g are different, the right hand side has only null terms, and the left hand side is zero as well since LfRg has a null row (for i with $$f(i)\notin g([m])$$). In the remaining case where the images of f and g are the same, say f([m]) = S = g([m]), we need to prove that

 * $$\det(L_fR_g)=\det((L_f)_{[m],S})\det((R_g)_{S,[m]}).\,$$

Let h be the unique increasing bijection [m] → S, and π,σ the permutations of [m] such that $$f=h\circ\pi^{-1}$$ and $$g=h\circ\sigma$$; then $$(L_f)_{[m],S}$$ is the permutation matrix for $\pi$, $$(R_g)_{S,[m]}$$ is the permutation matrix for σ, and LfRg is the permutation matrix for $$\pi\circ\sigma$$, and since the determinant of a permutation matrix equals the signature of the permutation, the identity follows from the fact that signatures are multiplicative.

Using multi-linearity with respect to both the rows of A and the columns of B in the proof is not necessary; one could use just one of them, say the former, and use that a matrix product LfB either consists of a permutation of the rows of Bf([m]),[m] (if f is injective), or has at least two equal rows.

Relation to the generalized Kronecker delta
As we have seen, the Cauchy–Binet formula is equivalent to the following:

\det(L_fR_g)=\sum_{S\in\tbinom{[n]}m} \det((L_f)_{[m],S})\det((R_g)_{S,[m]}), $$ where

L_f=\bigl((\delta_{f(i),j})_{i\in[m],j\in[n]}\bigr) \quad\text{and} \quad R_g=\bigl((\delta_{j,g(k)})_{j\in[n],k\in[m]}\bigr). $$

In terms of generalized Kronecker delta, we can derive the formula equivalent to the Cauchy–Binet formula:

\delta^{f(1) \dots f(m)}_{g(1) \dots g(m)} = \sum_{k:[m]\to[n] \atop k(1)<\dots<k(m)} \delta^{f(1) \dots f(m)}_{k(1) \dots k(m)} \delta^{k(1) \dots k(m)}_{g(1) \dots g(m)}. $$

Geometric interpretations
If A is a real m×n matrix, then det(A AT) is equal to the square of the m-dimensional volume of the parallelotope spanned in Rn by the m rows of A. Binet's formula states that this is equal to the sum of the squares of the volumes that arise if the parallelepiped is orthogonally projected onto the m-dimensional coordinate planes (of which there are $$\tbinom nm$$).

In the case m = 1 the parallelotope is reduced to a single vector and its volume is its length. The above statement then states that the square of the length of a vector is the sum of the squares of its coordinates; this is indeed the case by the definition of that length, which is based on the Pythagorean theorem.

In tensor algebra, given an inner product space $$V$$ of dimension n, the Cauchy–Binet formula defines an induced inner product on the exterior algebra $$\wedge^m V$$, namely: $$\langle v_1\wedge\cdots \wedge v_m, w_1\wedge\cdots \wedge w_m\rangle =\det\left( \langle v_i,w_j\rangle \right)_{i,j=1}^m .$$

Generalization
The Cauchy–Binet formula can be extended in a straightforward way to a general formula for the minors of the product of two matrices. Context for the formula is given in the article on minors, but the idea is that both the formula for ordinary matrix multiplication and the Cauchy–Binet formula for the determinant of the product of two matrices are special cases of the following general statement about the minors of a product of two matrices. Suppose that A is an m × n matrix, B is an n × p matrix, I is a subset of {1,...,m} with k elements and J is a subset of {1,...,p} with k elements. Then
 * $$[\mathbf{AB}]_{I,J} = \sum_{K} [\mathbf{A}]_{I,K} [\mathbf{B}]_{K,J}\,$$

where the sum extends over all subsets K of {1,...,n} with k elements.

Continuous version
A continuous version of the Cauchy–Binet formula, known as the Andréief-Heine identity or Andréief identity appears commonly in random matrix theory. It is stated as follows: let $$\left\{f_j(x)\right\}_{j=1}^{N}$$ and $$\left\{g_j(x)\right\}_{j=1}^{N}$$ be two sequences of integrable functions, supported on $$I$$. Then
 * $$\int_I \cdots \int_I \det \left[f_{j}(x_k)\right]_{j,k=1}^N \det \left[g_{j}(x_k)\right]_{j,k=1}^N dx_1 \cdots dx_N

= N!\, \det \left[\int_I f_j(x)g_k(x) dx\right]_{j,k=1}^{N}.$$

$$

Forrester describes how to recover the usual Cauchy–Binet formula as a discretisation of the above identity.

$$

title: Cauchy–Binet formula


text:
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