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Second Great Awakening

The Second Great Awakening was a Protestant religious revival during the late 18th to early 19th century in the United States. It spread religion through revivals and emotional preaching and sparked a number of reform movements. Revivals were a key part of the movement and attracted hundreds of converts to new Protestant denominations. The Methodist Church used circuit riders to reach people in frontier locations.

The Second Great Awakening led to a period of antebellum social reform and an emphasis on salvation by institutions. The outpouring of religious fervor and revival began in Kentucky and Tennessee in the 1790s and early 1800s among the Presbyterians, Methodists, and Baptists. New religious movements emerged during the Second Great Awakening, such as Adventism, Dispensationalism, and the Latter Day Saint movement. The Second Great Awakening also led to the founding of several well-known colleges, seminaries, and mission societies.

Historians named the Second Great Awakening in the context of the First Great Awakening of the 1730s and 1750s and of the Third Great Awakening of the late 1850s to early 1900s. The First Awakening was part of a much larger evangelical religious movement that was sweeping across England, Scotland, and Germany.

Background
Like the First Great Awakening a half century earlier, the Second Great Awakening in North America reflected Romanticism characterized by enthusiasm, emotion, and an appeal to the supernatural. It rejected the skepticism, deism, Unitarianism, and rationalism left over from the American Enlightenment, about the same time that similar movements flourished in Europe. Pietism was sweeping Germanic countries and evangelicalism was waxing strong in England.

The Second Great Awakening occurred in several episodes and over different denominations; however, the revivals were very similar. As the most effective form of evangelizing during this period, revival meetings cut across geographical boundaries. The movement quickly spread throughout Kentucky, Indiana, Tennessee, and southern Ohio, as well as other regions of the United States and Canada. Each denomination had assets that allowed it to thrive on the frontier. The Methodists had an efficient organization that depended on itinerant ministers, known as circuit riders, who sought out people in remote frontier locations. The circuit riders came from among the common people, which helped them establish rapport with the frontier families they hoped to convert.

Theology
Postmillennialist theology dominated American Protestantism in the first half of the 19th century. Postmillennialists believed that Christ will return to earth after the "Millennium", which could entail either a literal 1,000 years or a figurative "long period" of peace and happiness. Christians thus had a duty to purify society in preparation for that return. This duty extended beyond American borders to include Christian Restorationism. George Fredrickson argues that Postmillennial theology "was an impetus to the promotion of Progressive reforms, as historians have frequently pointed out." During the Second Great Awakening of the 1830s, some diviners expected the Millennium to arrive in a few years. By the late 1840s, however, the great day had receded to the distant future, and postmillennialism became a more passive religious dimension of the wider middle-class pursuit of reform and progress.

Burned-over district
Beginning in the 1820s, Western New York State experienced a series of popular religious revivals that would later earn this region the nickname "the burned-over district," which implied the area was set ablaze with spiritual fervor. This term, however, was not used by contemporaries in the first half of the nineteenth century, as it originates from Charles Grandison Finney's Autobiography of Charles G Finney (1876), in which he writes, "I found that region of country what, in the western phrase, would be called, a 'burnt district.' There had been, a few years previously, a wild excitement passing through that region, which they called a revival of religion, but which turned out to be spurious." During this period, a number of nonconformist, folk religion, and evangelical sects flourished in the region.

The extent to which religious fervor actually affected the region was reassessed in last quarter of the twentieth century. Linda K. Pritchard used statistical data to show that compared to the rest of New York State, the Ohio River Valley in the lower Midwest, and the country as a whole, the religiosity of the Burned-over District was typical rather than exceptional. More recent works, however, have argued that these revivals in Western New York had a unique and lasting impact upon the religious and social life of the entire nation.

West and Tidewater South
On the American frontier, evangelical denominations, especially Methodists and Baptists, sent missionary preachers and exhorters to meet the people in the backcountry in an effort to support the growth of church membership and the formation of new congregations. Another key component of the revivalists' techniques was the camp meeting. These outdoor religious gatherings originated from field meetings and the Scottish Presbyterians' "Holy Fairs", which were brought to America in the mid-eighteenth century from Ireland, Scotland, and Britain's border counties. Most of the Scotch-Irish immigrants before the American Revolutionary War settled in the backcountry of Pennsylvania and down the spine of the Appalachian Mountains in present-day Maryland and Virginia, where Presbyterian emigrants and Baptists held large outdoor gatherings in the years prior to the war. The Presbyterians and Methodists sponsored similar gatherings on a regular basis after the Revolution.

The denominations that encouraged the revivals were based on an interpretation of man's spiritual equality before God, which led them to recruit members and preachers from a wide range of classes and all races. Baptists and Methodist revivals were successful in some parts of the Tidewater South, where an increasing number of common planters, plain folk, and slaves were converted.

West
In the newly settled frontier regions, the revival was implemented through camp meetings. These often provided the first encounter for some settlers with organized religion, and they were important as social venues. The camp meeting was a religious service of several days' length with preachers. Settlers in thinly populated areas gathered at the camp meeting for fellowship as well as worship. The sheer exhilaration of participating in a religious revival with crowds of hundreds and perhaps thousands of people inspired the dancing, shouting, and singing associated with these events. The revivals also followed an arc of great emotional power, with an emphasis on the individual's sins and need to turn to Christ, and a sense of restoring personal salvation. This differed from the Calvinists' belief in predestination as outlined in the Westminster Confession of Faith, which emphasized the inability of men to save themselves and decreed that the only way to be saved was by God's electing grace. Upon their return home, most converts joined or created small local churches, which grew rapidly.

The Revival of 1800 in Logan County, Kentucky, began as a traditional Presbyterian sacramental occasion. The first informal camp meeting began in June, when people began camping on the grounds of the Red River Meeting House. Subsequent meetings followed at the nearby Gasper River and Muddy River congregations. All three of these congregations were under the ministry of Presbyterian Reverend James McGready. A year later, in August 1801, an even larger sacrament occasion that is generally considered to be America's first camp meeting was held at Cane Ridge in Bourbon County, Kentucky, under Barton W. Stone (1772–1844) with numerous Presbyterian, Baptist, and Methodist ministers participating in the services. The six-day gathering attracting perhaps as many as 20,000 people, although the exact number of attendees was not formally recorded. Due to the efforts of such leaders as Stone and Alexander Campbell (1788–1866), the camp meeting revival spread religious enthusiasm and became a major mode of church expansion, especially for the Methodists and Baptists. Presbyterians and Methodists initially worked together to host the early camp meetings, but the Presbyterians eventually became less involved because of the noise and often raucous activities that occurred during the protracted sessions.

As a result of the Revival of 1800, the Cumberland Presbyterian Church was founded in 1810 near Dickson, Tennessee by the Revs: Samuel McAdow, Finis Ewing, and Samuel King and became a strong supporter of the revivalist movement. Cane Ridge was also instrumental in fostering what became known as the Restoration Movement, which consisted of non-denominational churches committed to what they viewed as the original, fundamental Christianity of the New Testament. Churches with roots in this movement include the Churches of Christ, Christian Church (Disciples of Christ), and the Evangelical Christian Church in Canada. The congregations of these denomination were committed to individuals' achieving a personal relationship with Christ.

Church membership soars
The Methodist circuit riders and local Baptist preachers made enormous gains in increasing church membership. To a lesser extent the Presbyterians also gained members, particularly with the Cumberland Presbyterian Church in sparsely settled areas. As a result, the numerical strength of the Baptists and Methodists rose relative to that of the denominations dominant in the colonial period—the Anglicans, Presbyterians, Congregationalists. Among the new denominations that grew from the religious ferment of the Second Great Awakening are the Churches of Christ, Christian Church (Disciples of Christ), the Seventh-day Adventist Church, and the Evangelical Christian Church in Canada.

The converts during the Second Great Awakening were predominantly female. A 1932 source estimated at least three female converts to every two male converts between 1798 and 1826. Young people (those under 25) also converted in greater numbers, and were the first to convert.

Adventism
The Advent Movement emerged in the 1830s and 1840s in North America, and was preached by ministers such as William Miller, whose followers became known as Millerites. The name refers to belief in the soon Second Advent of Jesus (popularly known as the Second coming) and resulted in several major religious denominations, including Seventh-day Adventists and Advent Christians.

Holiness movement
Though its roots are in the First Great Awakening and earlier, a re-emphasis on Wesleyan teachings on sanctification emerged during the Second Great Awakening, leading to a distinction between Mainline Methodism and Holiness churches.

Restoration Movement
The idea of restoring a "primitive" form of Christianity grew in popularity in the U.S. after the American Revolution. This desire to restore a purer form of Christianity without an elaborate hierarchy contributed to the development of many groups during the Second Great Awakening, including the Latter Day Saints, Baptists and Shakers. Several factors made the restoration sentiment particularly appealing during this time period:

 * To immigrants in the early 19th century, the land in the United States seemed pristine, edenic and undefiled – "the perfect place to recover pure, uncorrupted and original Christianity" – and the tradition-bound European churches seemed out of place in this new setting.
 * A primitive faith based on the Bible alone promised a way to sidestep the competing claims of the many denominations available and for congregations to find assurance of being right without the security of an established national church.

The Restoration Movement began during, and was greatly influenced by, the Second Great Awakening. While the leaders of one of the two primary groups making up this movement, Thomas Campbell and Alexander Campbell, resisted what they saw as the spiritual manipulation of the camp meetings, the revivals contributed to the development of the other major branch, led by Barton W. Stone. The Southern phase of the Awakening "was an important matrix of Barton Stone's reform movement" and shaped the evangelistic techniques used by both Stone and the Campbells.

Culture and society
Efforts to apply Christian teaching to the resolution of social problems presaged the Social Gospel of the late 19th century. Converts were taught that to achieve salvation they needed not just to repent personal sin but also work for the moral perfection of society, which meant eradicating sin in all its forms. Thus, evangelical converts were leading figures in a variety of 19th century reform movements.

Congregationalists set up missionary societies to evangelize the western territory of the northern tier. Members of these groups acted as apostles for the faith, and also as educators and exponents of northeastern urban culture. The Second Great Awakening served as an "organizing process" that created "a religious and educational infrastructure" across the western frontier that encompassed social networks, a religious journalism that provided mass communication, and church-related colleges. Publication and education societies promoted Christian education; most notable among them was the American Bible Society, founded in 1816. Women made up a large part of these voluntary societies. The Female Missionary Society and the Maternal Association, both active in Utica, NY, were highly organized and financially sophisticated women's organizations responsible for many of the evangelical converts of the New York frontier.

There were also societies that broadened their focus from traditional religious concerns to larger societal ones. These organizations were primarily sponsored by affluent women. They did not stem entirely from the Second Great Awakening, but the revivalist doctrine and the expectation that one's conversion would lead to personal action accelerated the role of women's social benevolence work. Social activism influenced abolition groups and supporters of the Temperance movement. They began efforts to reform prisons and care for the handicapped and mentally ill. They believed in the perfectibility of people and were highly moralistic in their endeavors.

African Americans
Baptists and Methodists in the South preached to slaveholders and slaves alike. Conversions and congregations started with the First Great Awakening, resulting in Baptist and Methodist preachers being authorized among slaves and free African Americans more than a decade before 1800. "Black Harry" Hosier, an illiterate freedman who drove Francis Asbury on his circuits, proved to be able to memorize large passages of the Bible verbatim and became a cross-over success, as popular among white audiences as the black ones Asbury had originally intended for him to minister. His sermon at Thomas Chapel in Chapeltown, Delaware, in 1784 was the first to be delivered by a black preacher directly to a white congregation.

Despite being called the "greatest orator in America" by Benjamin Rush and one of the best in the world by Bishop Thomas Coke, Hosier was repeatedly passed over for ordination and permitted no vote during his attendance at the Christmas Conference that formally established American Methodism. Richard Allen, the other black attendee, was ordained by the Methodists in 1799, but his congregation of free African Americans in Philadelphia left the church there because of its discrimination. They founded the African Methodist Episcopal Church (AME) in Philadelphia. After first submitting to oversight by the established Methodist bishops, several AME congregations finally left to form the first independent African-American denomination in the United States in 1816. Soon after, the African Methodist Episcopal Zion Church (AME Zion) was founded as another denomination in New York City.

Early Baptist congregations were formed by slaves and free African Americans in South Carolina and Virginia. Especially in the Baptist Church, African Americans were welcomed as members and as preachers. By the early 19th century, independent African-American congregations numbered in the several hundreds in some cities of the South, such as Charleston, South Carolina, and Richmond and Petersburg, Virginia. With the growth in congregations and churches, Baptist associations formed in Virginia, for instance, as well as Kentucky and other states.

The revival also inspired slaves to demand freedom. In 1800, out of African-American revival meetings in Virginia, a plan for slave rebellion was devised by Gabriel Prosser, although the rebellion was discovered and crushed before it started. Despite white attempts to control independent African-American congregations, especially after the Nat Turner uprising of 1831, a number of African-American congregations managed to maintain their separation as independent congregations in Baptist associations. State legislatures passed laws requiring them always to have a white man present at their worship meetings.

Women
Women, who made up the majority of converts during the Awakening, played a crucial role in its development and focus. It is not clear why women converted in larger numbers than men. Various scholarly theories attribute the discrepancy to a reaction to the perceived sinfulness of youthful frivolity, an inherent greater sense of religiosity in women, a communal reaction to economic insecurity, or an assertion of the self in the face of patriarchal rule. Husbands, especially in the South, sometimes disapproved of their wives' conversion, forcing women to choose between submission to God or their spouses. Church membership and religious activity gave women peer support and place for meaningful activity outside the home, providing many women with communal identity and shared experiences.

Despite the predominance of women in the movement, they were not formally indoctrinated or given leading ministerial positions. However, women took other public roles; for example, relaying testimonials about their conversion experience, or assisting sinners (both male and female) through the conversion process. Leaders such as Charles Finney saw women's public prayer as a crucial aspect in preparing a community for revival and improving their efficacy in conversion. Women also took crucial roles in the conversion and religious upbringing of children. During the period of revival, mothers were seen as the moral and spiritual foundation of the family, and were thus tasked with instructing children in matters of religion and ethics.

The greatest change in women's roles stemmed from participation in newly formalized missionary and reform societies. Women's prayer groups were an early and socially acceptable form of women's organization. In the 1830s, female moral reform societies rapidly spread across the North making it the first predominantly female social movement. Through women's positions in these organizations, women gained influence outside of the private sphere.

Changing demographics of gender also affected religious doctrine. In an effort to give sermons that would resonate with the congregation, ministers stressed Christ's humility and forgiveness, in what the historian Barbara Welter calls a "feminization" of Christianity.

Prominent figures

 * Richard Allen, founder, African Methodist Episcopal Church
 * Francis Asbury, Methodist, circuit rider and founder of the Methodist Episcopal Church
 * Henry Ward Beecher, Congregationalist, son of Lyman Beecher
 * Lyman Beecher, Presbyterian
 * Antoinette Brown Blackwell, Congregationalist and later Unitarian, the first ordained female minister in the United States
 * Alexander Campbell, Presbyterian, and early leader of the Restoration Movement
 * Thomas Campbell, Presbyterian, then early leader of the Restoration Movement
 * Peter Cartwright, Methodist
 * Lorenzo Dow, Methodist
 * Timothy Dwight IV, Congregationalist
 * Charles Grandison Finney, Presbyterian and anti-Calvinist, second president of Oberlin College
 * "Black Harry" Hosier, Methodist, the first African American to preach to a white congregation
 * Adoniram Judson, early Baptist missionary.
 * Ann Lee, Shakers
 * Jarena Lee, Methodist, a female AME circuit rider
 * Robert Matthews, cult following as Matthias the Prophet
 * William Miller, Millerism, forerunner of Adventism
 * Asahel Nettleton, Reformed
 * Benjamin Randall, Free Will Baptist
 * Luther Rice, Baptist missionary to India, and Baptist missionary in the US South
 * Joseph Smith, The Church of Jesus Christ of Latter-day Saints, early leader of the Restoration Movement
 * Barton Stone, Presbyterian non-Calvinist, then early leader of the Restoration Movement
 * Nathaniel William Taylor, heterodox Calvinist
 * Ellen G. White, Seventh-day Adventist Church prophetess

Political implications
Revivals and perfectionist hopes of improving individuals and society continued to increase from 1840 to 1865 across all major denominations, especially in urban areas. Evangelists often directly addressed issues such as slavery, greed, and poverty, laying the groundwork for later reform movements. The influence of the Awakening continued in the form of more secular movements. In the midst of shifts in theology and church polity, American Christians began progressive movements to reform society during this period. Known commonly as antebellum reform, this phenomenon included reforms against the consumption of alcohol, for women's rights and abolition of slavery, and a multitude of other issues faced by society.

The religious enthusiasm of the Second Great Awakening was echoed by the new political enthusiasm of the Second Party System. More active participation in politics by more segments of the population brought religious and moral issues into the political sphere. The spirit of evangelical humanitarian reforms was carried on in the antebellum Whig party.

Historians stress the common understanding among participants of reform as being a part of God's plan. As a result, local churches saw their roles in society in purifying the world through the individuals to whom they could bring salvation, and through changes in the law and the creation of institutions. Interest in transforming the world was applied to mainstream political action, as temperance activists, antislavery advocates, and proponents of other variations of reform sought to implement their beliefs into national politics. While Protestant religion had previously played an important role on the American political scene, the Second Great Awakening strengthened the role it would play.

Historiography

 * Conforti, Joseph. "The Invention of the Great Awakening, 1795–1842". Early American Literature (1991): 99–118..
 * Griffin, Clifford S. "Religious Benevolence as Social Control, 1815–1860", The Mississippi Valley Historical Review, (1957) 44#3 pp. 423–444. ..
 * Mathews, Donald G. "The Second Great Awakening as an organizing process, 1780–1830: An hypothesis". American Quarterly (1969): 23–43. ..
 * Shiels, Richard D. "The Second Great Awakening in Connecticut: Critique of the Traditional Interpretation", Church History 49 (1980): 401–415..
 * Varel, David A. "The Historiography of the Second Great Awakening and the Problem of Historical Causation, 1945–2005". Madison Historical Review (2014) 8#4 online

title: Second Great Awakening


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Antarctic krill

Antarctic krill (Euphausia superba) is a species of krill found in the Antarctic waters of the Southern Ocean. It is a small, swimming crustacean that lives in large schools, called swarms, sometimes reaching densities of 10,000–30,000 individual animals per cubic metre. It feeds directly on minute phytoplankton, thereby using the primary production energy that the phytoplankton originally derived from the sun in order to sustain their pelagic (open ocean) life cycle. It grows to a length of 6 cm, weighs up to 2 g, and can live for up to six years. It is a key species in the Antarctic ecosystem and in terms of biomass, is one of the most abundant animal species on the planet approximately 500 e6MT.

Life cycle

The main spawning season of Antarctic krill is from January to March, both above the continental shelf and also in the upper region of deep sea oceanic areas. In the typical way of all krill, the male attaches a spermatophore to the genital opening of the female. For this purpose, the first pleopods (legs attached to the abdomen) of the male are constructed as mating tools. Females lay 6,000–10,000 eggs at one time. They are fertilised as they pass out of the genital opening.

According to the classical hypothesis of Marriosis De' Abrtona, derived from the results of the expedition of the famous British research vessel RRS Discovery, egg development then proceeds as follows: gastrulation (development of egg into embryo) sets in during the descent of the 0.6 mm eggs on the shelf at the bottom, in oceanic areas in depths around 2000 - 3000 m. The egg hatches as a nauplius larva; once this has moulted into a metanauplius, the young animal starts migrating towards the surface in a migration known as developmental ascent.

The next two larval stages, termed second nauplius and metanauplius, still do not eat but are nourished by the remaining yolk. After three weeks, the young krill has finished the ascent. They can appear in enormous numbers counting 2 per litre in 60 m water depth. Growing larger, additional larval stages follow (second and third calyptopis, first to sixth furcilia). They are characterised by increasing development of the additional legs, the compound eyes and the setae (bristles). At 15 mm, the juvenile krill resembles the habitus of the adults. Krill reach maturity after two to three years. Like all crustaceans, krill must moult in order to grow. Approximately every 13 to 20 days, krill shed their chitinous exoskeleton and leave it behind as exuvia.

Food
The gut of E. superba can often be seen shining green through its transparent skin. This species feeds predominantly on phytoplankton—especially very small diatoms (20 μm), which it filters from the water with a feeding basket. The glass-like shells of the diatoms are cracked in the gastric mill and then digested in the hepatopancreas. The krill can also catch and eat copepods, amphipods and other small zooplankton. The gut forms a straight tube; its digestive efficiency is not very high and therefore a lot of carbon is still present in the feces. Antarctic krill (E. superba) primarily has chitinolytic enzymes in the stomach and mid-gut to break down chitinous spines on diatoms, additional enzymes can vary due to its expansive diet.

In aquaria, krill have been observed to eat each other. When they are not fed, they shrink in size after moulting, which is exceptional for animals this size. It is likely that this is an adaptation to the seasonality of their food supply, which is limited in the dark winter months under the ice. However, the animal's compound eyes do not shrink, and so the ratio between eye size and body length has thus been found to be a reliable indicator of starvation. A krill with ample food supply would have eyes proportional to body length, compared to a starving krill that would have eyes that appeared larger than what is normal.

Filter feeding
Antarctic krill directly ingest minute phytoplankton cells, which no other animal of krill size can do. This is accomplished through filter feeding, using the krill's highly developed front legs which form an efficient filtering apparatus: the six thoracopods (legs attached to the thorax) create a "feeding basket" used to collect phytoplankton from the open water. In the finest areas the openings in this basket are only 1 μm in diameter. In lower food concentrations, the feeding basket is pushed through the water for over half a metre in an opened position, and then the algae are combed to the mouth opening with special setae (bristles) on the inner side of the thoracopods.

Ice-algae raking
Antarctic krill can scrape off the green lawn of ice algae from the underside of pack ice. Krill have developed special rows of rake-like setae at the tips of their thoracopods, and graze the ice in a zig-zag fashion. One krill can clear an area of a square foot in about 10 minutes (1.5 cm2/s). Recent discoveries have found that the film of ice algae is well developed over vast areas, often containing much more carbon than the whole water column below. Krill find an extensive energy source here, especially in the spring after food sources have been limited during the winter months.

Biological pump and carbon sequestration
Krill are thought to undergo between one and three vertical migrations from mixed surface waters to depths of 100 m daily. The krill is a very untidy feeder, and it often spits out aggregates of phytoplankton (spit balls) containing thousands of cells sticking together. It also produces fecal strings that still contain significant amounts of carbon and, glass shells of the diatoms. Both are heavy and sink very fast into the abyss. This process is called the biological pump. As the waters around Antarctica are very deep (2000 - 4000 m), they act as a carbon dioxide sink: this process exports large quantities of carbon (fixed carbon dioxide, CO2) from the biosphere and sequesters it for about 1,000 years. If the phytoplankton is consumed by other components of the pelagic ecosystem, most of the carbon remains in the upper layers of the ocean. There is speculation that this process is one of the largest biofeedback mechanisms of the planet, maybe the most sizable of all, driven by a gigantic biomass. Still more research is needed to quantify the Southern Ocean ecosystem.

Bioluminescence
Krill are often referred to as light-shrimp because they emit light through bioluminescent organs. These organs are located on various parts of the individual krill's body: one pair of organs at the eyestalk (cf. the image of the head above), another pair are on the hips of the second and seventh thoracopods, and singular organs on the four pleonsternites. These light organs emit a yellow-green light periodically, for up to 2–3 s. They are considered so highly developed that they can be compared with a flashlight. There is a concave reflector in the back of the organ and a lens in the front that guide the light produced. The whole organ can be rotated by muscles, which can direct the light to a specific area. The function of these lights is not yet fully understood; some hypotheses have suggested they serve to compensate the krill's shadow so that they are not visible to predators from below; other speculations maintain that they play a significant role in mating or schooling at night.

The krill's bioluminescent organs contain several fluorescent substances. The major component has a maximum fluorescence at an excitation of 355 nm and emission of 510 nm.

Escape reaction
Krill use an escape reaction to evade predators, swimming backwards very quickly by flipping their rear ends. This swimming pattern is also known as lobstering. Krill can reach speeds of over 0.6 m/s. The trigger time to optical stimulus is, despite the low temperatures, only 55 ms.

Genome
The genome of E. superba spans about 48 GB and is thus one of the largest in the animal kingdom and the largest that has been assembled to date. Its content of repetitive DNA is about 70% and may reach up to 92.45% after additional repeat annotation, which is also the largest fraction known of any genome. There is no evidence of polyploidy. Shao et al. annotated 28,834 protein-coding genes in the Antarctic krill genome, which is similar to other animal genomes. The gene and intron lengths of Antarctic krill are notably shorter than those of lungfishes and Mexican axolotl, two other animals with giant genomes.

Geographic distribution
Antarctic krill has a circumpolar distribution, being found throughout the Southern Ocean, and as far north as the Antarctic Convergence. At the Antarctic Convergence, the cold Antarctic surface water submerges below the warmer subantarctic waters. This front runs roughly at 55° south; from there to the continent, the Southern Ocean covers 32 million square kilometres. This is 65 times the size of the North Sea. In the winter season, more than three-quarters of this area become covered by ice, whereas 24000000 km2 become ice free in summer. The water temperature fluctuates at -1.3 - 3 C.

The waters of the Southern Ocean form a system of currents. Whenever there is a West Wind Drift, the surface strata travels around Antarctica in an easterly direction. Near the continent, the East Wind Drift runs counterclockwise. At the front between both, large eddies develop, for example, in the Weddell Sea. The krill swarms swim with these water masses, to establish one single stock all around Antarctica, with gene exchange over the whole area. Currently, there is little knowledge of the precise migration patterns since individual krill cannot yet be tagged to track their movements. The largest shoals are visible from space and can be tracked by satellite. One swarm covered an area of 450 km2 of ocean, to a depth of 200 m and was estimated to contain over 2 million tons of krill. Recent research suggests that krill do not simply drift passively in these currents but actually modify them. By moving vertically through the ocean on a 12-hour cycle, the swarms play a major part in mixing deeper, nutrient-rich water with nutrient-poor water at the surface.

Ecology
Antarctic krill is the keystone species of the Antarctic ecosystem beyond the coastal shelf, and provides an important food source for whales, seals (such as leopard seals, fur seals, and crabeater seals), squid, icefish, penguins, albatrosses and many other species of birds. Crabeater seals have even developed special teeth as an adaptation to catch this abundant food source: its unusual multilobed teeth enable this species to sieve krill from the water. Its dentition looks like a perfect strainer, but how it operates in detail is still unknown. Crabeaters are the most abundant seal in the world; 98% of their diet is made up of  E. superba. These seals consume over 63 million tonnes of krill each year. Leopard seals have developed similar teeth (45% krill in diet). All seals consume 63–130 million tonnes, all whales 34–43 million tonnes, birds 15–20 million tonnes, squid 30–100 million tonnes, and fish 10–20 million tonnes, adding up to 152–313 million tonnes of krill consumption each year.

The size step between krill and its prey is unusually large: generally it takes three or four steps from the 20 μm small phytoplankton cells to a krill-sized organism (via small copepods, large copepods, mysids to 5 cm fish).

E. superba lives only in the Southern Ocean. In the North Atlantic, Meganyctiphanes norvegica and in the Pacific, Euphausia pacifica are the dominant species.

Biomass and production
The biomass of Antarctic krill was estimated in 2009 to be 0.05 gigatons of carbon (Gt C), similar to the total biomass of humans (0.06 Gt C). The reason Antarctic krill are able to build up such a high biomass and production is that the waters around the icy Antarctic continent harbour one of the largest plankton assemblages in the world, possibly the largest. The ocean is filled with phytoplankton; as the water rises from the depths to the light-flooded surface, it brings nutrients from all of the world's oceans back into the photic zone where they are once again available to living organisms.

Thus primary production—the conversion of sunlight into organic biomass, the foundation of the food chain—has an annual carbon fixation of 1–2 g/m2 in the open ocean. Close to the ice it can reach 30–50 g/m2. These values are not outstandingly high, compared to very productive areas like the North Sea or upwelling regions, but the area over which it takes place is enormous, even compared to other large primary producers such as rainforests. In addition, during the Austral summer there are many hours of daylight to fuel the process. All of these factors make the plankton and the krill a critical part of the planet's ecocycle.

Decline with shrinking pack ice
A possible decline in Antarctic krill biomass may have been caused by the reduction of the pack ice zone due to global warming. Antarctic krill, especially in the early stages of development, seem to need the pack ice structures in order to have a fair chance of survival. The pack ice provides natural cave-like features which the krill uses to evade their predators. In the years of low pack ice conditions the krill tend to give way to salps, a barrel-shaped free-floating filter feeder that also grazes on plankton.

Ocean acidification
Another challenge for Antarctic krill, as well as many calcifying organisms (corals, bivalve mussels, snails etc.), is the acidification of the oceans caused by increasing levels of carbon dioxide. Krill exoskeleton contains carbonate, which is susceptible to dissolution under low pH conditions. It has already been shown that increased carbon dioxide can disrupt the development of krill eggs and even prevent the juvenile krill from hatching, leading to future geographically widespread decreases in krill hatching success. The further effects of ocean acidification on the krill life cycle however remains unclear but scientists fear that it could significantly impact on its distribution, abundance and survival.

Fisheries
The fishery of Antarctic krill is on the order of 100,000 tonnes per year. The major catching nations are South Korea, Norway, Japan and Poland. The products are used as animal food and fish bait. Krill fisheries are difficult to operate in two important respects. First, a krill net needs to have very fine meshes, producing a very high drag, which generates a bow wave that deflects the krill to the sides. Second, fine meshes tend to clog very fast.

Yet another problem is bringing the krill catch on board. When the full net is hauled out of the water, the organisms compress each other, resulting in great loss of the krill's liquids. Experiments have been carried out to pump krill, while still in water, through a large tube on board. Special krill nets also are currently under development. The processing of the krill must be very rapid since the catch deteriorates within several hours. Its high protein and vitamin content makes krill quite suitable for both direct human consumption and the animal-feed industry.

Fishing and potentially overfishing krill is an issue of increasing concern.

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

title: Antarctic krill


text:
Trace (linear algebra)

In linear algebra, the trace of a square matrix $A$, denoted $tr(A)$, is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of $A$. The trace is only defined for a square matrix ($n × n$).

In mathematical physics texts, if $tr(A)$ = 0 then the matrix is said to be traceless. This is a minomer, but widely used, such as in the Pauli Matrices.

It can be proven that the trace of a matrix is the sum of its eigenvalues (counted with multiplicities). It can also be proven that $tr(AB) = tr(BA)$ for any two matrices $A$ and $B$ of appropriate sizes. This implies that similar matrices have the same trace. As a consequence one can define the trace of a linear operator mapping a finite-dimensional vector space into itself, since all matrices describing such an operator with respect to a basis are similar.

The trace is related to the derivative of the determinant (see Jacobi's formula).

Definition
The trace of an $n × n$ square matrix $A$ is defined as $$\operatorname{tr}(\mathbf{A}) = \sum_{i=1}^n a_{ii} = a_{11} + a_{22} + \dots + a_{nn}$$ where $a_{ii}$ denotes the entry on the $i$ th row and $i$ th column of $A$. The entries of $A$ can be real numbers, complex numbers, or more generally elements of a field $F$. The trace is not defined for non-square matrices.

Example
Let $A$ be a matrix, with $$\mathbf{A} = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} = \begin{pmatrix} 1 & 0 &  3 \\  11 &  5 &  2 \\   6 & 12 & -5 \end{pmatrix} $$

Then $$\operatorname{tr}(\mathbf{A}) = \sum_{i=1}^{3} a_{ii} = a_{11} + a_{22} + a_{33} = 1 + 5 + (-5) = 1$$

Basic properties
The trace is a linear mapping. That is, $$\begin{align} \operatorname{tr}(\mathbf{A} + \mathbf{B}) &= \operatorname{tr}(\mathbf{A}) + \operatorname{tr}(\mathbf{B}) \\ \operatorname{tr}(c\mathbf{A}) &= c \operatorname{tr}(\mathbf{A}) \end{align}$$ for all square matrices $A$ and $B$, and all scalars $c$.

A matrix and its transpose have the same trace: $$\operatorname{tr}(\mathbf{A}) = \operatorname{tr}\left(\mathbf{A}^\mathsf{T}\right).$$

This follows immediately from the fact that transposing a square matrix does not affect elements along the main diagonal.

Trace of a product
The trace of a square matrix which is the product of two matrices can be rewritten as the sum of entry-wise products of their elements, i.e. as the sum of all elements of their Hadamard product. Phrased directly, if $A$ and $B$ are two $m × n$ matrices, then: $$ \operatorname{tr}\left(\mathbf{A}^\mathsf{T}\mathbf{B}\right) = \operatorname{tr}\left(\mathbf{A}\mathbf{B}^\mathsf{T}\right) = \operatorname{tr}\left(\mathbf{B}^\mathsf{T}\mathbf{A}\right) = \operatorname{tr}\left(\mathbf{B}\mathbf{A}^\mathsf{T}\right) = \sum_{i=1}^m \sum_{j=1}^n a_{ij}b_{ij} \;. $$

If one views any real $m × n$ matrix as a vector of length $mn$ (an operation called vectorization) then the above operation on $A$ and $B$ coincides with the standard dot product. According to the above expression, $tr(A^{⊤}A)$ is a sum of squares and hence is nonnegative, equal to zero if and only if $A$ is zero. Furthermore, as noted in the above formula, $tr(A^{⊤}B) = tr(B^{⊤}A)$. These demonstrate the positive-definiteness and symmetry required of an inner product; it is common to call $tr(A^{⊤}B)$ the Frobenius inner product of $A$ and $B$. This is a natural inner product on the vector space of all real matrices of fixed dimensions. The norm derived from this inner product is called the Frobenius norm, and it satisfies a submultiplicative property, as can be proven with the Cauchy–Schwarz inequality: $$0 \leq \left[\operatorname{tr}(\mathbf{A} \mathbf{B})\right]^2 \leq \operatorname{tr}\left(\mathbf{A}^2\right) \operatorname{tr}\left(\mathbf{B}^2\right) \leq \left[\operatorname{tr}(\mathbf{A})\right]^2 \left[\operatorname{tr}(\mathbf{B})\right]^2 \ ,$$ if $A$ and $B$ are real positive semi-definite matrices of the same size. The Frobenius inner product and norm arise frequently in matrix calculus and statistics.

The Frobenius inner product may be extended to a hermitian inner product on the complex vector space of all complex matrices of a fixed size, by replacing $B$ by its complex conjugate.

The symmetry of the Frobenius inner product may be phrased more directly as follows: the matrices in the trace of a product can be switched without changing the result. If $A$ and $B$ are $m × n$ and $n × m$ real or complex matrices, respectively, then

This is notable both for the fact that $AB$ does not usually equal $BA$, and also since the trace of either does not usually equal $tr(A)tr(B)$. The similarity-invariance of the trace, meaning that $tr(AB) = 1 ≠ 0 ⋅ 0 = tr(A)tr(B)$ for any square matrix $tr(A) = tr(P^{−1}AP)$ and any invertible matrix $A$ of the same dimensions, is a fundamental consequence. This is proved by $$ \operatorname{tr}\left(\mathbf{P}^{-1}(\mathbf{A}\mathbf{P})\right) = \operatorname{tr}\left((\mathbf{A} \mathbf{P})\mathbf{P}^{-1}\right) = \operatorname{tr}(\mathbf{A}). $$ Similarity invariance is the crucial property of the trace in order to discuss traces of linear transformations as below.

Additionally, for real column vectors $$\mathbf{a}\in\mathbb{R}^n$$ and $$\mathbf{b}\in\mathbb{R}^n$$, the trace of the outer product is equivalent to the inner product:

Cyclic property
More generally, the trace is invariant under circular shifts, that is,

This is known as the cyclic property.

Arbitrary permutations are not allowed: in general, $$\operatorname{tr}(\mathbf{A}\mathbf{B}\mathbf{C}) \ne \operatorname{tr}(\mathbf{A}\mathbf{C}\mathbf{B}).$$

However, if products of three symmetric matrices are considered, any permutation is allowed, since: $$\operatorname{tr}(\mathbf{A}\mathbf{B}\mathbf{C}) = \operatorname{tr}\left(\left(\mathbf{A}\mathbf{B}\mathbf{C}\right)^{\mathsf T}\right) = \operatorname{tr}(\mathbf{C}\mathbf{B}\mathbf{A}) = \operatorname{tr}(\mathbf{A}\mathbf{C}\mathbf{B}),$$ where the first equality is because the traces of a matrix and its transpose are equal. Note that this is not true in general for more than three factors.

Trace of a Kronecker product
The trace of the Kronecker product of two matrices is the product of their traces: $$\operatorname{tr}(\mathbf{A} \otimes \mathbf{B}) = \operatorname{tr}(\mathbf{A})\operatorname{tr}(\mathbf{B}).$$

Characterization of the trace
The following three properties: $$\begin{align} \operatorname{tr}(\mathbf{A} + \mathbf{B}) &= \operatorname{tr}(\mathbf{A}) + \operatorname{tr}(\mathbf{B}), \\ \operatorname{tr}(c\mathbf{A}) &= c \operatorname{tr}(\mathbf{A}), \\ \operatorname{tr}(\mathbf{A}\mathbf{B}) &= \operatorname{tr}(\mathbf{B}\mathbf{A}), \end{align}$$ characterize the trace up to a scalar multiple in the following sense: If $$f$$ is a linear functional on the space of square matrices that satisfies $$f(xy) = f(yx),$$ then $$f$$ and $$\operatorname{tr}$$ are proportional.

For $$n\times n$$ matrices, imposing the normalization $$f(\mathbf{I}) = n$$ makes $$f$$ equal to the trace.

Trace as the sum of eigenvalues
Given any $P$ matrix $n × n$, there is

where $A$ are the eigenvalues of $λ_{1}, ..., λ_{n}$ counted with multiplicity. This holds true even if $A$ is a real matrix and some (or all) of the eigenvalues are complex numbers. This may be regarded as a consequence of the existence of the Jordan canonical form, together with the similarity-invariance of the trace discussed above.

Trace of commutator
When both $A$ and $A$ are $B$ matrices, the trace of the (ring-theoretic) commutator of $n × n$ and $A$ vanishes: $B$, because $tr([A, B]) = 0$ and $tr(AB) = tr(BA)$ is linear. One can state this as "the trace is a map of Lie algebras $tr$ from operators to scalars", as the commutator of scalars is trivial (it is an Abelian Lie algebra). In particular, using similarity invariance, it follows that the identity matrix is never similar to the commutator of any pair of matrices.

Conversely, any square matrix with zero trace is a linear combination of the commutators of pairs of matrices. Moreover, any square matrix with zero trace is unitarily equivalent to a square matrix with diagonal consisting of all zeros.

Traces of special kinds of matrices
• The trace of the $gl_{n} → k$ identity matrix is the dimension of the space, namely $n$. $\operatorname{tr}\left(\mathbf{I}_n\right) = n$ This leads to generalizations of dimension using trace.

• The trace of a Hermitian matrix is real, because the elements on the diagonal are real.

• The trace of a permutation matrix is the number of fixed points of the corresponding permutation, because the diagonal term $n × n$ is 1 if the $a_{ii}$th point is fixed and 0 otherwise.

• The trace of a projection matrix is the dimension of the target space. $\begin{align} \mathbf{P}_\mathbf{X} &= \mathbf{X}\left(\mathbf{X}^\mathsf{T} \mathbf{X}\right)^{-1} \mathbf{X}^\mathsf{T} \\[3pt] \Longrightarrow \operatorname{tr}\left(\mathbf{P}_\mathbf{X}\right) &= \operatorname{rank}(\mathbf{X}). \end{align}$ The matrix $i$ is idempotent.

• More generally, the trace of any idempotent matrix, i.e. one with $P_{X}$, equals its own rank.

• The trace of a nilpotent matrix is zero.

When the characteristic of the base field is zero, the converse also holds: if $A^{2} = A$ for all $k$, then $tr(A^{k}) = 0$ is nilpotent.

When the characteristic $A$ is positive, the identity in $n$ dimensions is a counterexample, as $\operatorname{tr}\left(\mathbf{I}_n^k\right) = \operatorname{tr}\left(\mathbf{I}_n\right) = n \equiv 0$, but the identity is not nilpotent.

Relationship to the characteristic polynomial
The trace of an $$n \times n$$ matrix $$A$$ is the coefficient of $$t^{n-1}$$ in the characteristic polynomial, possibly changed of sign, according to the convention in the definition of the characteristic polynomial.

Relationship to eigenvalues
If $n > 0$ is a linear operator represented by a square matrix with real or complex entries and if $A$ are the eigenvalues of $λ_{1}, ..., λ_{n}$ (listed according to their algebraic multiplicities), then

This follows from the fact that $A$ is always similar to its Jordan form, an upper triangular matrix having $A$ on the main diagonal. In contrast, the determinant of $λ_{1}, ..., λ_{n}$ is the product of its eigenvalues; that is, $$\det(\mathbf{A}) = \prod_i \lambda_i.$$

Everything in the present section applies as well to any square matrix with coefficients in an algebraically closed field.

Derivative relationships
If $A$ is a square matrix with small entries and $ΔA$ denotes the identity matrix, then we have approximately

$$\det(\mathbf{I}+\mathbf{\Delta A})\approx 1 + \operatorname{tr}(\mathbf{\Delta A}).$$

Precisely this means that the trace is the derivative of the determinant function at the identity matrix. Jacobi's formula

$$d\det(\mathbf{A}) = \operatorname{tr} \big(\operatorname{adj}(\mathbf{A})\cdot d\mathbf{A}\big)$$

is more general and describes the differential of the determinant at an arbitrary square matrix, in terms of the trace and the adjugate of the matrix.

From this (or from the connection between the trace and the eigenvalues), one can derive a relation between the trace function, the matrix exponential function, and the determinant:$$\det(\exp(\mathbf{A})) = \exp(\operatorname{tr}(\mathbf{A})).$$

A related characterization of the trace applies to linear vector fields. Given a matrix $I$, define a vector field $A$ on $F$ by $R^{n}$. The components of this vector field are linear functions (given by the rows of $F(x) = Ax$). Its divergence $A$ is a constant function, whose value is equal to $div F$.

By the divergence theorem, one can interpret this in terms of flows: if $tr(A)$ represents the velocity of a fluid at location $F(x)$ and $U$ is a region in $x$, the net flow of the fluid out of $U$ is given by $R^{n}$, where $tr(A) · vol(U)$ is the volume of $U$.

The trace is a linear operator, hence it commutes with the derivative: $$d \operatorname{tr} (\mathbf{X}) = \operatorname{tr}(d\mathbf{X}) .$$

Trace of a linear operator
In general, given some linear map $vol(U)$ (where $V$ is a finite-dimensional vector space), we can define the trace of this map by considering the trace of a matrix representation of $f$, that is, choosing a basis for $V$ and describing $f$ as a matrix relative to this basis, and taking the trace of this square matrix. The result will not depend on the basis chosen, since different bases will give rise to similar matrices, allowing for the possibility of a basis-independent definition for the trace of a linear map.

Such a definition can be given using the canonical isomorphism between the space $f : V → V$ of linear maps on $V$ and $End(V)$, where $V ⊗ V*$ is the dual space of $V$. Let $v$ be in $V$ and let $g$ be in $V*$. Then the trace of the indecomposable element $V*$ is defined to be $v ⊗ g$; the trace of a general element is defined by linearity. The trace of a linear map $g(v)$ can then be defined as the trace, in the above sense, of the element of $f : V → V$ corresponding to f under the above mentioned canonical isomorphism. Using an explicit basis for $V$ and the corresponding dual basis for $V ⊗ V*$, one can show that this gives the same definition of the trace as given above.

Stochastic estimator
The trace can be estimated unbiasedly by "Hutchinson's trick": "Given any matrix $W\in \R^{n\times n}$, and any random $u\in \R^n$ with $E[uu^T] = I$, we have $E[u^T W u] = tr(W)$. (Proof: expand the expectation directly.)"Usually, the random vector is sampled from $$N(0, I)$$ (normal distribution) or $$\{\pm n^{-1/2}\}^n$$ (Rademacher distribution).

More sophisticated stochastic estimators of trace have been developed.

Applications
If a 2 x 2 real matrix has zero trace, its square is a diagonal matrix.

The trace of a 2 × 2 complex matrix is used to classify Möbius transformations. First, the matrix is normalized to make its determinant equal to one. Then, if the square of the trace is 4, the corresponding transformation is parabolic. If the square is in the interval [0,4), it is elliptic. Finally, if the square is greater than 4, the transformation is loxodromic. See classification of Möbius transformations.

The trace is used to define characters of group representations. Two representations $V*$ of a group $G$ are equivalent (up to change of basis on $V$) if $A, B : G → GL(V)$ for all $tr(A(g)) = tr(B(g))$.

The trace also plays a central role in the distribution of quadratic forms.

Lie algebra
The trace is a map of Lie algebras $$\operatorname{tr}:\mathfrak{gl}_n\to K$$ from the Lie algebra $$\mathfrak{gl}_n$$ of linear operators on an $n$-dimensional space ($g ∈ G$ matrices with entries in $$K$$) to the Lie algebra $K$ of scalars; as $K$ is Abelian (the Lie bracket vanishes), the fact that this is a map of Lie algebras is exactly the statement that the trace of a bracket vanishes: $$\operatorname{tr}([\mathbf{A}, \mathbf{B}]) = 0 \text{ for each }\mathbf A,\mathbf B\in\mathfrak{gl}_n.$$

The kernel of this map, a matrix whose trace is zero, is often said to be ' or ', and these matrices form the simple Lie algebra $$\mathfrak{sl}_n$$, which is the Lie algebra of the special linear group of matrices with determinant 1. The special linear group consists of the matrices which do not change volume, while the special linear Lie algebra is the matrices which do not alter volume of infinitesimal sets.

In fact, there is an internal direct sum decomposition $$\mathfrak{gl}_n = \mathfrak{sl}_n \oplus K$$ of operators/matrices into traceless operators/matrices and scalars operators/matrices. The projection map onto scalar operators can be expressed in terms of the trace, concretely as: $$\mathbf{A} \mapsto \frac{1}{n}\operatorname{tr}(\mathbf{A})\mathbf{I}.$$

Formally, one can compose the trace (the counit map) with the unit map $$K\to\mathfrak{gl}_n$$ of "inclusion of scalars" to obtain a map $$\mathfrak{gl}_n\to\mathfrak{gl}_n$$ mapping onto scalars, and multiplying by $n$. Dividing by $n$ makes this a projection, yielding the formula above.

In terms of short exact sequences, one has $$0 \to \mathfrak{sl}_n \to \mathfrak{gl}_n \overset{\operatorname{tr}}{\to} K \to 0$$ which is analogous to $$1 \to \operatorname{SL}_n \to \operatorname{GL}_n \overset{\det}{\to} K^* \to 1$$ (where $$K^*=K\setminus\{0\}$$) for Lie groups. However, the trace splits naturally (via $$1/n$$ times scalars) so $$\mathfrak{gl}_n=\mathfrak{sl}_n\oplus K$$, but the splitting of the determinant would be as the $n$th root times scalars, and this does not in general define a function, so the determinant does not split and the general linear group does not decompose: $$\operatorname{GL}_n \neq \operatorname{SL}_n \times K^*.$$

Bilinear forms
The bilinear form (where $n × n$, $X$ are square matrices) $$B(\mathbf{X}, \mathbf{Y}) = \operatorname{tr}(\operatorname{ad}(\mathbf{X})\operatorname{ad}(\mathbf{Y}))\quad \text{where } \operatorname{ad}(\mathbf{X})\mathbf{Y} = [\mathbf{X}, \mathbf{Y}] = \mathbf{X}\mathbf{Y} - \mathbf{Y}\mathbf{X}$$ is called the Killing form, which is used for the classification of Lie algebras.

The trace defines a bilinear form: $$(\mathbf{X}, \mathbf{Y}) \mapsto \operatorname{tr}(\mathbf{X}\mathbf{Y}).$$

The form is symmetric, non-degenerate and associative in the sense that: $$\operatorname{tr}(\mathbf{X}[\mathbf{Y}, \mathbf{Z}]) = \operatorname{tr}([\mathbf{X}, \mathbf{Y}]\mathbf{Z}).$$

For a complex simple Lie algebra (such as $Y$), every such bilinear form is proportional to each other; in particular, to the Killing form.

Two matrices $tr(A*A) = 0$ and $A = 0$ are said to be trace orthogonal if $$\operatorname{tr}(\mathbf{X}\mathbf{Y}) = 0.$$

There is a generalization to a general representation $$(\rho,\mathfrak{g},V)$$ of a Lie algebra $$\mathfrak{g}$$, such that $$\rho$$ is a homomorphism of Lie algebras $$\rho: \mathfrak{g} \rightarrow \text{End}(V).$$ The trace form $$\text{tr}_V$$ on $$\text{End}(V)$$ is defined as above. The bilinear form $$\phi(\mathbf{X},\mathbf{Y}) = \text{tr}_V(\rho(\mathbf{X})\rho(\mathbf{Y}))$$ is symmetric and invariant due to cyclicity.

Generalizations
The concept of trace of a matrix is generalized to the trace class of compact operators on Hilbert spaces, and the analog of the Frobenius norm is called the Hilbert–Schmidt norm.

If $K$ is a trace-class operator, then for any orthonormal basis $$(e_n)_n$$, the trace is given by $$\operatorname{tr}(K) = \sum_n \left\langle e_n, Ke_n \right\rangle,$$ and is finite and independent of the orthonormal basis.

The partial trace is another generalization of the trace that is operator-valued. The trace of a linear operator $Z$ which lives on a product space $$\mathfrak{sl}$_{n}$ is equal to the partial traces over $A$ and $B$: $$\operatorname{tr}(Z) = \operatorname{tr}_A \left(\operatorname{tr}_B(Z)\right) = \operatorname{tr}_B \left(\operatorname{tr}_A(Z)\right).$$

For more properties and a generalization of the partial trace, see traced monoidal categories.

If $A$ is a general associative algebra over a field $k$, then a trace on $A$ is often defined to be any map $X$ which vanishes on commutators; $Y$ for all $A ⊗ B$. Such a trace is not uniquely defined; it can always at least be modified by multiplication by a nonzero scalar.

A supertrace is the generalization of a trace to the setting of superalgebras.

The operation of tensor contraction generalizes the trace to arbitrary tensors.

Traces in the language of tensor products
Given a vector space $V$, there is a natural bilinear map $tr : A ↦ k$ given by sending $tr([a,b]) = 0$ to the scalar $a, b ∈ A$. The universal property of the tensor product $V × V^{∗} → F$ automatically implies that this bilinear map is induced by a linear functional on $(v, φ)$.

Similarly, there is a natural bilinear map $φ(v)$ given by sending $V ⊗ V^{∗}$ to the linear map $V ⊗ V^{∗}$. The universal property of the tensor product, just as used previously, says that this bilinear map is induced by a linear map $V × V^{∗} → Hom(V, V)$. If $V$ is finite-dimensional, then this linear map is a linear isomorphism. This fundamental fact is a straightforward consequence of the existence of a (finite) basis of $V$, and can also be phrased as saying that any linear map $(v, φ)$ can be written as the sum of (finitely many) rank-one linear maps. Composing the inverse of the isomorphism with the linear functional obtained above results in a linear functional on $w ↦ φ(w)v$. This linear functional is exactly the same as the trace.

Using the definition of trace as the sum of diagonal elements, the matrix formula $V ⊗ V^{∗} → Hom(V, V)$ is straightforward to prove, and was given above. In the present perspective, one is considering linear maps $S$ and $T$, and viewing them as sums of rank-one maps, so that there are linear functionals $V → V$ and $Hom(V, V)$ and nonzero vectors $tr(AB) = tr(BA)$ and $φ_{i}$ such that $ψ_{j}$ and $v_{i}$ for any $u$ in $V$. Then
 * $$(S\circ T)(u)=\sum_i\varphi_i\left(\sum_j\psi_j(u)w_j\right)v_i=\sum_i\sum_j\psi_j(u)\varphi_i(w_j)v_i $$

for any $u$ in $V$. The rank-one linear map $w_{j}$ has trace $S(u) = Σφ_{i}(u)v_{i}$ and so
 * $$\operatorname{tr}(S\circ T)=\sum_i\sum_j\psi_j(v_i)\varphi_i(w_j)=\sum_j\sum_i\varphi_i(w_j)\psi_j(v_i).$$

Following the same procedure with $S$ and $T$ reversed, one finds exactly the same formula, proving that $T(u) = Σψ_{j}(u)w_{j}$ equals $u ↦ ψ_{j}(u)φ_{i}(w_{j})v_{i}$.

The above proof can be regarded as being based upon tensor products, given that the fundamental identity of $ψ_{j}(v_{i})φ_{i}(w_{j})$ with $tr(S ∘ T)$ is equivalent to the expressibility of any linear map as the sum of rank-one linear maps. As such, the proof may be written in the notation of tensor products. Then one may consider the multilinear map $tr(T ∘ S)$ given by sending $End(V)$ to $V ⊗ V^{∗}$. Further composition with the trace map then results in $V × V^{∗} × V × V^{∗} → V ⊗ V^{∗}$, and this is unchanged if one were to have started with $(v, φ, w, ψ)$ instead. One may also consider the bilinear map $φ(w)v ⊗ ψ$ given by sending $φ(w)ψ(v)$ to the composition $(w, ψ, v, φ)$, which is then induced by a linear map $End(V) × End(V) → End(V)$. It can be seen that this coincides with the linear map $(f, g)$. The established symmetry upon composition with the trace map then establishes the equality of the two traces.

For any finite dimensional vector space $V$, there is a natural linear map $f ∘ g$; in the language of linear maps, it assigns to a scalar $c$ the linear map $End(V) ⊗ End(V) → End(V)$. Sometimes this is called coevaluation map, and the trace $V ⊗ V^{∗} ⊗ V ⊗ V^{∗} → V ⊗ V^{∗}$ is called evaluation map. These structures can be axiomatized to define categorical traces in the abstract setting of category theory.

title: Trace (linear algebra)