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Inertial confinement fusion

Inertial confinement fusion (ICF) is a fusion energy process that initiates nuclear fusion reactions by compressing and heating targets filled with fuel. The targets are small pellets, typically containing deuterium (2H) and tritium (3H).

Energy is deposited in the target's outer layer, which explodes outward. This produces a reaction force in the form of shock waves that travel through the target. The waves compress and heat it. Sufficiently powerful shock waves generate fusion.

ICF is one of two major branches of fusion energy research; the other is magnetic confinement fusion (MCF). When first proposed in the early 1970s, ICF appeared to be a practical approach to power production and the field flourished. Experiments demonstrated that the efficiency of these devices was much lower than expected. Throughout the 1980s and '90s, experiments were conducted in order to understand the interaction of high-intensity laser light and plasma. These led to the design of much larger machines that achieved ignition-generating energies.

The largest operational ICF experiment is the National Ignition Facility (NIF) in the US. In 2022, the NIF produced fusion, delivering 2.05 megajoules (MJ) of energy to the target which produced 3.15 MJ, the first time that an ICF device produced more energy than was delivered to the target.

Fusion basics
Fusion reactions combine smaller atoms to form larger ones. This occurs when two atoms (or ions, atoms stripped of their electrons) come close enough to each other that the nuclear force dominates the electrostatic force that otherwise keeps them apart. Overcoming electrostatic repulsion requires kinetic energy sufficient to overcome the Coulomb barrier or fusion barrier.

Less energy is needed to cause lighter nuclei to fuse, as they have less electrical charge and thus a lower barrier energy. Thus the barrier is lowest for hydrogen. Conversely, the nuclear force increases with the number of nucleons, so isotopes of hydrogen that contain additional neutrons reduce the required energy. The easiest fuel is a mixture of 2H, and 3H, known as D-T.

The odds of fusion occurring are a function of the fuel density and temperature and the length of time that the density and temperature are maintained. Even under ideal conditions, the chance that a D and T pair fuse is very small. Higher density and longer times allow more encounters among the atoms. This cross section is further dependent on individual ion energies. This combination, the fusion triple product, must reach the Lawson criterion, to reach ignition.

Thermonuclear devices
The first ICF devices were the hydrogen bombs invented in the early 1950s. A hydrogen bomb consists of two bombs in a single case. The first, the primary stage, is a fission-powered device normally using plutonium. When it explodes it gives off a burst of thermal X-rays that fill the interior of the specially designed bomb casing. These X-rays are absorbed by a special material surrounding the secondary stage, which consists mostly of the fusion fuel. The X-rays heat this material and cause it to explode. Due to Newton's Third Law, this causes the fuel inside to be driven inward, compressing and heating it. This causes the fusion fuel to reach the temperature and density where fusion reactions begin.

In the case of D-T fuel, most of the energy is released in the form of alpha particles and neutrons. Under normal conditions, an alpha can travel about 10 mm through the fuel, but in the ultra-dense conditions in the compressed fuel, they can travel about 0.01 mm before their electrical charge, interacting with the surrounding plasma, causes them to lose velocity. This means the majority of the energy released by the alphas is redeposited in the fuel. This transfer of kinetic energy heats the surrounding particles to the energies they need to undergo fusion. This process causes the fusion fuel to burn outward from the center. The electrically neutral neutrons travel longer distances in the fuel mass and do not contribute to this self-heating process. In a bomb, they are instead used to either breed tritium through reactions in a lithium-deuteride fuel, or are used to split additional fissionable fuel surrounding the secondary stage, often part of the bomb casing.

The requirement that the reaction has to be sparked by a fission bomb makes this method impractical for power generation. Not only would the fission triggers be expensive to produce, but the minimum size of such a bomb is large, defined roughly by the critical mass of the plutonium fuel used. Generally, it seems difficult to build efficient nuclear fusion devices much smaller than about 1 kiloton in yield, and the fusion secondary would add to this yield. This makes it a difficult engineering problem to extract power from the resulting explosions. Project PACER studied solutions to the engineering issues, but also demonstrated that it was not economically feasible. The cost of the bombs was far greater than the value of the resulting electricity.

Mechanism of action
The energy needed to overcome the Coulomb barrier corresponds to the energy of the average particle in a gas heated to 100 million K. The specific heat of hydrogen is about 14 Joule per gram-K, so considering a 1 milligram fuel pellet, the energy needed to raise the mass as a whole to this temperature is 1.4 megajoules (MJ).

In the more widely developed magnetic fusion energy (MFE) approach, confinement times are on the order of one second. However, plasmas can be sustained for minutes. In this case the confinement time represents the amount of time it takes for the energy from the reaction to be lost to the environment - through a variety of mechanisms. For a one second confinement, the density needed to meet the Lawson criterion is about 1014 particles per cubic centimetre (cc). For comparison, air at sea level has about 2.7 x 1019 particles/cc, so the MFE approach has been described as "a good vacuum".

Considering a 1 milligram drop of D-T fuel in liquid form, the size is about 1 mm and the density is about 4 x 1020/cc. Nothing holds the fuel together. Heat created by fusion events causes it to expand at the speed of sound, which leads to a confinement time around 2 x 10−10 seconds. At liquid density the required confinement time is about 2 x 10−7s. In this case only about 0.1 percent of the fuel fuses before the drop blows apart.

The rate of fusion reactions is a function of density, and density can be improved through compression. If the drop is compressed from 1 mm to 0.1 mm in diameter, the confinement time drops by the same factor of 10, because the particles have less distance to travel before they escape. However, the density, which is the cube of the dimensions, increases by 1,000 times. This means the overall rate of fusion increases 1,000 times while the confinement drops by 10 times, a 100-fold improvement. In this case 10% of the fuel undergoes fusion; 10% of 1 mg of fuel produces about 30 MJ of energy, 30 times the amount needed to compress it to that density.

The other key concept in ICF is that the entire fuel mass does not have to be raised to 100 million K. In a fusion bomb the reaction continues because the alpha particles released in the interior heat the fuel around it. At liquid density the alphas travel about 10 mm and thus their energy escapes the fuel. In the 0.1 mm compressed fuel, the alphas have a range of about 0.016 mm, meaning that they will stop within the fuel and heat it. In this case a "propagating burn" can be caused by heating only the center of the fuel to the needed temperature. This requires far less energy; calculations suggested 1 kJ is enough to reach the compression goal.

Some method is needed to heat the interior to fusion temperatures, and do so while when the fuel is compressed and the density is high enough. In modern ICF devices, the density of the compressed fuel mixture is as much as one-thousand times the density of water, or one-hundred times that of lead, around 1000 g/cm3. Much of the work since the 1970s has been on ways to create the central hot-spot that starts off the burning, and dealing with the many practical problems in reaching the desired density.

[[Image:Inertial confinement fusion.svg|thumb|center|600px|Schematic of the stages of inertial confinement fusion using lasers. The blue arrows represent radiation; orange is blowoff; yellow is inwardly transported thermal energy. 1. Laser beams or laser-produced X-rays rapidly heat the surface of the fusion target, forming a surrounding plasma envelope.

2. Fuel is compressed by the rocket-like blowoff of the hot surface material.

3. During the final part of the capsule implosion, the fuel core reaches 20 times the density of lead and ignites at 100,000,000 ˚C.

4. Thermonuclear burn spreads rapidly through the compressed fuel, yielding many times the input energy.]]

Heating concepts
Early calculations suggested that the amount of energy needed to ignite the fuel was very small, but this does not match subsequent experience.

Hot spot ignition

The initial solution to the heating problem involved deliberate "shaping" of the energy delivery. The idea was to use an initial lower-energy pulse to vaporize the capsule and cause compression, and then a very short, very powerful pulse near the end of the compression cycle. The goal is to launch shock waves into the compressed fuel that travel inward to the center. When they reach the center they meet the waves coming in from other sides. This causes a brief period where the density in the center reaches much higher values, over 800 g/cm3.

The central hot spot ignition concept was the first to suggest ICF was not only a practical route to fusion, but relatively simple. This led to numerous efforts to build working systems in the early 1970s. These experiments revealed unexpected loss mechanisms. Early calculations suggested about 4.5x107 J/g would be needed, but modern calculations place it closer to 108 J/g. Greater understanding led to complex shaping of the pulse into multiple time intervals.

Fast ignition
The fast ignition approach employs a separate laser to supply additional energy directly to the center of the fuel. This can be done mechanically, often using a small metal cone to puncture the outer fuel pellet wall to inject the energy into the center. In tests, this approach failed because the laser pulse had to reach the center at a precise moment, while the center is obscured by debris and free electrons from the compression pulse. It also has the disadvantage of requiring a second laser pulse, which generally involves a completely separate laser.

Shock ignition
Shock ignition is similar in concept to the hot-spot technique, but instead of achieving ignition via compression heating, a powerful shock wave is sent into the fuel at a later time through a combination of compression and shock heating. This increases the efficiency of the process while lowering the overall amount of power required.

Direct vs. indirect drive
In the simplest method of inertial confinement, the fuel is arranged as a sphere. This allows it to be compressed uniformly from all sides. To produce the inward force, the fuel is placed within a thin capsule that absorbs energy from the driver beams, causing the capsule shell to explode outward. The capsule shell is usually made of a lightweight plastic, and the fuel is deposited as a layer on the inside by injecting and freezing the gaseous fuel into the shell.

Shining the driver beams directly onto the fuel capsule is known as "direct drive". The implosion process must be extremely uniform in order to avoid asymmetry due to Rayleigh–Taylor instability and similar effects. For a beam energy of 1 MJ, the fuel capsule cannot be larger than about 2 mm before these effects disrupt the implosion symmetry. This limits the size of the laser beams to a diameter so narrow that it is difficult to achieve in practice.

Alternatively "indirect drive" illuminates a small cylinder of heavy metal, often gold or lead, known as a hohlraum. The beam energy heats the hohlraum until it emits X-rays. These X-rays fill the interior of the hohlraum and heat the capsule. The advantage of indirect drive is that the beams can be larger and less accurate. The disadvantage is that much of the delivered energy is used to heat the hohlraum until it is "X-ray hot", so the end-to-end energy efficiency is much lower than the direct drive method.

Challenges
The primary challenges with increasing ICF performance are:
 * Improving the energy delivered to the target
 * Controlling symmetry of the imploding fuel
 * Delaying fuel heating until sufficient density is achieved
 * Preventing premature mixing of hot and cool fuel by hydrodynamic instabilities
 * Achieving shockwave convergence at the fuel center

In order to focus the shock wave on the center of the target, the target must be made with great precision and sphericity with tolerances of no more than a few micrometres over its (inner and outer) surface. The lasers must be precisely targeted in space and time. Beam timing is relatively simple and is solved by using delay lines in the beams' optical path to achieve picosecond accuracy. The other major issue is so-called "beam-beam" imbalance and beam anisotropy. These problems are, respectively, where the energy delivered by one beam may be higher or lower than other beams impinging and of "hot spots" within a beam diameter hitting a target which induces uneven compression on the target surface, thereby forming Rayleigh-Taylor instabilities in the fuel, prematurely mixing it and reducing heating efficacy at the instant of maximum compression. The Richtmyer-Meshkov instability is also formed during the process due to shock waves.

These problems have been mitigated by beam smoothing techniques and beam energy diagnostics; however, RT instability remains a major issue. Modern cryogenic hydrogen ice targets tend to freeze a thin layer of deuterium on the inside of the shell while irradiating it with a low power infrared laser to smooth its inner surface and monitoring it with a microscope equipped camera, thereby allowing the layer to be closely monitored. Cryogenic targets filled with D-T are "self-smoothing" due to the small amount of heat created by tritium decay. This is referred to as "beta-layering".

In the indirect drive approach, the absorption of thermal x-rays by the target is more efficient than the direct absorption of laser light. However, the hohlraums take up considerable energy to heat, significantly reducing energy transfer efficiency. Most often, indirect drive hohlraum targets are used to simulate thermonuclear weapons tests due to the fact that the fusion fuel in weapons is also imploded mainly by X-ray radiation.

ICF drivers are evolving. Lasers have scaled up from a few joules and kilowatts to megajoules and hundreds of terawatts, using mostly frequency doubled or tripled light from neodymium glass amplifiers.

Heavy ion beams are particularly interesting for commercial generation, as they are easy to create, control, and focus. However, it is difficult to achieve the energy densities required to implode a target efficiently, and most ion-beam systems require the use of a hohlraum surrounding the target to smooth out the irradiation.

United States
ICF history began as part of the "Atoms For Peace" conference in 1957. This was an international, UN-sponsored conference between the US and the Soviet Union. Some thought was given to using a hydrogen bomb to heat a water-filled cavern. The resulting steam could then be used to power conventional generators, and thereby provide electrical power.

This meeting led to Operation Plowshare, formed in June 1957 and formally named in 1961. It included three primary concepts; energy generation under Project PACER, the use of nuclear explosions for excavation, and for fracking in the natural gas industry. PACER was directly tested in December 1961 when the 3 kt Project Gnome device was detonated in bedded salt in New Mexico. While the press looked on, radioactive steam was released from the drill shaft, at some distance from the test site. Further studies designed engineered cavities to replace natural ones, but Plowshare turned from bad to worse, especially after the failure of 1962's Sedan which produced significant fallout. PACER continued to receive funding until 1975, when a 3rd party study demonstrated that the cost of electricity from PACER would be ten times the cost of conventional nuclear plants.

Another outcome of Atoms For Peace was to prompt John Nuckolls to consider what happens on the fusion side of the bomb as fuel mass is reduced. This work suggested that at sizes on the order of milligrams, little energy would be needed to ignite the fuel, much less than a fission primary. He proposed building, in effect, tiny all-fusion explosives using a tiny drop of D-T fuel suspended in the center of a hohlraum. The shell provided the same effect as the bomb casing in an H-bomb, trapping x-rays inside to irradiate the fuel. The main difference is that the X-rays would be supplied by an external device that heated the shell from the outside until it was glowing in the x-ray region. The power would be delivered by a then-unidentified pulsed power source he referred to, using bomb terminology, as the "primary".

The main advantage to this scheme is the fusion efficiency at high densities. According to the Lawson criterion, the amount of energy needed to heat the D-T fuel to break-even conditions at ambient pressure is perhaps 100 times greater than the energy needed to compress it to a pressure that would deliver the same rate of fusion. So, in theory, the ICF approach could offer dramatically more gain. This can be understood by considering the energy losses in a conventional scenario where the fuel is slowly heated, as in the case of magnetic fusion energy; the rate of energy loss to the environment is based on the temperature difference between the fuel and its surroundings, which continues to increase as the fuel temperature increases. In the ICF case, the entire hohlraum is filled with high-temperature radiation, limiting losses.

Germany
In 1956 a meeting was organized at the Max Planck Institute in Germany by fusion pioneer Carl Friedrich von Weizsäcker. At this meeting Friedwardt Winterberg proposed the non-fission ignition of a thermonuclear micro-explosion by a convergent shock wave driven with high explosives. Further reference to Winterberg's work in Germany on nuclear micro explosions (mininukes) is contained in a declassified report of the former East German Stasi (Staatsicherheitsdienst).

In 1964 Winterberg proposed that ignition could be achieved by an intense beam of microparticles accelerated to a velocity of 1000 km/s. In 1968, he proposed to use intense electron and ion beams generated by Marx generators for the same purpose. The advantage of this proposal is that charged particle beams are not only less expensive than laser beams, but can entrap the charged fusion reaction products due to the strong self-magnetic beam field, drastically reducing the compression requirements for beam ignited cylindrical targets.

USSR
In 1967, research fellow Gurgen Askaryan published an article proposing the use of focused laser beams in the fusion of lithium deuteride or deuterium.

Early research
Through the late 1950s,  and collaborators at Lawrence Livermore National Laboratory (LLNL) completed computer simulations of the ICF concept. In early 1960, they performed a full simulation of the implosion of 1 mg of D-T fuel inside a dense shell. The simulation suggested that a 5 MJ power input to the hohlraum would produce 50 MJ of fusion output, a gain of 10x. This was before the laser and a variety of other possible drivers were considered, including pulsed power machines, charged particle accelerators, plasma guns, and hypervelocity pellet guns.

Two theoretical advances advanced the field. One came from new simulations that considered the timing of the energy delivered in the pulse, known as "pulse shaping", leading to better implosion. The second was to make the shell much larger and thinner, forming a thin shell as opposed to an almost solid ball. These two changes dramatically increased implosion efficiency and thereby greatly lowered the required compression energy. Using these improvements, it was calculated that a driver of about 1 MJ would be needed, a five-fold reduction. Over the next two years, other theoretical advancements were proposed, notably Ray Kidder's development of an implosion system without a hohlraum, the so-called "direct drive" approach, and Stirling Colgate and Ron Zabawski's work on systems with as little as 1 μg of D-T fuel.

The introduction of the laser in 1960 at Hughes Research Laboratories in California appeared to present a perfect driver mechanism. Starting in 1962, Livermore's director John S. Foster, Jr. and Edward Teller began a small ICF laser study. Even at this early stage the suitability of ICF for weapons research was well understood and was the primary reason for its funding. Over the next decade, LLNL made small experimental devices for basic laser-plasma interaction studies.

Development begins
In 1967 Kip Siegel started KMS Industries. In the early 1970s he formed KMS Fusion to begin development of a laser-based ICF system. This development led to considerable opposition from the weapons labs, including LLNL, who put forth a variety of reasons that KMS should not be allowed to develop ICF in public. This opposition was funnelled through the Atomic Energy Commission, which demanded funding. Adding to the background noise were rumours of an aggressive Soviet ICF program, new higher-powered CO2 and glass lasers, the electron beam driver concept, and the energy crisis which added impetus to many energy projects.

In 1972 John Nuckolls wrote a paper introducing ICF and suggesting that testbed systems could be made to generate fusion with drivers in the kJ range, and high-gain systems with MJ drivers.

In spite of limited resources and business problems, KMS Fusion successfully demonstrated IFC fusion on 1 May 1974. This success was soon followed by Siegel's death and the end of KMS Fusion a year later. By this point several weapons labs and universities had started their own programs, notably the solid-state lasers (Nd:glass lasers) at LLNL and the University of Rochester, and krypton fluoride excimer lasers systems at Los Alamos and the Naval Research Laboratory.

"High-energy" ICF
High-energy ICF experiments (multi-hundred joules per shot) began in the early 1970s, when better lasers appeared. Funding for fusion research was stimulated by energy crises produced rapid gains in performance, and inertial designs were soon reaching the same sort of "below break-even" conditions of the best MCF systems.

LLNL was, in particular, well funded and started a laser fusion development program. Their Janus laser started operation in 1974, and validated the approach of using Nd:glass lasers for high power devices. Focusing problems were explored in the Long path and Cyclops lasers, which led to the larger Argus laser. None of these were intended to be practical devices, but they increased confidence that the approach was valid. It was then believed that a much larger device of the Cyclops type could both compress and heat targets, leading to ignition. This misconception was based on extrapolation of the fusion yields seen from experiments utilizing the so-called "exploding pusher" fuel capsule. During the late 1970s and early 1980s the estimates for laser energy on target needed to achieve ignition doubled almost yearly as plasma instabilities and laser-plasma energy coupling loss modes were increasingly understood. The realization that exploding pusher target designs and single-digit kilojoule (kJ) laser irradiation intensities would never scale to high yields led to the effort to increase laser energies to the 100 kJ level in the ultraviolet band and to the production of advanced ablator and cryogenic DT ice target designs.

Shiva and Nova
One of the earliest large scale attempts at an ICF driver design was the Shiva laser, a 20-beam neodymium doped glass laser system at LLNL that started operation in 1978. Shiva was a "proof of concept" design intended to demonstrate compression of fusion fuel capsules to many times the liquid density of hydrogen. In this, Shiva succeeded, reaching 100 times the liquid density of deuterium. However, due to the laser's coupling with hot electrons, premature heating of the dense plasma was problematic and fusion yields were low. This failure to efficiently heat the compressed plasma pointed to the use of optical frequency multipliers as a solution that would frequency triple the infrared light from the laser into the ultraviolet at 351 nm. Schemes to efficiently triple the frequency of laser light discovered at the Laboratory for Laser Energetics in 1980 was experimented with in the 24 beam OMEGA laser and the NOVETTE laser, which was followed by the Nova laser design with 10 times Shiva's energy, the first design with the specific goal of reaching ignition.

Nova also failed, this time due to severe variation in laser intensity in its beams (and differences in intensity between beams) caused by filamentation that resulted in large non-uniformity in irradiation smoothness at the target and asymmetric implosion. The techniques pioneered earlier could not address these new issues. This failure led to a much greater understanding of the process of implosion, and the way forward again seemed clear, namely to increase the uniformity of irradiation, reduce hot-spots in the laser beams through beam smoothing techniques to reduce Rayleigh–Taylor instabilities and increase laser energy on target by at least an order of magnitude. Funding was constrained in the 1980s.

National Ignition Facility
The resulting 192-beam design, dubbed the National Ignition Facility, started construction at LLNL in 1997. NIF's main objective is to operate as the flagship experimental device of the so-called nuclear stewardship program, supporting LLNLs traditional bomb-making role. Completed in March 2009, NIF experiments set new records for power delivery by a laser. As of September 27, 2013, for the first time fusion energy generated was greater than the energy absorbed into deuterium–tritium fuel. In June, 2018 NIF announced record production of 54kJ of fusion energy output. On August 8, 2021 the NIF produced 1.3MJ of output, 25x higher than the 2018 result, generating 70% of the break-even definition of ignition - when energy out equals energy in. As of December 2022, the NIF claims to have become the first fusion experiment to achieve scientific breakeven on December 5, 2022, with an experiment producing 3.15 megajoules of energy from a 2.05 megajoule input of laser light (somewhat less than the energy needed to boil 1 kg of water) for an energy gain of about 1.5.

Fast ignition
Fast ignition may offer a way to directly heat fuel after compression, thus decoupling the heating and compression phases. In this approach, the target is first compressed "normally" using a laser system. When the implosion reaches maximum density (at the stagnation point or "bang time"), a second short, high-power petawatt (PW) laser delivers a single pulse to one side of the core, dramatically heating it and starting ignition.

The two types of fast ignition are the "plasma bore-through" method and the "cone-in-shell" method. In plasma bore-through, the second laser bores through the outer plasma of an imploding capsule, impinges on and heats the core. In the cone-in-shell method, the capsule is mounted on the end of a small high-z (high atomic number) cone such that the tip of the cone projects into the core. In this second method, when the capsule is imploded, the laser has a clear view of the core and does not use energy to bore through a 'corona' plasma. However, the presence of the cone affects the implosion process in significant ways that are not fully understood. Several projects are currently underway to explore fast ignition, including upgrades to the OMEGA laser at the University of Rochester and the GEKKO XII device in Japan.

HiPer is a proposed £500 million facility in the European Union. Compared to NIF's 2 MJ UV beams, HiPER's driver was planned to be 200 kJ and heater 70 kJ, although the predicted fusion gains are higher than NIF. It was to employ diode lasers, which convert electricity into laser light with much higher efficiency and run cooler. This allows them to operate at much higher frequencies. HiPER proposed to operate at 1 MJ at 1 Hz, or alternately 100 kJ at 10 Hz. The project's final update was in 2014. It was expected to offer a higher Q with a 10x reduction in construction costs times.

Other projects
The French Laser Mégajoule achieved its first experimental line in 2002, and its first target shots were conducted in 2014. The machine was roughly 75% complete as of 2016.

Using a different approach entirely is the z-pinch device. Z-pinch uses massive electric currents switched into a cylinder comprising extremely fine wires. The wires vaporize to form an electrically conductive, high current plasma. The resulting circumferential magnetic field squeezes the plasma cylinder, imploding it, generating a high-power x-ray pulse that can be used to implode a fuel capsule. Challenges to this approach include relatively low drive temperatures, resulting in slow implosion velocities and potentially large instability growth, and preheat caused by high-energy x-rays.

Shock ignition was proposed to address problems with fast ignition. Japan developed the KOYO-F design and laser inertial fusion test (LIFT) experimental reactor. In April 2017, clean energy startup Apollo Fusion began to develop a hybrid fusion-fission reactor technology.

In Germany, technology company Marvel Fusion is working on laser-initiated inertial confinement fusion. The startup adopted a short-pulsed high energy laser and the aneutronic fuel pB11. It was founded in Munich 2019. It works with Siemens Energy, TRUMPF, and Thales. The company partnered with Ludwig Maximilian University of Munich in July 2022.

In March 2022, Australian company HB11 announced fusion using non-thermal laser pB11, at a higher than predicted rate of alpha particle creation. Other companies include NIF-like Longview Fusion and fast-ignition origned Focused Energy.

Electricity generation
Inertial fusion energy (IFE) power plants have been studied since the late 1970s. These devices were to deliver multiple targets per second into the reaction chamber, using the resulting energy to drive a conventional steam turbine.

Technical challenges

Even if the many technical challenges in reaching ignition were all to be solved, practical problems abound. Given the 1 to 1.5% efficiency of the laser amplification process and that steam-driven turbine systems are typically about 35% efficient, fusion gains would have to be on the order of 125-fold just to energetically break even.

An order of magnitude improvement in laser efficiency may be possible through the use of designs that replace flash lamps with laser diodes that are tuned to produce most of their energy in a frequency range that is strongly absorbed. Initial experimental devices offer efficiencies of about 10%, and it is suggested that 20% is possible.

NIF uses about 330 MJ to produce the driver beams, producing an expected yield of about 20 MJ, with maximum credible yield of 45 MJ.

Power extraction
ICF systems face some of the secondary power extraction problems as MCF systems. One of the primary concerns is how to successfully remove heat from the reaction chamber without interfering with the targets and driver beams. Another concern is that the released neutrons react with the reactor structure, mechanically weakening it, and turning it intensely radioactive. Conventional metals such as steel would have a short lifetime and require frequent replacement of the core containment walls. Another concern is fusion afterdamp (debris left in the reaction chamber), which could interfere with subsequent shots, including helium ash produced by fusion, along with unburned hydrogen and other elements used in the fuel pellet. This problem is most troublesome with indirect drive systems. If the driver energy misses the fuel pellet completely and strikes the containment chamber, material could foul the interaction region, or the lenses or focusing elements.

One concept, as shown in the HYLIFE-II design, is to use a "waterfall" of FLiBe, a molten mix of fluoride salts of lithium and beryllium, which both protect the chamber from neutrons and carry away heat. The FLiBe is passed into a heat exchanger where it heats water for the turbines. The tritium produced by splitting lithium nuclei can be extracted in order to close the power plant's thermonuclear fuel cycle, a necessity for perpetual operation because tritium is rare and otherwise must be manufactured. Another concept, Sombrero, uses a reaction chamber built of carbon-fiber-reinforced polymer which has a low neutron cross section. Cooling is provided by a molten ceramic, chosen because of its ability to absorb the neutrons and its efficiency as a heat transfer agent.

Economic viability
Another factor working against IFE is the cost of the fuel. Even as Nuckolls was developing his earliest calculations, co-workers pointed out that if an IFE machine produces 50 MJ of fusion energy, a shot could produce perhaps 10 MJ (2.8 kWh) of energy. Wholesale rates for electrical power on the grid were about 0.3 cents/kWh at the time, which meant the monetary value of the shot was perhaps one cent. In the intervening 50 years the real price of power has remained about even, and the rate in 2012 in Ontario, Canada was about 2.8 cents/kWh. Thus, in order for an IFE plant to be economically viable, fuel shots would have to cost considerably less than ten cents in 2012 dollars.

Direct-drive systems avoid the use of a hohlraum and thereby may be less expensive in fuel terms. However, these systems still require an ablator, and the accuracy and geometrical considerations are critical. The direct-drive approach still may not be less expensive to operate.

Nuclear weapons
The hot and dense conditions encountered during an ICF experiment are similar to those in a thermonuclear weapon, and have applications to nuclear weapons programs. ICF experiments might be used, for example, to help determine how warhead performance degrades as it ages, or as part of a weapons design program. Retaining knowledge and expertise inside the nuclear weapons program is another motivation for pursuing ICF. Funding for the NIF in the United States is sourced from the Nuclear Weapons Stockpile Stewardship program, whose goals are oriented accordingly. It has been argued that some aspects of ICF research violate the Comprehensive Test Ban Treaty or the Nuclear Non-Proliferation Treaty. In the long term, despite the formidable technical hurdles, ICF research could lead to the creation of a "pure fusion weapon".

Neutron source
ICF has the potential to produce orders of magnitude more neutrons than spallation. Neutrons are capable of locating hydrogen atoms in molecules, resolving atomic thermal motion and studying collective excitations of photons more effectively than X-rays. Neutron scattering studies of molecular structures could resolve problems associated with protein folding, diffusion through membranes, proton transfer mechanisms, dynamics of molecular motors, etc. by modulating thermal neutrons into beams of slow neutrons. In combination with fissile materials, neutrons produced by ICF can potentially be used in Hybrid Nuclear Fusion designs to produce electric power.

title: Inertial confinement fusion


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Quadratic programming

Quadratic programming (QP) is the process of solving certain mathematical optimization problems involving quadratic functions. Specifically, one seeks to optimize (minimize or maximize) a multivariate quadratic function subject to linear constraints on the variables. Quadratic programming is a type of nonlinear programming.

"Programming" in this context refers to a formal procedure for solving mathematical problems. This usage dates to the 1940s and is not specifically tied to the more recent notion of "computer programming." To avoid confusion, some practitioners prefer the term "optimization" — e.g., "quadratic optimization."

Problem formulation
The quadratic programming problem with $n$ variables and $m$ constraints can be formulated as follows. Given: the objective of quadratic programming is to find an $n$-dimensional vector $c$, that will
 * a real-valued, $Q$-dimensional vector $n×n$,
 * an $m×n$-dimensional real symmetric matrix $A$,
 * an $b$-dimensional real matrix $m$, and
 * an $n$-dimensional real vector $x$,

 * {| cellspacing="10"

where $x^{T}$ denotes the vector transpose of $x$, and the notation $Ax ⪯ b$ means that every entry of the vector $Ax$ is less than or equal to the corresponding entry of the vector $b$ (component-wise inequality).
 * minimize
 * $$\tfrac{1}{2} \mathbf{x}^\mathrm{T} Q\mathbf{x} + \mathbf{c}^\mathrm{T} \mathbf{x}$$
 * subject to
 * $$A \mathbf{x} \preceq \mathbf{b},$$
 * }
 * $$A \mathbf{x} \preceq \mathbf{b},$$
 * }

Constrained least squares
As a special case when $Q$ is symmetric positive-definite, the cost function reduces to least squares:
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where $Q = R^{T}R$ follows from the Cholesky decomposition of $Q$ and $c = −R^{T} d$. Conversely, any such constrained least squares program can be equivalently framed as a quadratic programming problem, even for a generic non-square $R$ matrix.
 * minimize
 * $$\tfrac{1}{2} \| R \mathbf{x} - \mathbf{d}\|^2 $$
 * subject to
 * $$A \mathbf{x} \preceq \mathbf{b},$$
 * }
 * $$A \mathbf{x} \preceq \mathbf{b},$$
 * }

Generalizations
When minimizing a function $f$ in the neighborhood of some reference point $x_{0}$, $Q$ is set to its Hessian matrix $H(f(x_{0}))$ and $c$ is set to its gradient $∇f(x_{0})$. A related programming problem, quadratically constrained quadratic programming, can be posed by adding quadratic constraints on the variables.

Solution methods
For general problems a variety of methods are commonly used, including

 * interior point,
 * active set,
 * augmented Lagrangian,
 * conjugate gradient,
 * gradient projection,
 * extensions of the simplex algorithm.

In the case in which $Q$ is positive definite, the problem is a special case of the more general field of convex optimization.

Equality constraints
Quadratic programming is particularly simple when $Q$ is positive definite and there are only equality constraints; specifically, the solution process is linear. By using Lagrange multipliers and seeking the extremum of the Lagrangian, it may be readily shown that the solution to the equality constrained problem

 * $$\text{Minimize} \quad \tfrac{1}{2} \mathbf{x}^\mathrm{T} Q\mathbf{x} + \mathbf{c}^\mathrm{T} \mathbf{x}$$

 * $$\text{subject to} \quad E\mathbf{x} =\mathbf{d}$$

is given by the linear system

\begin{bmatrix} Q & E^\top \\ E & 0 \end{bmatrix} \begin{bmatrix} \mathbf x \\ \lambda \end{bmatrix} = \begin{bmatrix} -\mathbf c \\ \mathbf d \end{bmatrix} $$

where $λ$ is a set of Lagrange multipliers which come out of the solution alongside $x$.

The easiest means of approaching this system is direct solution (for example, LU factorization), which for small problems is very practical. For large problems, the system poses some unusual difficulties, most notably that the problem is never positive definite (even if $Q$ is), making it potentially very difficult to find a good numeric approach, and there are many approaches to choose from dependent on the problem.

If the constraints don't couple the variables too tightly, a relatively simple attack is to change the variables so that constraints are unconditionally satisfied. For example, suppose $d = 0$ (generalizing to nonzero is straightforward). Looking at the constraint equations:

 * $$E\mathbf{x} = 0$$

introduce a new variable $y$ defined by

 * $$Z \mathbf{y} = \mathbf x$$

where $y$ has dimension of $x$ minus the number of constraints. Then

 * $$E Z \mathbf{y} = \mathbf 0$$

and if $Z$ is chosen so that $EZ = 0$ the constraint equation will be always satisfied. Finding such $Z$ entails finding the null space of $E$, which is more or less simple depending on the structure of $E$. Substituting into the quadratic form gives an unconstrained minimization problem:

 * $$\tfrac{1}{2} \mathbf{x}^\top Q\mathbf{x} + \mathbf{c}^\top \mathbf{x} \quad \implies \quad

\tfrac{1}{2} \mathbf{y}^\top Z^\top Q Z \mathbf{y} + \left(Z^\top \mathbf{c}\right)^\top \mathbf{y}$$

the solution of which is given by:

 * $$Z^\top Q Z \mathbf{y} = -Z^\top \mathbf{c}$$

Under certain conditions on $Q$, the reduced matrix $Z^{T}QZ$ will be positive definite. It is possible to write a variation on the conjugate gradient method which avoids the explicit calculation of $Z$.

Lagrangian duality
The Lagrangian dual of a quadratic programming problem is also a quadratic programming problem. To see this let us focus on the case where $c = 0$ and $Q$ is positive definite. We write the Lagrangian function as
 * $$L(x,\lambda) = \tfrac{1}{2} x^\top Qx + \lambda^\top (Ax-b). $$

Defining the (Lagrangian) dual function $g(λ)$ as $$g(\lambda) = \inf_{x} L(x,\lambda) $$, we find an infimum of $L$, using $$\nabla_{x} L(x,\lambda)=0$$ and positive-definiteness of $Q$:

 * $$x^* = -Q^{-1} A^\top \lambda.$$

Hence the dual function is
 * $$g(\lambda) = -\tfrac{1}{2} \lambda^\top AQ^{-1}A^\top \lambda - \lambda^\top b,$$

and so the Lagrangian dual of the quadratic programming problem is

 * $$\text{maximize}_{\lambda\geq 0} \quad -\tfrac{1}{2} \lambda^\top AQ^{-1} A^\top \lambda - \lambda^\top b.$$

Besides the Lagrangian duality theory, there are other duality pairings (e.g. Wolfe, etc.).

Convex quadratic programming
For positive definite $Q$, when the problem is convex, the ellipsoid method solves the problem in (weakly) polynomial time.

Ye and Tse present a polynomial-time algorithm, which extends Karmarkar's algorithm from linear programming to convex quadratic programming. On a system with n variables and L input bits, their algorithm requires O(L n) iterations, each of which can be done using O(L n3) arithmetic operations, for a total runtime complexity of O(L2 n4).

Kapoor and Vaidya present another algorithm, which requires O(L * log L * n3.67 * log n) arithmetic operations.

Non-convex quadratic programming
If $Q$ is indefinite, (so the problem is non-convex) then the problem is NP-hard. A simple way to see this is to consider the non-convex quadratic constraint xi2 = xi. This constraint is equivalent to requiring that xi is in {0,1}, that is, xi is a binary integer variable. Therefore, such constraints can be used to model any integer program with binary variables, which is known to be NP-hard.

Moreover, these non-convex problems might have several stationary points and local minima. In fact, even if $Q$ has only one negative eigenvalue, the problem is (strongly) NP-hard.

Moreover, finding a KKT point of a non-convex quadratic program is CLS-hard.

Mixed-integer quadratic programming
There are some situations where one or more elements of the vector $x$ will need to take on integer values. This leads to the formulation of a mixed-integer quadratic programming (MIQP) problem. Applications of MIQP include water resources and the construction of index funds.

Extensions
Polynomial optimization is a more general framework, in which the constraints can be polynomial functions of any degree, not only 2.

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

title: Quadratic programming


text:
Positive semidefinite

In mathematics, positive semidefinite may refer to:
 * Positive semidefinite function
 * Positive semidefinite matrix
 * Positive semidefinite quadratic form
 * Positive semidefinite bilinear form

title: Positive semidefinite


text:
Definite matrix

In mathematics, a symmetric matrix $$\ M\ $$ with real entries is positive-definite if the real number $$\ \mathbf{x}^\top M \mathbf{x}\ $$ is positive for every nonzero real column vector $$\ \mathbf{x}\ ,$$ where $$\ \mathbf{x}^\top\ $$ is the row vector transpose of $$\ \mathbf{x} ~.$$ More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number $$\ \mathbf{z}^* M \mathbf{z}\ $$ is positive for every nonzero complex column vector $$\ \mathbf{z}\ ,$$ where $$\ \mathbf{z}^*\ $$ denotes the conjugate transpose of $$\ \mathbf{z} ~.$$

Positive semi-definite matrices are defined similarly, except that the scalars $$\ \mathbf{x}^\top M \mathbf{x}\ $$ and $$\ \mathbf{z}^* M \mathbf{z}\ $$ are required to be positive or zero (that is, not negative). Negative-definite and negative semi-definite matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is sometimes called indefinite.

Ramifications
It follows from the above definitions that a matrix is positive-definite if and only if it is the matrix of a positive-definite quadratic form or Hermitian form. In other words, a matrix is positive-definite if and only if it defines an inner product.

Positive-definite and positive-semidefinite matrices can be characterized in many ways, which may explain the importance of the concept in various parts of mathematics. A matrix $M$ is positive-definite if and only if it satisfies any of the following equivalent conditions. A matrix is positive semi-definite if it satisfies similar equivalent conditions where "positive" is replaced by "nonnegative", "invertible matrix" is replaced by "matrix", and the word "leading" is removed.
 * $$\ M\ $$ is congruent with a diagonal matrix with positive real entries.
 * $$\ M\ $$ is symmetric or Hermitian, and all its eigenvalues are real and positive.
 * $$\ M\ $$ is symmetric or Hermitian, and all its leading principal minors are positive.
 * There exists an invertible matrix $$\ B\ $$ with conjugate transpose $$\ B^*\ $$ such that $$\ M = B^* B ~.$$

Positive-definite and positive-semidefinite real matrices are at the basis of convex optimization, since, given a function of several real variables that is twice differentiable, then if its Hessian matrix (matrix of its second partial derivatives) is positive-definite at a point $$\ p\ ,$$ then the function is convex near $p$, and, conversely, if the function is convex near $$\ p\ ,$$ then the Hessian matrix is positive-semidefinite at $$\ p ~.$$

The set of positive definite matrices is an open convex cone, while the set of positive semi-definite matrices is a closed convex cone.

Some authors use more general definitions of definiteness, including some non-symmetric real matrices, or non-Hermitian complex ones.

Definitions
In the following definitions, $$\ \mathbf{x}^\top\ $$ is the transpose of $$\ \mathbf{x}\ ,$$ $$\ \mathbf{z}^*\ $$ is the conjugate transpose of $$\ \mathbf{z}\ ,$$ and $$\ \mathbf{0}\ $$ denotes the $n$ dimensional zero-vector.

Definitions for real matrices
An $$n \times n$$ symmetric real matrix $$\ M\ $$ is said to be positive-definite if $$\ \mathbf{x}^\top M\ \mathbf{x} > 0\ $$ for all non-zero $$\ \mathbf{x}\ $$ in $$\ \mathbb{R}^n ~.$$ Formally,

An $$\ n \times n\ $$ symmetric real matrix $$\ M\ $$ is said to be positive-semidefinite or non-negative-definite if $$\ \mathbf{x}^\top M\ \mathbf{x} \geq 0\ $$ for all $$\ \mathbf{x}\ $$ in $$\ \mathbb{R}^n ~.$$ Formally,

An $$\ n \times n\ $$ symmetric real matrix $$\ M\ $$ is said to be negative-definite if $$\ \mathbf{x}^\top M\ \mathbf{x} < 0\ $$ for all non-zero $$\ \mathbf{x}\ $$ in $$\ \R^n ~.$$ Formally,

An $$\ n \times n\ $$ symmetric real matrix $$\ M\ $$ is said to be negative-semidefinite or non-positive-definite if $$\ \mathbf{x}^\top M\ \mathbf{x} \leq 0\ $$ for all $$\ \mathbf{x}\ $$ in $$\ \mathbb{R}^n ~.$$ Formally,

An $$n \times n$$ symmetric real matrix which is neither positive semidefinite nor negative semidefinite is called indefinite.

Definitions for complex matrices
The following definitions all involve the term $$\ \mathbf{z}^* M\ \mathbf{z} ~.$$ Notice that this is always a real number for any Hermitian square matrix $$\ M ~.$$

An $$\ n \times n\ $$ Hermitian complex matrix $$\ M\ $$ is said to be positive-definite if $$\ \mathbf{z}^* M\ \mathbf{z} > 0\ $$ for all non-zero $$\ \mathbf{z}\ $$ in $$\ \mathbb{C}^n ~.$$ Formally,

An $$\ n \times n\ $$ Hermitian complex matrix $$\ M\ $$ is said to be positive semi-definite or non-negative-definite if $$\ \mathbf{z}^* M\ \mathbf{z} \geq 0\ $$ for all $$\ \mathbf{z}\ $$ in $$\ \mathbb{C}^n ~.$$ Formally,

An $$\ n \times n\ $$ Hermitian complex matrix $$\ M\ $$ is said to be negative-definite if $$\ \mathbf{z}^* M\ \mathbf{z} < 0\ $$ for all non-zero $$\ \mathbf{z}\ $$ in $$\ \mathbb{C}^n ~.$$ Formally,

An $$\ n \times n\ $$ Hermitian complex matrix $$\ M\ $$ is said to be negative semi-definite or non-positive-definite if $$\ \mathbf{z}^* M\ \mathbf{z} \leq 0\ $$ for all $$\ \mathbf{z}\ $$ in $$\ \mathbb{C}^n ~.$$ Formally,

An $$\ n \times n\ $$ Hermitian complex matrix which is neither positive semidefinite nor negative semidefinite is called indefinite.

Consistency between real and complex definitions
Since every real matrix is also a complex matrix, the definitions of "definiteness" for the two classes must agree.

For complex matrices, the most common definition says that $$\ M\ $$ is positive-definite if and only if $$\ \mathbf{z}^* M\ \mathbf{z}\ $$ is real and positive for every non-zero complex column vectors $$\mathbf{z} ~.$$ This condition implies that $$M$$ is Hermitian (i.e. its transpose is equal to its conjugate), since $$\mathbf{z}^* M\ \mathbf{z}$$ being real, it equals its conjugate transpose $$\ \mathbf{z}^*\ M^*\ \mathbf{z}\ $$ for every $$\ \mathbf{z}\ ,$$ which implies $$\ M = M^* ~.$$

By this definition, a positive-definite real matrix $$\ M\ $$ is Hermitian, hence symmetric; and $$\ \mathbf{z}^\top M\ \mathbf{z}\ $$ is positive for all non-zero real column vectors $$\ \mathbf{z} ~.$$ However the last condition alone is not sufficient for $$\ M\ $$ to be positive-definite. For example, if $$\ M = \begin{bmatrix} ~1~ & ~1~ \\ -1~ & ~1~ \end{bmatrix},$$

then for any real vector $$\ \mathbf{z}\ $$ with entries $$\ a\ $$ and $$\ b\ $$ we have $$\ \mathbf{z}^\top M\ \mathbf{z} = \left(a + b\right)a + \left(-a + b\right) b = a^2 + b^2\ ,$$ which is always positive if $$\ \mathbf{z}\ $$ is not zero. However, if $$\ \mathbf{z}\ $$ is the complex vector with entries $1$ and $$\ i\ ,$$ one gets

$$\mathbf{z}^* M\ \mathbf{z} = \begin{bmatrix} ~1~ & -i~ \end{bmatrix}\ M\ \begin{bmatrix} ~1~ \\ ~i~ \end{bmatrix} = \begin{bmatrix} ~1 + i~ & ~1 - i~ \end{bmatrix}\ \begin{bmatrix} ~1~ \\ ~i~ \end{bmatrix} = 2 + 2i ~.$$

which is not real. Therefore, $$\ M\ $$ is not positive-definite.

On the other hand, for a symmetric real matrix $$\ M\ ,$$ the condition "$$\ \mathbf{z}^\top M\ \mathbf{z} > 0\ $$ for all nonzero real vectors $$\ \mathbf{z}\ $$ does imply that $$\ M\ $$ is positive-definite in the complex sense.

Notation
If a Hermitian matrix $$\ M\ $$ is positive semi-definite, one sometimes writes $$\ M \succeq 0\ $$ and if $$\ M\ $$ is positive-definite one writes $$\ M \succ 0 ~.$$ To denote that $$\ M\ $$ is negative semi-definite one writes $$\ M \preceq 0\ $$ and to denote that $$\ M\ $$ is negative-definite one writes $$\ M \prec 0 ~.$$

The notion comes from functional analysis where positive semidefinite matrices define positive operators. If two matrices $$\ A\ $$ and $$\ B\ $$ satisfy $$\ B - A \succeq 0\ ,$$ we can define a non-strict partial order $$\ B \succeq A\ $$ that is reflexive, antisymmetric, and transitive; It is not a total order, however, as $$\ B - A\ ,$$ in general, may be indefinite.

A common alternative notation is $$\ M \geq 0\ ,$$ $$\ M > 0\ ,$$ $$\ M \leq 0\ ,$$ and $$\ M < 0\ $$ for positive semi-definite and positive-definite, negative semi-definite and negative-definite matrices, respectively. This may be confusing, as sometimes nonnegative matrices (respectively, nonpositive matrices) are also denoted in this way.

Eigenvalues
Let $$M$$ be an $$n \times n$$ Hermitian matrix (this includes real symmetric matrices). All eigenvalues of $$M$$ are real, and their sign characterize its definiteness:
 * $$M$$ is positive definite if and only if all of its eigenvalues are positive.
 * $$M$$ is positive semi-definite if and only if all of its eigenvalues are non-negative.
 * $$M$$ is negative definite if and only if all of its eigenvalues are negative
 * $$M$$ is negative semi-definite if and only if all of its eigenvalues are non-positive.
 * $$M$$ is indefinite if and only if it has both positive and negative eigenvalues.

Let $$\ P D P^{-1}\ $$ be an eigendecomposition of $$\ M\ ,$$ where $$\ P\ $$ is a unitary complex matrix whose columns comprise an orthonormal basis of eigenvectors of $$\ M\ ,$$ and $$\ D\ $$ is a real diagonal matrix whose main diagonal contains the corresponding eigenvalues. The matrix $$\ M\ $$ may be regarded as a diagonal matrix $$\ D\ $$ that has been re-expressed in coordinates of the (eigenvectors) basis $$\ P ~.$$ Put differently, applying $$M$$ to some vector $$\ \mathbf{z}\ ,$$ giving $$\ M \mathbf{z}\ ,$$ is the same as changing the basis to the eigenvector coordinate system using $$\ P^{-1}\ ,$$ giving $$\ P^{-1} \mathbf{z}\ ,$$ applying the stretching transformation $$\ D\ $$ to the result, giving $$\ D P^{-1} \mathbf{z}\ ,$$ and then changing the basis back using $$\ P\ ,$$ giving $$\ P D P^{-1} \mathbf{z} ~.$$

With this in mind, the one-to-one change of variable $$\ \mathbf{y} = P\mathbf{z}\ $$ shows that $$\ \mathbf{z}^* M\mathbf{z}\ $$ is real and positive for any complex vector $$\ \mathbf{z}\ $$ if and only if $$\ \mathbf{y}^* D \mathbf{y}\ $$ is real and positive for any $$\ y\ ;$$ in other words, if $$\ D\ $$ is positive definite. For a diagonal matrix, this is true only if each element of the main diagonal – that is, every eigenvalue of $$\ M\ $$ – is positive. Since the spectral theorem guarantees all eigenvalues of a Hermitian matrix to be real, the positivity of eigenvalues can be checked using Descartes' rule of alternating signs when the characteristic polynomial of a real, symmetric matrix $$\ M\ $$ is available.

Decomposition
Let $$\ M\ $$ be an $$\ n \times n\ $$ Hermitian matrix. $$\ M\ $$ is positive semidefinite if and only if it can be decomposed as a product $$\ M = B^* B\ $$ of a matrix $$\ B\ $$ with its conjugate transpose.

When $$\ M\ $$ is real, $$\ B\ $$ can be real as well and the decomposition can be written as $$\ M = B^\top B ~.$$

$$M$$ is positive definite if and only if such a decomposition exists with $$\ B\ $$ invertible. More generally, $$\ M\ $$ is positive semidefinite with rank $$\ k\ $$ if and only if a decomposition exists with a $$\ k \times n\ $$ matrix $$\ B\ $$ of full row rank (i.e. of rank $$\ k\ $$). Moreover, for any decomposition $$\ M = B^* B\ ,$$ $$\ \operatorname{rank}(M) = \operatorname{rank}(B) ~.$$

$$

The columns $$\ b_1, \dots, b_n\ $$ of $$\ B\ $$ can be seen as vectors in the complex or real vector space $$\ \mathbb{R}^k\ ,$$ respectively. Then the entries of $$M$$ are inner products (that is dot products, in the real case) of these vectors $$\ M_{ij} = \langle b_i, b_j\rangle ~.$$ In other words, a Hermitian matrix $$\ M\ $$ is positive semidefinite if and only if it is the Gram matrix of some vectors $$\ b_1, \dots, b_n ~.$$ It is positive definite if and only if it is the Gram matrix of some linearly independent vectors. In general, the rank of the Gram matrix of vectors $$\ b_1, \dots, b_n\ $$ equals the dimension of the space spanned by these vectors.

Uniqueness up to unitary transformations
The decomposition is not unique: if $$\ M = B^* B\ $$ for some $$\ k \times n\ $$ matrix $$\ B\ $$ and if $$\ Q\ $$ is any unitary $$k \times k$$ matrix (meaning $$\ Q^* Q = Q Q^* = I\ $$), then $$\ M = B^* B = B^* Q^* Q B = A^*\ $$ for $$\ A = Q B ~.$$

However, this is the only way in which two decompositions can differ: The decomposition is unique up to unitary transformations. More formally, if $$\ A\ $$ is a $$\ k \times n\ $$ matrix and $$\ B\ $$ is a $$\ \ell \times n\ $$ matrix such that $$\ A^* A = B^* B\ ,$$ then there is a $$\ \ell \times k\ $$ matrix $$Q$$ with orthonormal columns (meaning $$\ Q^* Q = I_{k \times k}\ $$) such that $$\ B = Q A ~.$$ When $$\ell = k$$ this means $$Q$$ is unitary.

This statement has an intuitive geometric interpretation in the real case: let the columns of $$A$$ and $$B$$ be the vectors $$a_1,\dots,a_n$$ and $$\ b_1, \dots, b_n\ $$ in $$\ \mathbb{R}^k ~.$$ A real unitary matrix is an orthogonal matrix, which describes a rigid transformation (an isometry of Euclidean space $$\mathbb{R}^k$$) preserving the 0 point (i.e. rotations and reflections, without translations). Therefore, the dot products $$a_i \cdot a_j$$ and $$b_i \cdot b_j$$ are equal if and only if some rigid transformation of $$\mathbb{R}^k$$ transforms the vectors $$a_1,\dots,a_n$$ to $$b_1,\dots,b_n$$ (and 0 to 0).

Square root
A Hermitian matrix $$\ M\ $$ is positive semidefinite if and only if there is a positive semidefinite matrix $$\ B\ $$ (in particular $$\ B\ $$ is Hermitian, so $$\ B^* = B\ $$) satisfying $$\ M = B B ~.$$ This matrix $$\ B\ $$ is unique, is called the non-negative square root of $$\ M\ ,$$ and is denoted with $$\ B = M^\frac{1}{2} ~.$$ When $$M$$ is positive definite, so is $$\ M^\frac{1}{2}\ ,$$ hence it is also called the positive square root of $$\ M ~.$$

The non-negative square root should not be confused with other decompositions $$\ M = B^*B ~.$$ Some authors use the name square root and $$M^\frac{1}{2}$$ for any such decomposition, or specifically for the Cholesky decomposition, or any decomposition of the form $$\ M = B B\ ;$$ others only use it for the non-negative square root.

If $$\ M > N > 0\ $$ then $$\ M^\frac{1}{2} > N^\frac{1}{2} > 0 ~.$$

Cholesky decomposition
A Hermitian positive semidefinite matrix $$\ M\ $$ can be written as $$\ M = L L^*\ ,$$ where $$\ L\ $$ is lower triangular with non-negative diagonal (equivalently $$M = B^*B$$ where $$\ B = L^*\ $$ is upper triangular); this is the Cholesky decomposition. If $$\ M\ $$ is positive definite, then the diagonal of $$\ L\ $$ is positive and the Cholesky decomposition is unique. Conversely if $$\ L\ $$ is lower triangular with nonnegative diagonal then $$\ L L^*\ $$ is positive semidefinite. The Cholesky decomposition is especially useful for efficient numerical calculations. A closely related decomposition is the LDL decomposition, $$\ M = L D L^*\ ,$$ where $$D$$ is diagonal and $$\ L\ $$ is lower unitriangular.

Other characterizations
Let $$\ M\ $$ be an $$\ n \times n\ $$ real symmetric matrix, and let $$\ B_1(M) \equiv \{ \mathbf{x} \in \mathbb{R}^n : \mathbf{x}^\top M\ \mathbf{x} \leq 1\}\ $$ be the "unit ball" defined by $$\ M ~.$$ Then we have the following

Let $$M$$ be an $$\ n \times n\ $$ Hermitian matrix. The following properties are equivalent to $$\ M\ $$ being positive definite:
 * $$\ B_1( \mathbf{v}\ \mathbf{v}^\top )$$ is a solid slab sandwiched between $$\ \pm \{ \mathbf{w}: \langle \mathbf{w}, \mathbf{v}\rangle = 1 \} ~.$$
 * $$\ M \succeq 0\ $$ if and only if $$\ B_1(M)\ $$ is an ellipsoid, or an ellipsoidal cylinder.
 * $$\ M \succ 0\ $$ if and only if $$\ B_1(M)\ $$ is bounded, that is, it is an ellipsoid.
 * If $$\ N \succ 0\ ,$$ then $$\ M \succeq N\ $$ if and only if $$\ B_1(M) \subseteq B_1(N)\ ;$$ $$\ M \succ N\ $$ if and only if $$\ B_1(M) \subseteq \operatorname{int}\!\bigl(\ B_1(N)\ \bigr) ~.$$
 * If $$\ N \succ 0\ ,$$ then $$\ M \succeq \frac{ \mathbf{v}\ \mathbf{v}^\top }{\ \mathbf{v}^\top N\ \mathbf{v}\ }\ $$ for all $$v \neq 0$$ if and only if $\ B_1(M) \subset \bigcap_{ \mathbf{v}^\top N\ \mathbf{v} = 1 } B_1(\mathbf{v} \mathbf{v}^\top) ~.$ So, since the polar dual of an ellipsoid is also an ellipsoid with the same principal axes, with inverse lengths, we have $$\ B_1(N^{-1}) = \bigcap_{\mathbf{v}^\top N\ \mathbf{v} = 1} B_1(\mathbf{v}\ \mathbf{v}^\top) = \bigcap_{ \mathbf{v}^\top N\ \mathbf{v} = 1 } \{ \mathbf{w}: |\langle \mathbf{w}, \mathbf{v}\rangle| \leq 1 \} ~.$$ That is, if $$\ N\ $$ is positive-definite, then $$\ M \succeq \frac{ \mathbf{v} \mathbf{v}^\top }{\ \mathbf{v}^\top N\ \mathbf{v}\ }\ $$ for all $$\ \mathbf{v} \neq \mathbf{0}\ $$ if and only if $$\ M \succeq N^{-1} ~.$$
 * The associated sesquilinear form is an inner product: The sesquilinear form defined by $$M$$ is the function $$\ \langle \cdot, \cdot \rangle\ $$ from $$\ \mathbb{C}^n \times \mathbb{C}^n\ $$ to $$\ \mathbb{C}^n\ $$ such that $$\ \langle \mathbf{x}, \mathbf{y} \rangle \equiv \mathbf{y}^* M\ \mathbf{x}\ $$ for all $$\ \mathbf{x}\ $$ and $$\ \mathbf{y}\ $$ in $$\ \mathbb{C}^n\ ,$$ where $$\ \mathbf{y}^*\ $$ is the conjugate transpose of $$\ \mathbf{y} ~.$$ For any complex matrix $$\ M\ ,$$ this form is linear in $$x$$ and semilinear in $$\ \mathbf{y} ~.$$ Therefore, the form is an inner product on $$\ \mathbb{C}^n\ $$ if and only if $$\ \langle \mathbf{z}, \mathbf{z} \rangle\ $$ is real and positive for all nonzero $$\ \mathbf{z}\ ;$$ that is if and only if $$\ M\ $$ is positive definite. (In fact, every inner product on $$\ \mathbb{C}^n\ $$ arises in this fashion from a Hermitian positive definite matrix.)
 * Its leading principal minors are all positive: The $k$th leading principal minor of a matrix $$\ M\ $$ is the determinant of its upper-left $$\ k \times k\ $$ sub-matrix. It turns out that a matrix is positive definite if and only if all these determinants are positive. This condition is known as Sylvester's criterion, and provides an efficient test of positive definiteness of a symmetric real matrix. Namely, the matrix is reduced to an upper triangular matrix by using elementary row operations, as in the first part of the Gaussian elimination method, taking care to preserve the sign of its determinant during pivoting process. Since the $k$th leading principal minor of a triangular matrix is the product of its diagonal elements up to row $$\ k\ ,$$ Sylvester's criterion is equivalent to checking whether its diagonal elements are all positive.  This condition can be checked each time a new row $$\ k\ $$ of the triangular matrix is obtained.

A positive semidefinite matrix is positive definite if and only if it is invertible. A matrix $$\ M\ $$ is negative (semi)definite if and only if $$\ -M\ $$ is positive (semi)definite.

Quadratic forms
The (purely) quadratic form associated with a real $$\ n \times n\ $$ matrix $$\ M\ $$ is the function $$\ Q : \mathbb{R}^n \to \mathbb{R}\ $$ such that $$\ Q(\mathbf{x}) = \mathbf{x}^\top M \mathbf{x}\ $$ for all $$\ \mathbf{x} ~.$$ $$\ M\ $$ can be assumed symmetric by replacing it with $$\ \tfrac{1}{2} \left(M + M^\top \right)\ ,$$ since any asymetric part will be zeroed-out in the double-sided product.

A symmetric matrix $$\ M\ $$ is positive definite if and only if its quadratic form is a strictly convex function.

More generally, any quadratic function from $$\ \mathbb{R}^n\ $$ to $$\mathbb{R}$$ can be written as $$\ \mathbf{x}^\top M \mathbf{x} + \mathbf{b}^\top \mathbf{x} + c\ $$ where $$\ M\ $$ is a symmetric $$\ n \times n\ $$ matrix, $$\ \mathbf{b}\ $$ is a real $n$ vector, and $$\ c\ $$ a real constant. In the $$\ n = 1\ $$ case, this is a parabola, and just like in the $$\ n = 1\ $$ case, we have

Theorem: This quadratic function is strictly convex, and hence has a unique finite global minimum, if and only if $$\ M\ $$ is positive definite.

Proof: If $$\ M\ $$ is positive definite, then the function is strictly convex. Its gradient is zero at the unique point of $$\ M^{-1} \mathbf{b}\ ,$$ which must be the global minimum since the function is strictly convex. If $$\ M\ $$ is not positive definite, then there exists some vector $$\ \mathbf{v}\ $$ such that $$\ \mathbf{v}^\top M \mathbf{v} \leq 0\ ,$$ so the function $$\ f(t) \equiv ( t \mathbf{v} )^\top M ( t\mathbf{v} ) + b^\top (t \mathbf{v}) + c\ $$ is a line or a downward parabola, thus not strictly convex and not having a global minimum.

For this reason, positive definite matrices play an important role in optimization problems.

Simultaneous diagonalization
One symmetric matrix and another matrix that is both symmetric and positive definite can be simultaneously diagonalized. This is so although simultaneous diagonalization is not necessarily performed with a similarity transformation. This result does not extend to the case of three or more matrices. In this section we write for the real case. Extension to the complex case is immediate.

Let $$\ M\ $$ be a symmetric and $$\ N\ $$ a symmetric and positive definite matrix. Write the generalized eigenvalue equation as $$\ \left(M - \lambda N\right)\mathbf{x} = 0\ $$ where we impose that $$\ \mathbf{x}\ $$ be normalized, i.e. $$\ \mathbf{x}^\top N \mathbf{x} = 1 ~.$$ Now we use Cholesky decomposition to write the inverse of $$\ N\ $$ as $$\ Q^\top Q ~.$$ Multiplying by $$\ Q\ $$ and letting $$\ \mathbf{x} = Q^\top \mathbf{y}\ ,$$ we get $$\ Q \left(M - \lambda N\right) Q^\top \mathbf{y} = 0\ ,$$ which can be rewritten as $$\ \left(Q M Q^\top \right)\mathbf{y} = \lambda \mathbf{y}\ $$ where $$\mathbf{y}^\top \mathbf{y} = 1 ~.$$ Manipulation now yields $$MX = NX\Lambda$$ where $$X$$ is a matrix having as columns the generalized eigenvectors and $$\Lambda$$ is a diagonal matrix of the generalized eigenvalues. Now premultiplication with $$X^\top$$ gives the final result: $$\ X^\top MX = \Lambda\ $$ and $$\ X^\top N X = I\ ,$$ but note that this is no longer an orthogonal diagonalization with respect to the inner product where $$\ \mathbf{y}^\top \mathbf{y} = 1 ~.$$ In fact, we diagonalized $$\ M\ $$ with respect to the inner product induced by $$\ N ~.$$

Note that this result does not contradict what is said on simultaneous diagonalization in the article Diagonalizable matrix, which refers to simultaneous diagonalization by a similarity transformation. Our result here is more akin to a simultaneous diagonalization of two quadratic forms, and is useful for optimization of one form under conditions on the other.

Induced partial ordering
For arbitrary square matrices $$\ M\ ,$$ $$\ N\ $$ we write $$\ M \ge N\ $$ if $$\ M - N \ge 0\ $$ i.e., $$\ M - N\ $$ is positive semi-definite. This defines a partial ordering on the set of all square matrices. One can similarly define a strict partial ordering $$\ M > N ~.$$ The ordering is called the Loewner order.

Inverse of positive definite matrix
Every positive definite matrix is invertible and its inverse is also positive definite. If $$\ M \geq N > 0\ $$ then $$\ N^{-1} \geq M^{-1} > 0 ~.$$ Moreover, by the min-max theorem, the $k$th largest eigenvalue of $$\ M\ $$ is greater than or equal to the $k$th largest eigenvalue of $$\ N ~.$$

Scaling
If $$\ M\ $$ is positive definite and $$\ r > 0\ $$ is a real number, then $$\ r M\ $$ is positive definite.

Addition

 * If $$M$$ and $$N$$ are positive-definite, then the sum $$M + N$$ is also positive-definite.
 * If $$M$$ and $$N$$ are positive-semidefinite, then the sum $$M + N$$ is also positive-semidefinite.
 * If $$M$$ is positive-definite and $$N$$ is positive-semidefinite, then the sum $$M + N$$ is also positive-definite.

Multiplication

 * If $$\ M\ $$ and $$\ N\ $$ are positive definite, then the products $$\ M N M\ $$ and $$NMN$$ are also positive definite. If $$\ M N = N M\ ,$$ then $$\ M N\ $$ is also positive definite.
 * If $$\ M\ $$ is positive semidefinite, then $$\ A^* M A\ $$ is positive semidefinite for any (possibly rectangular) matrix $$\ A ~.$$ If $$\ M\ $$ is positive definite and $$A$$ has full column rank, then $$\ A^* M A\ $$ is positive definite.

Trace
The diagonal entries $$\ m_{ii}\ $$ of a positive-semidefinite matrix are real and non-negative. As a consequence the trace, $$\ \operatorname{tr}(M) \ge 0 ~.$$ Furthermore, since every principal sub-matrix (in particular, 2-by-2) is positive semidefinite, $$\ \left|m_{ij}\right| \leq \sqrt{m_{ii}m_{jj}} \quad \forall i, j\ $$

and thus, when $$\ n \ge 1\ ,$$ $$ \max_{i,j} \left|m_{ij}\right| \leq \max_i m_{ii}$$

An $$n \times n$$ Hermitian matrix $$M$$ is positive definite if it satisfies the following trace inequalities: $$\ \operatorname{tr}(M) > 0 \quad \mathrm{and} \quad \frac{(\operatorname{tr}(M))^2}{\operatorname{tr}(M^2)} > n-1 ~.$$

Another important result is that for any $$\ M\ $$ and $$N$$ positive-semidefinite matrices, $$\ \operatorname{tr}(MN) \ge 0 ~.$$ This follows by writing $$\ \operatorname{tr}(MN) = \operatorname{tr}(M^\frac{1}{2}N M^\frac{1}{2}) ~.$$ The matrix $$M^\frac{1}{2}N M^\frac{1}{2}$$ is positive-semidefinite and thus has non-negative eigenvalues, whose sum, the trace, is therefore also non-negative.

Hadamard product
If $$\ M, N \geq 0\ ,$$ although $$\ M N\ $$ is not necessary positive semidefinite, the Hadamard product is, $$\ M \circ N \geq 0\ $$ (this result is often called the Schur product theorem).

Regarding the Hadamard product of two positive semidefinite matrices $$\ M = (m_{ij}) \geq 0\ ,$$ $$\ N \geq 0\ ,$$ there are two notable inequalities:
 * Oppenheim's inequality: $$\ \det(M \circ N) \geq \det (N) \prod\nolimits_i m_{ii} ~.$$
 * $$\ \det(M \circ N) \geq \det(M) \det(N) ~.$$

Kronecker product
If $$\ M, N \geq 0\ ,$$ although $$\ M N\ $$ is not necessary positive semidefinite, the Kronecker product $$M \otimes N \geq 0 ~.$$

Frobenius product
If $$\ M, N \geq 0\ ,$$ although $$\ M N\ $$ is not necessary positive semidefinite, the Frobenius inner product $$\ M : N \geq 0\ $$ (Lancaster–Tismenetsky, The Theory of Matrices, p. 218).

Convexity
The set of positive semidefinite symmetric matrices is convex. That is, if $$\ M\ $$ and $$\ N\ $$ are positive semidefinite, then for any $$\ \alpha\ $$ between $0$ and $1$, $$\ \alpha M + \left(1 - \alpha\right) N\ $$ is also positive semidefinite. For any vector $$\ \mathbf{x}\ $$: $$\ \mathbf{x}^\top \left(\alpha M + \left(1 - \alpha\right)N\right)\mathbf{x} = \alpha \mathbf{x}^\top M\mathbf{x} + (1 - \alpha) \mathbf{x}^\top N\mathbf{x} \geq 0 ~.$$

This property guarantees that semidefinite programming problems converge to a globally optimal solution.

Relation with cosine
The positive-definiteness of a matrix $$A$$ expresses that the angle $$\ \theta\ $$ between any vector $$\ \mathbf{x}\ $$ and its image $$\ A \mathbf{x}\ $$ is always $$\ -\pi / 2 < \theta < +\pi / 2\ :$$

$$\ \cos\theta = \frac{ \mathbf{x}^\top A\mathbf{x} }{\lVert \mathbf{x} \rVert \lVert A\mathbf{x} \rVert} = \frac{\langle \mathbf{x}, A\mathbf{x} \rangle}{\lVert \mathbf{x} \rVert \lVert A\mathbf{x} \rVert}, \theta = \theta(\mathbf{x}, A \mathbf{x}) \equiv \widehat{\left(\mathbf{x},A\mathbf{x}\right)} \equiv\ $$ the angle between $$\ \mathbf{x}\ $$ and $$\ A\mathbf{x} ~.$$

Further properties

 * 1) If $$M$$ is a symmetric Toeplitz matrix, i.e. the entries $$m_{ij}$$ are given as a function of their absolute index differences: $$\ m_{ij} = h(|i-j|)\ ,$$ and the strict inequality $\sum_{j \neq 0} \left|h(j)\right| < h(0)$  holds, then $$M$$ is strictly positive definite.
 * 2) Let $$M > 0$$ and $$N$$ Hermitian. If $$MN + NM \ge 0$$ (resp., $$MN + NM > 0$$) then $$N \ge 0$$ (resp., $$N > 0$$).
 * 3) If $$\ M > 0\ $$ is real, then there is a $$\ \delta > 0\ $$ such that $$\ M > \delta I\ ,$$ where $$\ I\ $$ is the identity matrix.
 * 4) If $$M_k$$ denotes the leading $$\ k \times k\ $$ minor, $$\ \det\left(M_k\right)/\det\left(M_{k-1}\right)\ $$ is the $k$th pivot during LU decomposition.
 * 5) A matrix is negative definite if its $k$th order leading principal minor is negative when $$\ k\ $$ is odd, and positive when $$\ k\ $$ is even.
 * 6) If $$\ M\ $$ is a real positive definite matrix, then there exists a positive real number $$\ m\ $$ such that for every vector $$\ \mathbf{v}\ ,$$ $$\ \mathbf{v}^\top M\ \mathbf{v} \geq m\ \|\mathbf{v}\|_2^{\ \!2} ~.$$
 * 7) A Hermitian matrix is positive semidefinite if and only if all of its principal minors are nonnegative. It is however not enough to consider the leading principal minors only, as is checked on the diagonal matrix with entries $0$ and $−1.$

Block matrices and submatrices
A positive $$\ 2n \times 2n\ $$ matrix may also be defined by blocks: $$\ M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}\ $$

where each block is $$\ n \times n ~,$$ By applying the positivity condition, it immediately follows that $$\ A\ $$ and $$\ D\ $$ are hermitian, and $$\ C = B^* ~.$$

We have that $$\ \mathbf{z}^* M\mathbf{z} \ge 0\ $$ for all complex $$\ \mathbf{z}\ ,$$ and in particular for $$\ \mathbf{z} = [\mathbf{v}, 0]^\top ~.$$ Then $$\ \begin{bmatrix} \mathbf{v}^* & 0 \end{bmatrix} \begin{bmatrix} A & B \\ B^* & D \end{bmatrix} \begin{bmatrix} \mathbf{v} \\ 0 \end{bmatrix} = \mathbf{v}^* A\mathbf{v} \ge 0 ~.$$

A similar argument can be applied to $$\ D\ ,$$ and thus we conclude that both $$\ A\ $$ and $$\ D\ $$ must be positive definite. The argument can be extended to show that any principal submatrix of $$\ M\ $$ is itself positive definite.

Converse results can be proved with stronger conditions on the blocks, for instance, using the Schur complement.

Local extrema
A general quadratic form $$f(\mathbf{x})$$ on $$n$$ real variables $$x_1, \ldots, x_n$$ can always be written as $$\mathbf{x}^\top M \mathbf{x}$$ where $$\mathbf{x}$$ is the column vector with those variables, and $$M$$ is a symmetric real matrix. Therefore, the matrix being positive definite means that $$f$$ has a unique minimum (zero) when $$\mathbf{x}$$ is zero, and is strictly positive for any other $$\ \mathbf{x} ~.$$

More generally, a twice-differentiable real function $$f$$ on $$n$$ real variables has local minimum at arguments $$x_1, \ldots, x_n$$ if its gradient is zero and its Hessian (the matrix of all second derivatives) is positive semi-definite at that point. Similar statements can be made for negative definite and semi-definite matrices.

Covariance
In statistics, the covariance matrix of a multivariate probability distribution is always positive semi-definite; and it is positive definite unless one variable is an exact linear function of the others. Conversely, every positive semi-definite matrix is the covariance matrix of some multivariate distribution.

Extension for non-Hermitian square matrices
The definition of positive definite can be generalized by designating any complex matrix $$\ M\ $$ (e.g. real non-symmetric) as positive definite if $$\ \mathcal{R_e} \left\{\ \mathbf{z}^* M \mathbf{z}\right\} > 0\ $$ for all non-zero complex vectors $$\ \mathbf{z}\ ,$$ where $$\ \mathcal{R_e}\{\ c\ \}\ $$ denotes the real part of a complex number $$\ c ~.$$ Only the Hermitian part $\ \frac{1}{2}\left(M + M^*\right)\ $ determines whether the matrix is positive definite, and is assessed in the narrower sense above. Similarly, if $$\ \mathbf{x}\ $$ and $$\ M\ $$ are real, we have $$\ \mathbf{x}^\top M \mathbf{x} > 0\ $$ for all real nonzero vectors $$\ \mathbf{x}\ $$ if and only if the symmetric part $\ \frac{1}{2}\left(M + M^\top \right)\ $ is positive definite in the narrower sense. It is immediately clear that $\ \mathbf{x}^\top M \mathbf{x} = \sum_{ij} x_i M_{ij} x_j\ $ is insensitive to transposition of $$\ M ~.$$

Consequently, a non-symmetric real matrix with only positive eigenvalues does not need to be positive definite. For example, the matrix $$M = \left[\begin{smallmatrix} 4 & 9 \\ 1 & 4 \end{smallmatrix}\right]$$ has positive eigenvalues yet is not positive definite; in particular a negative value of $$\mathbf{x}^\top M \mathbf{x}$$ is obtained with the choice $$\mathbf{x} = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right] $$ (which is the eigenvector associated with the negative eigenvalue of the symmetric part of $M$).

In summary, the distinguishing feature between the real and complex case is that, a bounded positive operator on a complex Hilbert space is necessarily Hermitian, or self adjoint. The general claim can be argued using the polarization identity. That is no longer true in the real case.

Heat conductivity matrix
Fourier's law of heat conduction, giving heat flux $$\ \mathbf{q}\ $$ in terms of the temperature gradient $$\ \mathbf{g} = \nabla T\ $$ is written for anisotropic media as $$\ \mathbf{q} = -K \mathbf{g}\ ,$$ in which $$\ K\ $$ is the symmetric thermal conductivity matrix. The negative is inserted in Fourier's law to reflect the expectation that heat will always flow from hot to cold. In other words, since the temperature gradient $$\ \mathbf{g}\ $$ always points from cold to hot, the heat flux $$\ \mathbf{q}\ $$ is expected to have a negative inner product with $$\ \mathbf{g}\ $$ so that $$\ \mathbf{q}^\top \mathbf{g} < 0 ~.$$ Substituting Fourier's law then gives this expectation as $$\ \mathbf{g}^\top K\mathbf{g} > 0\ ,$$ implying that the conductivity matrix should be positive definite.

title: Definite matrix


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Magnoliaceae

The Magnoliaceae are a flowering plant family, the magnolia family,  in the order Magnoliales. It consists of two genera: Magnolia and Liriodendron (tulip trees).

Unlike most angiosperms, whose flower parts are in whorls (rings), the Magnoliaceae have their stamens and pistils in spirals on a conical receptacle. This arrangement is found in some fossil plants and is believed to be a basal or early condition for angiosperms. The flowers also have parts not distinctly differentiated into sepals and petals, while angiosperms that evolved later tend to have distinctly differentiated sepals and petals. The poorly differentiated perianth parts that occupy both positions are known as tepals.

The family has about 219 species and ranges across subtropical eastern North America, Mexico and Central America, the West Indies, tropical South America, southern and eastern India, Sri Lanka, Indochina, Malesia, China, Japan, and Korea.

Genera
The number of genera in Magnoliaceae is a subject of debate. Up to 17 have been recognized, including Alcimandra, Lirianthe, Manglietia, Michelia, Pachylarnax, Parakmeria, Talauma and Yulania. However, many recent studies have opted to merge all genera within subfamily Magnolioideae into the genus Magnolia. Thus, Magnoliaceae would include only two extant genera, Magnolia and Liriodendron.

Description
The monophyly of Magnoliaceae is supported by a number of shared morphological characters among the various genera in the family. Most have bisexual flowers (with the exception of Kmeria and some species of Magnolia section Gynopodium), showy, fragrant, radial, and with an elongated receptacle. Leaves are alternate, simple, and sometimes lobed. The inflorescence is a solitary, showy flower with indistinguishable petals and sepals. Sepals range from six to many; stamens are numerous and feature short filaments which are poorly differentiated from the anthers. Carpels are usually numerous, distinct, and on an elongated receptacle or torus. The fruit is an etaerio of follicles which usually become closely appressed as they mature and open along the abaxial surface. Seeds have a fleshy coat, aril, and color that ranges from red to orange (except Liriodendron). Magnoliaceae flowers are beetle pollinated, except for Liriodendron, which is bee pollinated. The carpels of Magnolia flowers are especially thick to avoid damage by beetles that land, crawl, and feast on them. The seeds of Magnolioideae are bird-dispersed, while the seeds of Liriodendron are wind-dispersed.

Biogeography
Due to its great age, the geographical distribution of the Magnoliaceae has become disjunct or fragmented as a result of major geologic events such as ice ages, continental drift, and mountain formation. This distribution pattern has isolated some species, while keeping others in close contact. Extant species of the Magnoliaceae are widely distributed in temperate and tropical Asia from the Himalayas to Japan and southwest through Malaysia and New Guinea. Asia is home to about two-thirds of the species in Magnoliaceae, with the remainder of the family spread across the Americas with temperate species extending into southern Canada and tropical elements extending into Brazil and the West Indies.

Foundational Taxonomic and Systematics Research (18th-19th century)
The earliest botanical description of the Magnoliaceae as a family is in Antonii Laurentii de Jussieu's Genera Plantarum, which describes eight genera included within the family (Euryandra, Drymis, Illicium, Michelia, Magnolia, Talauma, Liriodendrum, and Mayna) as well as four genera closely related to the family (Dillenia, Curatella, Ochna, and Quassia). Bentham and Hooker's Genera Plantarum, almost a century later, sorts the family's genera into three tribes: the Wintereae, including the genera Drimys and Illicium, the Magnolieae, including the genera Talauma, Magnolia, Manglieta, Michelia, and Liriodendron, and the Schizandreae, including the genera Schizandra and Kadsura. In his following work Adansonia, Baillon recognizes Bentham and Hooker's changes and additions but proposes an alternative taxonomy where he sets aside the Tulipier genus and include all remaining genera under one Magnolieae tribe. From this basic separation, scholars have continued to debate the systematics of the family.

Modern Systematics Research (20th-21st century)
Dandy's taxonomic proposal in 1927 sets aside the genus Liriodendron as a part of the subfamily Liriodendreae and includes Bentham and Hooker's four genera in addition to four more (Kmeria, Pachylarnax, Alcimandra, and Elmerrillia) within the Magnolieae tribe. Dandy's model with eleven genera was widely accepted until molecular evidence brought it into question (Figlar, 2019). Qiu et al. analyzed molecular data in 1995 to investigate the divergences within and between East Asian and East North American species of Magnolia, presenting molecular evidence which shows that Dandy's section Rytidospermum is not monophyletic. Azuma et al. employ both molecular phylogeny and parsimonious mapping of the chemistry of floral scents in 1999 to propose a phylogenetic tree where, unlike Dandy's taxonomy, they include Michelia species within the Magnolia genus as a sister group to the subgenus Yulania and also find that the section Rytidospermum is not monophyletic, placing some of its members in a clade with the section Oyama.

The most recent research on the family continues the debate over the genera of the family. Wang et al.'s study analyzes complete chloroplast genome sequences of 86 species in the Magnoliaceae and supports a phylogeny with fifteen major clades, two subfamilies, two genera, and fifteen sections, maintaining Magnolia's classification as one monophyletic genus. Dong et al. also place Magnolia as the sole genus of the subfamily Magnolioideae made up of fifteen sections. However, Yang et al. and Zhao et al. work with phylogenies of the Magnoliaceae that recognize several genera in the Magnolioideae.

Consensus and Debates Today
Although phylogenetic trees of the Magnoliaceae still include anywhere from 2 to 17 genera, the broad generic concept (where one genus, Magnolia, is in the Magnolioideae) is largely accepted as a practical construction upheld by molecular and morphological evidence. Even as debates over rank persist, monophyletic groups are largely established with opportunities for further research into endangered and extinct species. The family's place as early angiosperms means that research into its taxonomy and evolutionary history contributes to our broader understanding of the evolution of plant life.

The development of DNA sequencing at the end of the 20th century had a profound impact on the research of phylogenetic relationships within the family. The employment of ndhF and cpDNA sequences has refuted many of the traditionally accepted phylogenetic relationships within the Magnoliaceae. For example, the genera Magnolia and Michelia were shown to be paraphyletic when the remaining four genera of the Magnolioideae are split out. In fact, even many of the subgenera (Magnolia subg. Magnolia, Magnolia subg. Talauma) have been found to be paraphyletic. Although no completely resolved phylogeny for the family has yet been determined, these technological advances have allowed systematists to broadly circumscribe major lineages.

Economic significance
As a whole, the Magnoliaceae are not an economically significant family. With the exception of ornamental cultivation, the economic significance of magnolias is generally confined to the use of wood from certain timber species and the use of bark and flowers from several species believed to possess medicinal qualities. The wood of the American tuliptree, Liriodendron tulipifera and the wood of the cucumbertree magnolia, Magnolia acuminata, and, to a lesser degree, that of the Frasier magnolia, Magnolia fraseri, are harvested and marketed collectively as "yellow poplar." This is a lightweight and exceptionally fine-grained wood, lending itself to precision woodworking for purposes such as pipe organ building.

Magnolias have a rich cultural tradition in China, where references to their healing qualities go back thousands of years. The Chinese have long used the bark of Magnolia officinalis, a magnolia native to the mountains of China with large leaves and fragrant white flowers, as a remedy for cramps, abdominal pain, nausea, diarrhea, and indigestion. Certain magnolia flowers, such as the buds of Magnolia liliiflora, have been used to treat chronic respiratory and sinus infections and lung congestion. Recently, magnolia bark has become incorporated into alternative medicine in the west, where tablets made from the bark of M. officinalis have been marketed as an aid for anxiety, allergies, asthma, and weight loss. Compounds found in magnolia bark might have antibacterial and antifungal properties, but no large-scale study on the health effects of magnolia bark or flowers has yet been conducted.

title: Magnoliaceae


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Cannoli

Cannoli is a Sicilian pastry consisting of a tube-shaped shell of fried pastry dough, filled with a sweet, creamy filling containing ricotta cheese. Its size ranges from 9 to 20 cm. In mainland Italy, it is commonly known as cannolo siciliano (lit. 'Sicilian cannoli').

In culinary traditions across Sicily, regional variations in cannoli fillings reflect local preferences and ingredient availability. In Palermo, cannoli are decorated with candied orange zest, adding a citrusy sweetness to the filling. In Catania, chopped pistachios are favored, adding a distinctive nutty flavor and texture. Ramacca is known for its purple artichokes, which also feature as filling in some cannoli recipes.

Etymology
In English, cannoli is used for the singular and the plural, but the plural cannolis is also common. In Italian, cannoli is plural; the singular is cannolo (cannolu, ), a diminutive meaning 'little tube', from canna, 'cane' or 'tube'. This form is uncommon in English and not even mentioned in most dictionaries.

History
Some food historians place the origins of cannoli in 827–1091 in Caltanissetta, Sicily, by the concubines of princes looking to capture their attention. This period marks the Arab rule of the island, known then as the Emirate of Sicily, giving rise to the theory that the etymology stemmed from the Arabic word qanawāt, 'tubes', in reference to their tube-shaped shells. During this time, the Arabs influenced Sicilian baking with the introduction of candied fruits, pistachios, and cinnamon. They also introduced the technique of combining nuts and fruits with sugar and honey.

Gaetano Basile claims that cannoli come from the Palermo and Messina areas and were historically prepared as a treat during Carnival season, possibly as a fertility symbol. The dessert eventually became a year-round staple in Sicily.

Some similar desserts in Middle Eastern tradition include "Zaynab's fingers" (أصابع زينب), which are filled with nuts, and qanawāt (قنوات), deep-fried dough tubes filled with various sweets, which were a popular pastry.

title: Cannoli