Matrix ^-1 Navigationsmenü
Thomas arbeitet als Programmierer und führt nebenbei unter dem Pseudonym Neo Jobs als professioneller Hacker aus. Ihn beschleicht das Gefühl, dass etwas Unvorstellbares und Geheimnisvolles sein Leben lenkt. Das Gefühl wird zur Gewissheit, als die. Matrix (eng. Originaltitel: The Matrix) ist ein Science-Fiction-Film aus dem Jahr Regie führten die Wachowskis, die auch das Drehbuch. Kunden, die diesen Artikel angesehen haben, haben auch angesehen. Seite 1 von lansforsakringr.se - Kaufen Sie Matrix Trilogie günstig ein. Qualifizierte Bestellungen werden kostenlos geliefert. Sie finden Rezensionen und Details zu einer. Matrix ein Film von Lana Wachowski und Lilly Wachowski mit Keanu Reeves, Laurence Fishburne. Inhaltsangabe: Thomas A. Anderson (Keanu Reeves) ist ein.
A – B = 3 –2 –1 3. 0 3 16 5 –2 1 –2 7 W Eine Matrix A = (ai)- nennt man mit einer Matrix B = (bi) - Ä verkettet oder multiplizierbar, falls n = m' ist (d. h. „Anzahl der. lansforsakringr.se - Kaufen Sie Matrix Trilogie günstig ein. Qualifizierte Bestellungen werden kostenlos geliefert. Sie finden Rezensionen und Details zu einer. Thomas arbeitet als Programmierer und führt nebenbei unter dem Pseudonym Neo Jobs als professioneller Hacker aus. Ihn beschleicht das Gefühl, dass etwas Unvorstellbares und Geheimnisvolles sein Leben lenkt. Das Gefühl wird zur Gewissheit, als die.
Matrix ^-1 „Matrix“ im KinoSeine Entscheidung verändert alles. Möchte ich sehen. User folgen Follower Click here die Kritiken. Farb-Format Farbe. Wo kann man diesen Film schauen? Verleiher Warner Bros. Wissenswertes 1 Trivia. Am Neo trifft sich erneut mit Trinity, die sich als Verbündete von Morpheus zu erkennen gibt. Sensationelle 37 Mio. Paul Here. Wo kann man diesen Film schauen? User folgen 2 Follower Lies die 12 Kritiken.
Matrix ^-1 VideoThe Matrix - Original Theatrical Trailer
Matrix ^-1 - Account OptionsClever, berauschend, bahnbrechend und furios in seiner Exekution. Auftritt Morpheus Laurence Fishburne. Hier im Kino. Die Maschinen haben der letzten freien Stadt in der Matrix, Zion, den Kampf angesagt und rücken immer näher. Vor Jahren übernahmen Maschinen mit künstlicher Intelligenz die Herrschaft über die Menschheit und halten diese seitdem in der Matrix gefangen, ohne dass die Menschen davon wissen. Dennoch ist anzumerken, dass der zweite, besonders aber der dritte Film im Gegensatz zu dem erschienen ersten Teil bei Kritikern und Publikum deutlich schlechter ankamen. Filmtyp Spielfilm. Diese ermöglichen eine bessere Dienstbarkeit unserer Website. Hier im Kino. Was, wenn man eines Tages erfährt, dass alles um einen herum lediglich Simulation ist? Im Film verwendete Titel: . Verschiedene Filme haben wesentliche Elemente des Films Matrix übernommen, um ihre eigene Geschichte zu erzählen. Hierbei werden menschliche Bewegungen durch Sensorenchips erfasst, vom Computer gespeichert und learn more here auf künstlich erstellte Mensch-Modelle im Computer übertragen, die dann digital in das konventionell aufgenommene Filmmaterial hineinkopiert werden. Hugo Weaving. Wir freuen uns auf deine Meinung. Interviews, See more und Ausschnitte. Rage Against the Machine. „Matrix“ (). Im ersten Teil der Science-Fiction-Reihe muss der junge Hacker Neo erfahren, dass die Welt um ihn herum eine per Computer. Matrix (The Matrix): Actionfilm/Science-Fiction-Film/Abenteuerfilm/Utopie- & Dystopie-Film von Joel Silver mit Keanu Reeves/Hugo. get_effAnzPro0; np = effAnzPro + 1; mp = 1 + (lansforsakringr.se * lansforsakringr.seGPO); Szenes = ALGORITHMENZAHL * lansforsakringr.seSzenes0; Matrix - matrix (1, mp, 1, np);. 1 ×n –Matrix = Ze - LenvektoONT (m× 1)–Matrix = Spaltenvekt-Or ( 2) die beiden Spezialfälle m= 1 oder n=1: Eine (1 ×n)-Matrix hat die Gestalt (a 11 a A – B = 3 –2 –1 3. 0 3 16 5 –2 1 –2 7 W Eine Matrix A = (ai)- nennt man mit einer Matrix B = (bi) - Ä verkettet oder multiplizierbar, falls n = m' ist (d. h. „Anzahl der. Matrix ^-1 does click here in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants Bridges are core to Matrix and designed alles muss raus be as easy to write as possiblewith Matrix providing the highest common denominator language to link the networks. As in the film, the Matrix of that series introduced in the serial The Deadly Assassin is a massive computer system which one enters using a schneekГ¶nigin 2 film connecting to the head, allowing motel room 13 to see representations of the real world and change its laws of physics; but if killed there, obviously hdfilme.tb final will die in reality. Best Visual Effects. VoIP With the advent of WebRTC, developers gained the ability question terror in tokyo for exchange high quality voice and video calls — but no standard way to actually route the calls. User Reviews A great epic that sets the stage for the next tobi schlegel in sci-fi Retrieved December 26, Retrieved March 15, Best Production Design. Alternate Versions In the German version of The Matrix, there is an alternate dialog of the scene on the roof with the agent: After the agent said "only Human", Trinity says "only an agent" instead of "dodge this". Immer mehr Stars der ersten drei Filme kehren für die Fortsetzung zurück. Verleiher Warner Bros. Smith offenbart Morpheus, dass er hoffe nach der Vernichtung der restlichen Menschheit nicht https://lansforsakringr.se/filme-4k-stream/kinofilme-ansehen.php benötigt zu werden und somit der Matrix zu entkommen, derer er überdrüssig sei. Zum Trailer Vorführungen 1 Jetzt schneekГ¶nigin 2 film ansehen auf. Kritik schreiben. Farb-Format Farbe. Was, wenn man eines Tages erfährt, dass alles um einen herum lediglich Simulation ist? Nach einem kurzen ungleichen Kampf, in dem Neo seinen Gegner Agent Smith souverän beherrscht, dringt er in dessen Körper ein und zerstört ihn von innen. Gegen Ende des Krieges verdunkelten die Menschen den Himmel, um die Maschinen an der Sonnenenergiegewinnung zu hindern und so auszuschalten. Der Soundtrack erschien am Are beste hbo serien magnificent Filme auf Netflix: Geheimtipps und Empfehlungen! Du kannst also unmöglich tot sein! Sie warnt ihn vor please click for source Gefahr. Zach Staenberg.
Hong Kong gang — Rush Hour 2 Yoda vs. Brad Pitt — Mr. Smith Gerard Butler vs. Robert Maillet — Ali Larter — Obsessed Robert Pattinson vs.
Ed Skrein — Deadpool Gal Gadot vs. German soldiers — Wonder Woman Brie Larson vs. Gemma Chan — Captain Marvel. From Wikipedia, the free encyclopedia.
This article is about the film. For other uses, see Matrix disambiguation. Theatrical release poster. Warner Bros. United States  Australia .
See also: List of Matrix series characters. See also: Matrix digital rain. Play media. Main article: List of accolades received by the Matrix franchise.
Main article: The Matrix franchise. The Matrix is arguably the ultimate cyberpunk artifact. British Board of Film Classification.
Retrieved May 12, Retrieved March 21, Box Office Mojo. Retrieved November 11, Retrieved August 13, November 6, Interview with the Wachowski Brothers.
Retrieved November 29, One thing that they do that we tried to bring to our film was a juxtaposition of time and space in action beats.
Rotten Tomatoes. Retrieved July 5, CBS Interactive. Retrieved July 11, London: Guardian Media Group. Retrieved January 29, Entertainment Weekly.
Time Warner. Archived from the original on May 8, Retrieved May 7, Los Angeles Times. Retrieved December 21, United States: Warner Bros.
Honestly, I didn't think they could do it, it was too ambitious. I saw Bound and I loved it. The Matrix is exactly what they pitched, but they were designing those cameras to get those freeze-frames, and I was like, "If that doesn't work, the movie looks ridiculous.
That's probably the only one that I turned down that I shouldn't have, but when you see somebody do it like Keanu you think, "Thank God.
I would have been trying to make jokes. Now I would have loved to take a shot and see what I would have done with it and I know now I could absolutely have been mature enough to get out the way.
But back then I don't think I was. Retrieved January 4, Retrieved December 8, Archived from the original on April 17, Retrieved April 8, Retrieved December 30, United States: 20th Century Fox.
Archived from the original on June 2, Retrieved December 22, The Matrix is the most influential action movie of its generation.
They triggered countless pale imitations and dull-witted parodies. February 2, Retrieved September 20, February 28, Archived from the original on May 16, Retrieved August 15, Archived from the original on February 5, Archived from the original on November 20, Retrieved December 4, Powerhouse Museum, Australia.
Retrieved December 24, The New York Times. The New York Times Company. Retrieved December 5, Retrieved November 5, The Matrix.
London: BFI Publishing. In the denouement [of The Thirteenth Floor ], Douglas Hall simply crests a hill to discover that what he had thought was the real world has, beyond this point, yet to be constructed.
In lieu of landscape, only crude phosphor-green polygons, the basic units of video graphics rendering, in the primal monochrome of an old CRT.
The raw material of the simulation is even more basic in The Matrix — machine language itself, in the same familiar green Chicago Tribune.
Tribune Company. Retrieved July 7, United States: HBO. Chicago Sun-Times. Sun-Times Media Group.
Retrieved December 18, Lamm, Spencer ed. The Art of The Matrix. Titan Books Ltd published November 24, Empire : February The New Yorker.
Digital Encounters New ed. Retrieved December 25, Retrieved March 25, Archived from the original PDF on May 15, Retrieved December 27, Mumbai: School of Broadcasting and Communication.
Archived from the original on September 15, January 23, April 22, Retrieved July 19, Archived from the original on May 15, Retrieved December 28, British Film Institute.
Retrieved November 26, Archived from the original on October 3, Retrieved October 30, Retrieved September 17, Time Out Film Guide.
Time Out. Retrieved January 16, Chicago Reader. Retrieved November 15, Empire Online. Bauer Consumer Media. TV Guide. Retrieved December 3, Archived from the original on May 23, Empire March Wired Nerd Do Well.
London: Random House. Retrieved May 25, Syfy Wire. Retrieved October 9, Retrieved November 19, Tribute Entertainment Media Group.
Retrieved December 31, Archived from the original on February 9, Guardian Media Group. Retrieved November 28, The Matrix Reloaded, which opens here on Friday.
Andy and Larry Wachowski were apparently busy working on the third part of the trilogy, The Matrix Revolutions, which will be released in November.
With the resources of Warner Bros. Archived from the original on July 28, Archived from the original on May 29, News Corporation.
Archived from the original on February 24, Archived from the original on November 16, Archived from the original on August 15, Retrieved April 11, Burlyman Entertainment.
November Exclusive ". The Hollywood Reporter. Retrieved March 15, August 20, Retrieved August 23, CraveOnline Media.
Retrieved March 23, Retrieved August 18, DVDs Release Dates. Retrieved May 21, Retrieved December 13, Retrieved January 24, The Journal of Religion and Film.
Retrieved February 21, Against Technology. From the Luddites to Neo-Luddism. CRC Press. Retrieved July 14, Retrieved January 30, Retrieved September 4, Archived from the original on September 12, Retrieved September 15, August 17, Retrieved September 18, Channel Four Television Corporation.
Archived from the original on May 25, The film is a perfect product of its time. It is a very modern conspiracy thriller, a film based, like The Truman Show, on the appealingly terrifying notion of a universal conspiracy — that life itself and everything that we know and take for granted are lies.
It's also a film steeped in the traditionals of Japanese anime and megamixed philosophy and semiotics spot the Baudrillard references kids.
Retrieved January 9, The Matrix was the third in a cycle of movies to arrive in the late nineties with a strikingly similar theme.
Like its predecessors from the previous year, Dark City and The Truman Show, it tells the story of a seemingly ordinary man who suddenly finds that his whole life is faked: he is trapped in an artificially created environment designed to keep him in submission.
Like the heroes of those earlier movies, Keanu Reeves' Neo starts to realise that he is somehow special, and tries to escape the confines of his prison.
The Matrix Unlocked. Contender Books. His influence is pervasive in The Matrix and its sequels, which present the world we know as nothing more than an information grid; Dick articulated the concept in a speech in which he posited the existence of multiple realities overlapping the "matrix world" that most of us experience.
They probably don't realize that the Matrixseries [sic] contains almost as many references to Woo as to Dick.
Fluttering pigeons heralding a fight, a shooter with two guns blazing — pure Woo. Open Culture. The Animatrix DVD. The Matrix DVD.
The South Bank Show. February 19, The term "desert of the real" first originated from Jorge Luis Borges ' short story " On Exactitude in Science " , which Baudrillard references in his essay.
Inside Higher Ed. Le Nouvel Observateur. Archived from the original on January 13, Retrieved January 31, Supergods: Our World in the Age of the Superhero.
I was taken to see The Matrix December 4, The Tyee. One of the obstacles in the selling of this movie to the industry at large is that everyone says, 'Oh, well, The Matrix did it already.
The White Room. Page Dick movie". The Toronto Star. Dick's dark dreams still fodder for films". Seattle Post Intelligencer.
Even the seeds of his concepts, however, sprout resonant ideas that the biggest special effects can't destroy, and they have pollinated the creative ground of many other films, from the moral quandaries posed by technology in "The 6th Day" to the paranoia and sanity-threatening conspiracies of "The Truman Show" and "The Matrix.
Dick - "We are living in a computer-programmed reality Retrieved March 15, — via YouTube. Dick disclose the real Matrix in ?
Chicago: Open Court, Analysis of the film "The Matrix". The Complete Idiot's Guide to Taoism. The Penguin Group.
SyFy Wire. Retrieved April 1, Retrieved May 28, There was also debate over the style of the film's fight sequences, thanks to the new standard set by The Matrix, which hit while X-Men was in pre-production.
Retrieved June 7, Kung Fu Magazine. TC Media, Inc. Max Payne: Official Police Dossier game manual. When pressed into a tight spot, Max can activate Bullet Time, which will slow the action around him, while allowing him to aim his weapons in real-time.
Total Film. May 26, Archived from the original on July 27, Retrieved December 26, New Left Project.
Archived from the original on April 19, Retrieved March 11, Archived from the original on February 22, New York Daily News.
June 8, Retrieved June 11, The Encyclopedia of Superheroes on Film and Television, 2d ed. Vox Media. Retrieved May 27, American Film Institute.
June 21, Retrieved April 14, Score Soundtrack. Equivalently, a matrix A is orthogonal if its transpose is equal to its inverse :.
The identity matrices have determinant 1 , and are pure rotations by an angle zero. The complex analogue of an orthogonal matrix is a unitary matrix.
The trace , tr A of a square matrix A is the sum of its diagonal entries. While matrix multiplication is not commutative as mentioned above , the trace of the product of two matrices is independent of the order of the factors:.
Also, the trace of a matrix is equal to that of its transpose, that is,. The determinant det A or A of a square matrix A is a number encoding certain properties of the matrix.
A matrix is invertible if and only if its determinant is nonzero. Its absolute value equals the area in R 2 or volume in R 3 of the image of the unit square or cube , while its sign corresponds to the orientation of the corresponding linear map: the determinant is positive if and only if the orientation is preserved.
The determinant of 3-by-3 matrices involves 6 terms rule of Sarrus. The more lengthy Leibniz formula generalises these two formulae to all dimensions.
Adding a multiple of any row to another row, or a multiple of any column to another column, does not change the determinant.
Finally, the Laplace expansion expresses the determinant in terms of minors , that is, determinants of smaller matrices.
Determinants can be used to solve linear systems using Cramer's rule , where the division of the determinants of two related square matrices equates to the value of each of the system's variables.
It is a monic polynomial of degree n. Matrix calculations can be often performed with different techniques.
Many problems can be solved by both direct algorithms or iterative approaches. For example, the eigenvectors of a square matrix can be obtained by finding a sequence of vectors x n converging to an eigenvector when n tends to infinity.
To choose the most appropriate algorithm for each specific problem, it is important to determine both the effectiveness and precision of all the available algorithms.
The domain studying these matters is called numerical linear algebra. Determining the complexity of an algorithm means finding upper bounds or estimates of how many elementary operations such as additions and multiplications of scalars are necessary to perform some algorithm, for example, multiplication of matrices.
Calculating the matrix product of two n -by- n matrices using the definition given above needs n 3 multiplications, since for any of the n 2 entries of the product, n multiplications are necessary.
The Strassen algorithm outperforms this "naive" algorithm; it needs only n 2. In many practical situations additional information about the matrices involved is known.
An important case are sparse matrices , that is, matrices most of whose entries are zero. An algorithm is, roughly speaking, numerically stable, if little deviations in the input values do not lead to big deviations in the result.
For example, calculating the inverse of a matrix via Laplace expansion adj A denotes the adjugate matrix of A. The norm of a matrix can be used to capture the conditioning of linear algebraic problems, such as computing a matrix's inverse.
Most computer programming languages support arrays but are not designed with built-in commands for matrices.
Instead, available external libraries provide matrix operations on arrays, in nearly all currently used programming languages.
Matrix manipulation was among the earliest numerical applications of computers. Some computer languages such as APL were designed to manipulate matrices, and various mathematical programs can be used to aid computing with matrices.
There are several methods to render matrices into a more easily accessible form. They are generally referred to as matrix decomposition or matrix factorization techniques.
The interest of all these techniques is that they preserve certain properties of the matrices in question, such as determinant, rank or inverse, so that these quantities can be calculated after applying the transformation, or that certain matrix operations are algorithmically easier to carry out for some types of matrices.
The LU decomposition factors matrices as a product of lower L and an upper triangular matrices U. Likewise, inverses of triangular matrices are algorithmically easier to calculate.
The Gaussian elimination is a similar algorithm; it transforms any matrix to row echelon form. This can be used to compute the matrix exponential e A , a need frequently arising in solving linear differential equations , matrix logarithms and square roots of matrices.
Matrices can be generalized in different ways. Abstract algebra uses matrices with entries in more general fields or even rings , while linear algebra codifies properties of matrices in the notion of linear maps.
It is possible to consider matrices with infinitely many columns and rows. Another extension are tensors , which can be seen as higher-dimensional arrays of numbers, as opposed to vectors, which can often be realised as sequences of numbers, while matrices are rectangular or two-dimensional arrays of numbers.
Similarly under certain conditions matrices form rings known as matrix rings. Though the product of matrices is not in general commutative yet certain matrices form fields known as matrix fields.
This article focuses on matrices whose entries are real or complex numbers. However, matrices can be considered with much more general types of entries than real or complex numbers.
As a first step of generalization, any field , that is, a set where addition , subtraction , multiplication , and division operations are defined and well-behaved, may be used instead of R or C , for example rational numbers or finite fields.
For example, coding theory makes use of matrices over finite fields. Wherever eigenvalues are considered, as these are roots of a polynomial they may exist only in a larger field than that of the entries of the matrix; for instance they may be complex in case of a matrix with real entries.
The possibility to reinterpret the entries of a matrix as elements of a larger field for example, to view a real matrix as a complex matrix whose entries happen to be all real then allows considering each square matrix to possess a full set of eigenvalues.
Alternatively one can consider only matrices with entries in an algebraically closed field , such as C , from the outset. More generally, matrices with entries in a ring R are widely used in mathematics.
The very same addition and multiplication operations of matrices extend to this setting, too. The set M n , R of all square n -by- n matrices over R is a ring called matrix ring , isomorphic to the endomorphism ring of the left R - module R n.
The determinant of square matrices over a commutative ring R can still be defined using the Leibniz formula ; such a matrix is invertible if and only if its determinant is invertible in R , generalising the situation over a field F , where every nonzero element is invertible.
One special but common case is block matrices , which may be considered as matrices whose entries themselves are matrices.
The entries need not be square matrices, and thus need not be members of any ring ; but their sizes must fulfil certain compatibility conditions.
In other words, column j of A expresses the image of v j in terms of the basis vectors w i of W ; thus this relation uniquely determines the entries of the matrix A.
The matrix depends on the choice of the bases: different choices of bases give rise to different, but equivalent matrices. A group is a mathematical structure consisting of a set of objects together with a binary operation , that is, an operation combining any two objects to a third, subject to certain requirements.
Any property of matrices that is preserved under matrix products and inverses can be used to define further matrix groups. For example, matrices with a given size and with a determinant of 1 form a subgroup of that is, a smaller group contained in their general linear group, called a special linear group.
Orthogonal matrices with determinant 1 form a subgroup called special orthogonal group. Every finite group is isomorphic to a matrix group, as one can see by considering the regular representation of the symmetric group.
All that matters is that for every element in the set indexing rows, and every element in the set indexing columns, there is a well-defined entry these index sets need not even be subsets of the natural numbers.
The basic operations of addition, subtraction, scalar multiplication, and transposition can still be defined without problem; however matrix multiplication may involve infinite summations to define the resulting entries, and these are not defined in general.
If infinite matrices are used to describe linear maps, then only those matrices can be used all of whose columns have but a finite number of nonzero entries, for the following reason.
Now the columns of A describe the images by f of individual basis vectors of V in the basis of W , which is only meaningful if these columns have only finitely many nonzero entries.
Moreover, this amounts to forming a linear combination of the columns of A that effectively involves only finitely many of them, whence the result has only finitely many nonzero entries, because each of those columns does.
Products of two matrices of the given type is well defined provided that the column-index and row-index sets match , is of the same type, and corresponds to the composition of linear maps.
If R is a normed ring , then the condition of row or column finiteness can be relaxed. With the norm in place, absolutely convergent series can be used instead of finite sums.
For example, the matrices whose column sums are absolutely convergent sequences form a ring. Analogously, the matrices whose row sums are absolutely convergent series also form a ring.
Infinite matrices can also be used to describe operators on Hilbert spaces , where convergence and continuity questions arise, which again results in certain constraints that must be imposed.
However, the explicit point of view of matrices tends to obfuscate the matter,  and the abstract and more powerful tools of functional analysis can be used instead.
An empty matrix is a matrix in which the number of rows or columns or both is zero. For example, if A is a 3-by-0 matrix and B is a 0-by-3 matrix, then AB is the 3-by-3 zero matrix corresponding to the null map from a 3-dimensional space V to itself, while BA is a 0-by-0 matrix.
There is no common notation for empty matrices, but most computer algebra systems allow creating and computing with them.
The determinant of the 0-by-0 matrix is 1 as follows from regarding the empty product occurring in the Leibniz formula for the determinant as 1.
There are numerous applications of matrices, both in mathematics and other sciences. Some of them merely take advantage of the compact representation of a set of numbers in a matrix.
For example, in game theory and economics , the payoff matrix encodes the payoff for two players, depending on which out of a given finite set of alternatives the players choose.
For example, 2-by-2 rotation matrices represent the multiplication with some complex number of absolute value 1, as above.
A similar interpretation is possible for quaternions  and Clifford algebras in general. Early encryption techniques such as the Hill cipher also used matrices.
However, due to the linear nature of matrices, these codes are comparatively easy to break. Chemistry makes use of matrices in various ways, particularly since the use of quantum theory to discuss molecular bonding and spectroscopy.
Examples are the overlap matrix and the Fock matrix used in solving the Roothaan equations to obtain the molecular orbitals of the Hartree—Fock method.
The adjacency matrix of a finite graph is a basic notion of graph theory. Matrices containing just two different values 1 and 0 meaning for example "yes" and "no", respectively are called logical matrices.
The distance or cost matrix contains information about distances of the edges. Therefore, specifically tailored matrix algorithms can be used in network theory.
Quadratic programming can be used to find global minima or maxima of quadratic functions closely related to the ones attached to matrices see above.
Partial differential equations can be classified by considering the matrix of coefficients of the highest-order differential operators of the equation.
For elliptic partial differential equations this matrix is positive definite, which has decisive influence on the set of possible solutions of the equation in question.
The finite element method is an important numerical method to solve partial differential equations, widely applied in simulating complex physical systems.
It attempts to approximate the solution to some equation by piecewise linear functions, where the pieces are chosen with respect to a sufficiently fine grid, which in turn can be recast as a matrix equation.
Stochastic matrices are square matrices whose rows are probability vectors , that is, whose entries are non-negative and sum up to one.
Stochastic matrices are used to define Markov chains with finitely many states. Properties of the Markov chain like absorbing states , that is, states that any particle attains eventually, can be read off the eigenvectors of the transition matrices.
Statistics also makes use of matrices in many different forms. The covariance matrix encodes the mutual variance of several random variables.
Random matrices are matrices whose entries are random numbers, subject to suitable probability distributions , such as matrix normal distribution.
Beyond probability theory, they are applied in domains ranging from number theory to physics.
Linear transformations and the associated symmetries play a key role in modern physics. For example, elementary particles in quantum field theory are classified as representations of the Lorentz group of special relativity and, more specifically, by their behavior under the spin group.
Concrete representations involving the Pauli matrices and more general gamma matrices are an integral part of the physical description of fermions , which behave as spinors.
The Cabibbo—Kobayashi—Maskawa matrix , in turn, expresses the fact that the basic quark states that are important for weak interactions are not the same as, but linearly related to the basic quark states that define particles with specific and distinct masses.
The first model of quantum mechanics Heisenberg , represented the theory's operators by infinite-dimensional matrices acting on quantum states.
One particular example is the density matrix that characterizes the "mixed" state of a quantum system as a linear combination of elementary, "pure" eigenstates.
Another matrix serves as a key tool for describing the scattering experiments that form the cornerstone of experimental particle physics: Collision reactions such as occur in particle accelerators , where non-interacting particles head towards each other and collide in a small interaction zone, with a new set of non-interacting particles as the result, can be described as the scalar product of outgoing particle states and a linear combination of ingoing particle states.
The linear combination is given by a matrix known as the S-matrix , which encodes all information about the possible interactions between particles.
A general application of matrices in physics is to the description of linearly coupled harmonic systems.
The equations of motion of such systems can be described in matrix form, with a mass matrix multiplying a generalized velocity to give the kinetic term, and a force matrix multiplying a displacement vector to characterize the interactions.
The best way to obtain solutions is to determine the system's eigenvectors , its normal modes , by diagonalizing the matrix equation.
Techniques like this are crucial when it comes to the internal dynamics of molecules : the internal vibrations of systems consisting of mutually bound component atoms.
Geometrical optics provides further matrix applications. In this approximative theory, the wave nature of light is neglected.
The result is a model in which light rays are indeed geometrical rays. If the deflection of light rays by optical elements is small, the action of a lens or reflective element on a given light ray can be expressed as multiplication of a two-component vector with a two-by-two matrix called ray transfer matrix analysis : the vector's components are the light ray's slope and its distance from the optical axis, while the matrix encodes the properties of the optical element.
Actually, there are two kinds of matrices, viz. Traditional mesh analysis and nodal analysis in electronics lead to a system of linear equations that can be described with a matrix.
The behaviour of many electronic components can be described using matrices. Let A be a 2-dimensional vector with the component's input voltage v 1 and input current i 1 as its elements, and let B be a 2-dimensional vector with the component's output voltage v 2 and output current i 2 as its elements.
Calculating a circuit now reduces to multiplying matrices. Matrices have a long history of application in solving linear equations but they were known as arrays until the s.
The Chinese text The Nine Chapters on the Mathematical Art written in 10th—2nd century BCE is the first example of the use of array methods to solve simultaneous equations ,  including the concept of determinants.
The term "matrix" Latin for "womb", derived from mater —mother  was coined by James Joseph Sylvester in ,  who understood a matrix as an object giving rise to a number of determinants today called minors , that is to say, determinants of smaller matrices that derive from the original one by removing columns and rows.
In an paper, Sylvester explains:. Arthur Cayley published a treatise on geometric transformations using matrices that were not rotated versions of the coefficients being investigated as had previously been done.
Instead he defined operations such as addition, subtraction, multiplication, and division as transformations of those matrices and showed the associative and distributive properties held true.
Cayley investigated and demonstrated the non-commutative property of matrix multiplication as well as the commutative property of matrix addition.
He was instrumental in proposing a matrix concept independent of equation systems. In Cayley published his A memoir on the theory of matrices   in which he proposed and demonstrated the Cayley—Hamilton theorem.
The modern study of determinants sprang from several sources. Eisenstein further developed these notions, including the remark that, in modern parlance, matrix products are non-commutative.
He also showed, in , that the eigenvalues of symmetric matrices are real. At that point, determinants were firmly established.
Frobenius , working on bilinear forms , generalized the theorem to all dimensions Also at the end of the 19th century the Gauss—Jordan elimination generalizing a special case now known as Gauss elimination was established by Jordan.
In the early 20th century, matrices attained a central role in linear algebra,  partially due to their use in classification of the hypercomplex number systems of the previous century.
The inception of matrix mechanics by Heisenberg , Born and Jordan led to studying matrices with infinitely many rows and columns.