LexiconShiftInnovations/SinhalaWikipediaArticles huggingface dataset:
st) matrix contains information about distances of the edges.[63] These concepts can be applied to websites connected hyperlinks or cities connected by roads etc., in which case (unless the road network is extremely dense) the matrices tend to be sparse, i.e., contain few nonzero entries. Therefore, specifically tailored matrix algorithms can be used in network theory.විශ්ලේෂණය සහ ජ්‍යාමිතියThe Hessian matrix of a differentiable function ƒ: Rn → R consists of the second derivatives of ƒ with respect to the several coordinate directions, i.e.[64]H(f)=[∂2f∂xi∂xj].{\displaystyle H(f)=\left[{\frac {\partial ^{2}f}{\partial x_{i}\,\partial x_{j}}}\right].}At the saddle point (x = 0, y = 0) (red) of the function f(x,−y) = x2 − y2, the Hessian matrix [200−2]{\displaystyle {\begin{bmatrix}2&0\\0&-2\end{bmatrix}}} is indefinite.It encodes information about the local growth behaviour of the function:
given a critical point x = (x1, ..., xn), i.e., a point where the first partial derivatives ∂f/∂xi{\displaystyle \partial f/\partial x_{i}} of ƒ vanish, the function has a local minimum if the Hessian matrix is positive definite. Quadratic programming can be used to find global minima or maxima of quadratic functions closely related to the ones attached to matrices (see above).[65]Another matrix frequently used in geometrical situations is the Jacobi matrix of a differentiable map f: Rn → Rm. If f1, ..., fm denote the components of f, then the Jacobi matrix is defined as [66]J(f)=[∂fi∂xj]1≤i≤m,1≤j≤n.{\displaystyle J(f)=\left[{\frac {\partial f_{i}}{\partial x_{j}}}\right]_{1\leq i\leq m,1\leq j\leq n}.}If n > m, and if the rank of the Jacobi matrix attains its maximal value m, f is locally invertible at that point, by the implicit function theorem.[67]Partial differential equations can be classified by con
sidering 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.[68]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.[69]සම්භාවිතා සිද්ධාන්ත සහ සංඛ්‍යානයTwo different Markov chains. The chart depicts the number of particles (of a total of 1000) in state "2". Both limiting values can be determined from the transition matrices, which are given by [.70.31]{\displaystyle {\begin{bmatrix}.7&0\\.3&1\end{bmatrix}}} (red) and [.7.
2.3.8]{\displaystyle {\begin{bmatrix}.7&.2\\.3&.8\end{bmatrix}}} (black).Stochastic matrices are square matrices whose rows are probability vectors, i.e., whose entries sum up to one. Stochastic matrices are used to define Markov chains with finitely many states.[70] A row of the stochastic matrix gives the probability distribution for the next position of some particle currently in the state that corresponds to the row. Properties of the Markov chain like absorbing states, i.e., states that any particle attains eventually, can be read off the eigenvectors of the transition matrices.[71]Statistics also makes use of matrices in many different forms.[72] Descriptive statistics is concerned with describing data sets, which can often be represented in matrix form, by reducing the amount of data. The covariance matrix encodes the mutual variance of several random variables.[73] Another technique using matrices are linear least squares, a method that approximates a finit
e set of pairs (x1, y1), (x2, y2), ..., (xN, yN), by a linear functionyi ≈ axi + b, i = 1, ..., Nwhich can be formulated in terms of matrices, related to the singular value decomposition of matrices.[74]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.[75][76]සමමිති සහ භෞතික විද්‍යාව තුළ පරිණාමණවැඩිදුර තොරතුරු: Symmetry in physicsLinear 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.
[77] For the three lightest quarks, there is a group-theoretical representation involving the special unitary group SU(3); for their calculations, physicists use a convenient matrix representation known as the Gell-Mann matrices, which are also used for the SU(3) gauge group that forms the basis of the modern description of strong nuclear interactions, quantum chromodynamics. 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.[78]ක්වොන්ටම් ගති ස්වභාවයේ රේඛීය සංයෝජනThe first model of quantum mechanics (Heisenberg, 1925) represented the theory's operators by infinite-dimensional matrices acting on quantum states.[79] This is also referred to as matrix mechanics. One particular example is the density matrix that characterizes the "mixed" state of a quantum system as a lin
ear combination of elementary, "pure" eigenstates.[80]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.[81]සාමාන්‍ය ක්‍රමA 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 d
isplacement 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.[82] They are also needed for describing mechanical vibrations, and oscillations in electrical circuits.[83]ජ්‍යාමිතික දෘශ්ටි විද්‍යාවGeometrical 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: the vector's components are the light ray's slope and its distanc
e from the optical axis, while the matrix encodes the properties of the optical element. Actually, there are two kinds of matrices, viz. a refraction matrix describing the refraction at a lens surface, and a translation matrix, describing the translation of the plane of reference to the next refracting surface, where another refraction matrix applies.The optical system, consisting of a combination of lenses and/or reflective elements, is simply described by the matrix resulting from the product of the components' matrices.[84]ඉලෙක්ට්‍රෝනික විද්‍යාවTraditional mesh analysis in electronics leads 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 v1 and input current i1 as its elements, and let B be a 2-dimensional vector with the component's output voltage v2 and output current i2 as its elements. Then the behaviou
r of the electronic component can be described by  B = H · A, where H is a 2 x 2 matrix containing one impedance element (h12), one admittance element (h21) and two dimensionless elements (h11 and h22). Calculating a circuit now reduces to multiplying matrices.ඉතිහාසයMatrices have a long history of application in solving linear equations. The Chinese text The Nine Chapters on the Mathematical Art (Jiu Zhang Suan Shu), from between 300 BC and AD 200, is the first example of the use of matrix methods to solve simultaneous equations,[85] including the concept of determinants, over 1000 years before its publication by the Japanese mathematician Seki in 1683[තහවුරු කර නොමැත] and the German mathematician Leibniz in 1693. Cramer presented his rule in 1750.Early matrix theory emphasized determinants more strongly than matrices and an independent matrix concept akin to the modern notion emerged only in 1858, with Cayley's Memoir on the theory of matrices.[86][87] The term "matrix"
(Latin for "womb", derived from mater—mother[88]) was coined by Sylvester, 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.[89] In a 1851 paper, Sylvester explains:  I have in previous papers defined a "Matrix" as a rectangular array of terms, out of which different systems of determinants may be engendered as from the womb of a common parent.[90]The study of determinants sprang from several sources.[91] Number-theoretical problems led Gauss to relate coefficients of quadratic forms, i.e., expressions such as x2 + xy − 2y2, and linear maps in three dimensions to matrices. Eisenstein further developed these notions, including the remark that, in modern parlance, matrix products are non-commutative. Cauchy was the first to prove general statements about determinants, using as definition of the determinant of a matrix A = [ai
,j] the following: replace the powers ajk by ajk in the polynomiala1a2⋯an∏i<j(aj−ai),{\displaystyle a_{1}a_{2}\cdots a_{n}\prod _{i<j}(a_{j}-a_{i})\;,}where Π denotes the product of the indicated terms. He also showed, in 1829, that the eigenvalues of symmetric matrices are real.[92] Jacobi studied "functional determinants"—later called Jacobi determinants by Sylvester—which can be used to describe geometric transformations at a local (or infinitesimal) level, see above; Kronecker's Vorlesungen über die Theorie der Determinanten[93] and Weierstrass' Zur Determinantentheorie,[94] both published in 1903, first treated determinants axiomatically, as opposed to previous more concrete approaches such as the mentioned formula of Cauchy. At that point, determinants were firmly established.Many theorems were first established for small matrices only, for example the Cayley–Hamilton theorem was proved for 2×2 matrices by Cayley in the afor
ementioned memoir, and by Hamilton for 4×4 matrices. Frobenius, working on bilinear forms, generalized the theorem to all dimensions (1898). 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.[95] 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.[96] Later, von Neumann carried out the mathematical formulation of quantum mechanics, by further developing functional analytic notions such as linear operators on Hilbert spaces, which, very roughly speaking, correspond to Euclidean space, but with an infinity of independent directions.ලෝක ඉතිහාසයේ ගණිතය තුළ න්‍යසයන්හි වෙනත් භාවිතThe word has been used in unusual ways by a
t least two authors of historical importance.Bertrand Russell and Alfred North Whitehead in their Principia Mathematica (1910–1913) use the word matrix in the context of their Axiom of reducibility. They proposed this axiom as a means to reduce any function to one of lower type, successively, so that at the “bottom” (0 order) the function is identical to its extension:“Let us give the name of matrix to any function, of however many variables, which does not involve any apparent variables. Then any possible function other than a matrix is derived from a matrix by means of generalization, i.e., by considering the proposition which asserts that the function in question is true with all possible values or with some value of one of the arguments, the other argument or arguments remaining undetermined”.[97]For example a function Φ(x, y) of two variables x and y can be reduced to a collection of functions of a single variable, e.g., y, by “considering” the function for all possible values
of “individuals” ai substituted in place of variable x. And then the resulting collection of functions of the single variable y, i.e., ∀ai: Φ(ai, y), can be reduced to a “matrix” of values by “considering” the function for all possible values of “individuals” bi substituted in place of variable y:∀bj∀ai: Φ(ai, bj).Alfred Tarski in his 1946 Introduction to Logic used the word “matrix” synonymously with the notion of truth table as used in mathematical logic.[98]See alsoMathematics ද්වාරයAlgebraic multiplicityGeometric multiplicityGram-Schmidt processList of matricesMatrix calculusPeriodic matrix setTensorසටහන්↑ සැකිල්ල:Harvard citations. Alternative references for this book include සැකිල්ල:Harvard citations and සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල
්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ This is immediate from the definition of matrix multiplication.tr⁡(AB)=∑i=1m∑j=1nAijBji=tr⁡(BA).{\displaystyle \scriptstyle \operatorname {tr} ({\mathsf {AB}})=\sum _{i=1}^{m}\sum _{j=1}^{n}A_{ij}B_{ji}=\operatorname {tr} ({\mathsf {BA}}).}↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ For example, Mathematica
, see සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ See any standard reference in group.↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ See any reference in representation theory or group representation.↑ See the item "Matrix" in සැකිල්ල:Harvard citations↑ "Empty Matrix: A matrix is empty if either its row or column dimension is zero", Glossary සංරක්ෂණය කළ පිටපත 2009-04-29 at the Wayback Machine, O-Matrix v6 User Guide↑ "A matrix having at least one dimension equal to zero is called an empty matrix", MATLAB Data Structures සංරක
්ෂණය කළ පිටපත 2009-12-28 at the Wayback Machine↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations. For a more advanced, and more general statement see සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations. See also stiffness method.↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ Healy, Michael (1986), Matrices for Statistics, Oxford University Press, ISBN 9780198507024 ↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ see සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard
citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations cited by සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑  Merriam–Webster dictionary, Merriam–Webster, http://www.merriam-webster.com/dictionary/matrix, ප්‍රතිෂ්ඨාපනය April, 20th 2009 ↑ Per the OED the first usage of the word "matrix" with respect to mathematics appears in J. J. Sylvester in London, Edinb. & Dublin Philos. Mag. 37 (1850), p. 369:  "We ‥commence‥ with an oblong arrangement of terms consisting, suppose, of m lines and n columns. This will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants by fixing upon a number p, and selecting at will p lines and p columns, the squares corresponding to which may be termed determinants of the pth order.↑ The Collected Mathematical Papers of James Joseph Sylvester: 1837–1853, Paper 37, p. 247↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැ
කිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ සැකිල්ල:Harvard citations↑ Alfred North Whitehead and Bertrand Russell (1913) Principia Mathematica to *56, Cambridge at the University Press, Cambridge UK (republished 1962) cf page 162ff.↑ Tarski, Alfred 1946 Introduction to Logic and the Methodology of Deductive Sciences, Dover Publications, Inc, New York NY, ISBN 0-486-28462-X.↑ Eigen means "own" in ජර්මානු and in Dutch.↑ Additionally, the group is required to be closed in the general linear group.↑ "Not much of matrix theory carries over to infinite-dimensional spaces, and what does is not so useful, but it sometimes helps." සැකිල්ල:Harvard citationsනිර්දේශArnold, V. I.; Cooke, Roger (1992), Ordinary differential equations, Berlin, New York: Springer-Verlag, ISBN 978-3-540-54813-3 Artin, Michael (1991), Algebra, Prentice Hall, ISBN 978-0-89871-510-1 Association for Computing Machinery (1979), Computer Graphics, Tata Mc
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