DefinitionA ''matrix'' is a rectangular array of s (or other mathematical objects), called the ''entries'' of the matrix. Matrices are subject to standard such as and . Most commonly, a matrix over a ''F'' is a rectangular array of of ''F''. A real matrix and a complex matrix are matrices whose entries are respectively s or s. More general types of entries are discussed . For instance, this is a real matrix: : The numbers, symbols, or expressions in the matrix are called its ''entries'' or its ''elements''. The horizontal and vertical lines of entries in a matrix are called ''rows'' and ''columns'', respectively.
SizeThe size of a matrix is defined by the number of rows and columns it contains. There is no limit to the numbers of rows and columns a matrix (in the usual sense) can have as long as they are positive integers. A matrix with ''m'' rows and ''n'' columns is called an ''m''×''n'' matrix, or ''m''-by-''n'' matrix, while ''m'' and ''n'' are called its ''dimensions''. For example, the matrix A above is a 3×2 matrix. Matrices with a single row are called ''s'', and those with a single column are called '. A matrix with the same number of rows and columns is called a '. A matrix with an infinite number of rows or columns (or both) is called an . In some contexts, such as , it is useful to consider a matrix with no rows or no columns, called an .
NotationMatrices are commonly written in s or : : The specifics of symbolic matrix notation vary widely, with some prevailing trends. Matrices are usually symbolized using letters (such as A in the examples above), while the corresponding letters, with two subscript indices (e.g., ''a'', or ''a''), represent the entries. In addition to using upper-case letters to symbolize matrices, many authors use a special , commonly boldface upright (non-italic), to further distinguish matrices from other mathematical objects. An alternative notation involves the use of a double-underline with the variable name, with or without boldface style (as in the case of ). The entry in the ''i''-th row and ''j''-th column of a matrix A is sometimes referred to as the ''i'',''j'', (''i'',''j''), or (''i'',''j'')th entry of the matrix, and most commonly denoted as ''a'', or ''a''. Alternative notations for that entry are ''A'' 'i,j''or ''A''. For example, the (1,3) entry of the following matrix A is 5 (also denoted ''a'', ''a'', ''A'' '1,3''or ''A''): : Sometimes, the entries of a matrix can be defined by a formula such as ''a'' = ''f''(''i'', ''j''). For example, each of the entries of the following matrix A is determined by the formula ''a'' = ''i'' − ''j''. : In this case, the matrix itself is sometimes defined by that formula, within square brackets or double parentheses. For example, the matrix above is defined as A = 'i''−''j'' or A = ((''i''−''j'')). If matrix size is ''m'' × ''n'', the above-mentioned formula ''f''(''i'', ''j'') is valid for any ''i'' = 1, ..., ''m'' and any ''j'' = 1, ..., ''n''. This can be either specified separately, or indicated using ''m'' × ''n'' as a subscript. For instance, the matrix A above is 3 × 4, and can be defined as A = 'i'' − ''j''(''i'' = 1, 2, 3; ''j'' = 1, ..., 4), or A = 'i'' − ''j'' Some programming languages utilize doubly subscripted arrays (or arrays of arrays) to represent an ''m''-×-''n'' matrix. Some programming languages start the numbering of array indexes at zero, in which case the entries of an ''m''-by-''n'' matrix are indexed by and . This article follows the more common convention in mathematical writing where enumeration starts from 1. An asterisk is occasionally used to refer to whole rows or columns in a matrix. For example, ''a'' refers to the i row of A, and ''a'' refers to the j column of A. The of all ''m''-by-''n'' real matrices is often denoted or The set of all ''m''-by-''n'' matrices matrices over another or over a , is similarly denoted or If , that is, in the case of , one does not repeat the dimension: or Often, is used in place of
Basic operationsThere are a number of basic operations that can be applied to modify matrices, called ''matrix addition'', ''scalar multiplication'', ''transposition'', ''matrix multiplication'', ''row operations'', and ''submatrix''.
Addition, scalar multiplication, and transpositionFamiliar properties of numbers extend to these operations of matrices: for example, addition is , that is, the matrix sum does not depend on the order of the summands: A+B=B+A. The transpose is compatible with addition and scalar multiplication, as expressed by (''c''A) = ''c''(A) and (A+B)=A+B. Finally, (A)=A.
Matrix multiplication''Multiplication'' of two matrices is defined if and only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If A is an ''m''-by-''n'' matrix and B is an ''n''-by-''p'' matrix, then their ''matrix product'' AB is the ''m''-by-''p'' matrix whose entries are given by of the corresponding row of A and the corresponding column of B: : where 1 ≤ ''i'' ≤ ''m'' and 1 ≤ ''j'' ≤ ''p''. For example, the underlined entry 2340 in the product is calculated as : Matrix multiplication satisfies the rules (AB)C = A(BC) (), and (A + B)C = AC + BC as well as C(A + B) = CA + CB (left and right ), whenever the size of the matrices is such that the various products are defined. The product AB may be defined without BA being defined, namely if A and B are ''m''-by-''n'' and ''n''-by-''k'' matrices, respectively, and Even if both products are defined, they generally need not be equal, that is: :AB ≠ BA, In other words, matrix multiplication is not , in marked contrast to (rational, real, or complex) numbers, whose product is independent of the order of the factors. An example of two matrices not commuting with each other is: : whereas : Besides the ordinary matrix multiplication just described, other less frequently used operations on matrices that can be considered forms of multiplication also exist, such as the and the . They arise in solving matrix equations such as the .
Row operationsThere are three types of row operations: # row addition, that is adding a row to another. # row multiplication, that is multiplying all entries of a row by a non-zero constant; # row switching, that is interchanging two rows of a matrix; These operations are used in several ways, including solving s and finding s.
SubmatrixA submatrix of a matrix is obtained by deleting any collection of rows and/or columns. For example, from the following 3-by-4 matrix, we can construct a 2-by-3 submatrix by removing row 3 and column 2: : The and cofactors of a matrix are found by computing the of certain submatrices. A principal submatrix is a square submatrix obtained by removing certain rows and columns. The definition varies from author to author. According to some authors, a principal submatrix is a submatrix in which the set of row indices that remain is the same as the set of column indices that remain. Other authors define a principal submatrix as one in which the first ''k'' rows and columns, for some number ''k'', are the ones that remain; this type of submatrix has also been called a leading principal submatrix.
Linear equationsMatrices can be used to compactly write and work with multiple linear equations, that is, systems of linear equations. For example, if A is an ''m''-by-''n'' matrix, x designates a column vector (that is, ''n''×1-matrix) of ''n'' variables ''x'', ''x'', ..., ''x'', and b is an ''m''×1-column vector, then the matrix equation : is equivalent to the system of linear equations : Using matrices, this can be solved more compactly than would be possible by writing out all the equations separately. If ''n'' = ''m'' and the equations are , then this can be done by writing : where A is the of A. If A has no inverse, solutions—if any—can be found using its .
Linear transformationsMatrices and matrix multiplication reveal their essential features when related to ''linear transformations'', also known as ''linear maps''. A real ''m''-by-''n'' matrix A gives rise to a linear transformation R → R mapping each vector x in R to the (matrix) product Ax, which is a vector in R. Conversely, each linear transformation ''f'': R → R arises from a unique ''m''-by-''n'' matrix A: explicitly, the of A is the ''i'' coordinate of ''f''(e), where e = (0,...,0,1,0,...,0) is the with 1 in the ''j'' position and 0 elsewhere. The matrix A is said to represent the linear map ''f'', and A is called the ''transformation matrix'' of ''f''. For example, the 2×2 matrix : can be viewed as the transform of the into a with vertices at , , , and . The parallelogram pictured at the right is obtained by multiplying A with each of the column vectors , and in turn. These vectors define the vertices of the unit square. The following table shows several 2×2 real matrices with the associated linear maps of R. The blue original is mapped to the green grid and shapes. The origin (0,0) is marked with a black point. Under the between matrices and linear maps, matrix multiplication corresponds to of maps: if a ''k''-by-''m'' matrix B represents another linear map ''g'': R → R, then the composition is represented by BA since :(''g'' ∘ ''f'')(x) = ''g''(''f''(x)) = ''g''(Ax) = B(Ax) = (BA)x. The last equality follows from the above-mentioned associativity of matrix multiplication. The A is the maximum number of row vectors of the matrix, which is the same as the maximum number of linearly independent column vectors. Equivalently it is the of the of the linear map represented by A. The states that the dimension of the of a matrix plus the rank equals the number of columns of the matrix.
Square matrixA is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order ''n.'' Any two square matrices of the same order can be added and multiplied. The entries ''a'' form the of a square matrix. They lie on the imaginary line that runs from the top left corner to the bottom right corner of the matrix.
Diagonal and triangular matrixIf all entries of A below the main diagonal are zero, A is called an ''upper ''. Similarly if all entries of ''A'' above the main diagonal are zero, A is called a ''lower triangular matrix''. If all entries outside the main diagonal are zero, A is called a .
Identity matrixThe ''identity matrix'' I of size ''n'' is the ''n''-by-''n'' matrix in which all the elements on the are equal to 1 and all other elements are equal to 0, for example, : It is a square matrix of order ''n'', and also a special kind of . It is called an identity matrix because multiplication with it leaves a matrix unchanged: :AI = IA = A for any ''m''-by-''n'' matrix A. A nonzero scalar multiple of an identity matrix is called a ''scalar'' matrix. If the matrix entries come from a field, the scalar matrices form a group, under matrix multiplication, that is isomorphic to the multiplicative group of nonzero elements of the field.
Symmetric or skew-symmetric matrixA square matrix A that is equal to its transpose, that is, A = A, is a . If instead, A is equal to the negative of its transpose, that is, A = −A, then A is a . In complex matrices, symmetry is often replaced by the concept of , which satisfy A = A, where the star or denotes the of the matrix, that is, the transpose of the of A. By the , real symmetric matrices and complex Hermitian matrices have an ; that is, every vector is expressible as a of eigenvectors. In both cases, all eigenvalues are real. This theorem can be generalized to infinite-dimensional situations related to matrices with infinitely many rows and columns, see .
Invertible matrix and its inverseA square matrix A is called ' or ''non-singular'' if there exists a matrix B such that :AB = BA = I , where I is the ''n''×''n'' with 1s on the and 0s elsewhere. If B exists, it is unique and is called the ' of A, denoted A.
Definite matrixA symmetric real matrix is called if the associated : has a positive value for every nonzero vector in . If only yields negative values then is ; if does produce both negative and positive values then is . If the quadratic form yields only non-negative values (positive or zero), the symmetric matrix is called ''positive-semidefinite'' (or if only non-positive values, then negative-semidefinite); hence the matrix is indefinite precisely when it is neither positive-semidefinite nor negative-semidefinite. A symmetric matrix is positive-definite if and only if all its eigenvalues are positive, that is, the matrix is positive-semidefinite and it is invertible. The table at the right shows two possibilities for 2-by-2 matrices. Allowing as input two different vectors instead yields the associated to : :. In the case of complex matrices, the same terminology and result apply, with ''symmetric matrix'', ''quadratic form'', ''bilinear form'', and ''transpose'' replaced respectively by , , , and .
Orthogonal matrixAn ''orthogonal matrix'' is a with entries whose columns and rows are s (that is, vectors). Equivalently, a matrix A is orthogonal if its is equal to its : : which entails : where I is the of size ''n''. An orthogonal matrix A is necessarily (with inverse ), (), and (). The of any orthogonal matrix is either or . A ''special orthogonal matrix'' is an orthogonal matrix with +1. As a , every orthogonal matrix with determinant is a pure without reflection, i.e., the transformation preserves the orientation of the transformed structure, while every orthogonal matrix with determinant reverses the orientation, i.e., is a composition of a pure and a (possibly null) rotation. The identity matrices have determinant , and are pure rotations by an angle zero. The analogue of an orthogonal matrix is a .
TraceThe , tr(A) of a square matrix A is the sum of its diagonal entries. While matrix multiplication is not commutative as mentioned , the trace of the product of two matrices is independent of the order of the factors: : tr(AB) = tr(BA). This is immediate from the definition of matrix multiplication: : It follows that the trace of the product of more than two matrices is independent of s of the matrices, however this does not in general apply for arbitrary permutations (for example, tr(ABC) ≠ tr(BAC), in general). Also, the trace of a matrix is equal to that of its transpose, that is, :tr(A) = tr(A).
DeterminantThe ''determinant'' of a square matrix A (denoted det(A) or , A, ) is a number encoding certain properties of the matrix. A matrix is invertible its determinant is nonzero. Its equals the area (in R) or volume (in R) 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 2-by-2 matrices is given by : The determinant of 3-by-3 matrices involves 6 terms (). The more lengthy generalises these two formulae to all dimensions. The determinant of a product of square matrices equals the product of their determinants: :det(AB) = det(A) · det(B). Adding a multiple of any row to another row, or a multiple of any column to another column does not change the determinant. Interchanging two rows or two columns affects the determinant by multiplying it by −1. Using these operations, any matrix can be transformed to a lower (or upper) triangular matrix, and for such matrices, the determinant equals the product of the entries on the main diagonal; this provides a method to calculate the determinant of any matrix. Finally, the expresses the determinant in terms of , that is, determinants of smaller matrices. This expansion can be used for a recursive definition of determinants (taking as starting case the determinant of a 1-by-1 matrix, which is its unique entry, or even the determinant of a 0-by-0 matrix, which is 1), that can be seen to be equivalent to the Leibniz formula. Determinants can be used to solve s using , where the division of the determinants of two related square matrices equates to the value of each of the system's variables.
Eigenvalues and eigenvectorsA number λ and a non-zero vector v satisfying : are called an ''eigenvalue'' and an ''eigenvector'' of A, respectively. The number λ is an eigenvalue of an ''n''×''n''-matrix A if and only if A−λI is not invertible, which is to : The polynomial ''p'' in an ''X'' given by evaluation of the determinant det(''X''I−A) is called the of A. It is a of ''n''. Therefore the polynomial equation ''p''(λ)=0 has at most ''n'' different solutions, that is, eigenvalues of the matrix. They may be complex even if the entries of A are real. According to the , ''p''(A) = 0, that is, the result of substituting the matrix itself into its own characteristic polynomial yields the .
Computational aspectsMatrix 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 of vectors x to an eigenvector when ''n'' tends to . 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 . As with other numerical situations, two main aspects are the of algorithms and their . Determining the complexity of an algorithm means finding s or estimates of how many elementary operations such as additions and multiplications of scalars are necessary to perform some algorithm, for example, . Calculating the matrix product of two ''n''-by-''n'' matrices using the definition given above needs ''n'' multiplications, since for any of the ''n'' entries of the product, ''n'' multiplications are necessary. The outperforms this "naive" algorithm; it needs only ''n'' multiplications. A refined approach also incorporates specific features of the computing devices. In many practical situations additional information about the matrices involved is known. An important case are , that is, matrices most of whose entries are zero. There are specifically adapted algorithms for, say, solving linear systems Ax = b for sparse matrices A, such as the . 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 of A) :A = adj(A) / det(A) may lead to significant rounding errors if the determinant of the matrix is very small. The can be used to capture the of linear algebraic problems, such as computing a matrix's inverse. Most computer s 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. The original had built-in commands for matrix arithmetic on arrays from its implementation in 1964. As early as the 1970s, some engineering desktop computers such as the had . Some computer languages such as were designed to manipulate matrices, and can be used to aid computing with matrices.
DecompositionThere 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 factors matrices as a product of lower (L) and an upper (U). Once this decomposition is calculated, linear systems can be solved more efficiently, by a simple technique called . Likewise, inverses of triangular matrices are algorithmically easier to calculate. The ''Gaussian elimination'' is a similar algorithm; it transforms any matrix to . Both methods proceed by multiplying the matrix by suitable , which correspond to and adding multiples of one row to another row. expresses any matrix A as a product UDV, where U and V are and D is a diagonal matrix. The or ''diagonalization'' expresses A as a product VDV, where D is a diagonal matrix and V is a suitable invertible matrix. If A can be written in this form, it is called . More generally, and applicable to all matrices, the Jordan decomposition transforms a matrix into , that is to say matrices whose only nonzero entries are the eigenvalues λ to λ of A, placed on the main diagonal and possibly entries equal to one directly above the main diagonal, as shown at the right. Given the eigendecomposition, the ''n'' power of A (that is, ''n''-fold iterated matrix multiplication) can be calculated via :A = (VDV) = VDVVDV...VDV = VDV and the power of a diagonal matrix can be calculated by taking the corresponding powers of the diagonal entries, which is much easier than doing the exponentiation for A instead. This can be used to compute the ''e'', a need frequently arising in solving s, s and . To avoid numerically situations, further algorithms such as the can be employed.
Abstract algebraic aspects and generalizationsMatrices can be generalized in different ways. Abstract algebra uses matrices with entries in more general or even , 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 is s, which can be seen as higher-dimensional arrays of numbers, as opposed to vectors, which can often be realized as sequences of numbers, while matrices are rectangular or two-dimensional arrays of numbers. Matrices, subject to certain requirements tend to form known as matrix groups. Similarly under certain conditions matrices form known as s. Though the product of matrices is not in general commutative yet certain matrices form known as s.
Matrices with more general entriesThis 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 , that is, a where , , , and operations are defined and well-behaved, may be used instead of R or C, for example s or s. For example, makes use of matrices over finite fields. Wherever s 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 the 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 , such as C, from the outset. More generally, matrices with entries in a ''R'' are widely used in mathematics. Rings are a more general notion than fields in that a division operation need not exist. The very same addition and multiplication operations of matrices extend to this setting, too. The set M(''n'', ''R'') (also denoted M''n''(R)) of all square ''n''-by-''n'' matrices over ''R'' is a ring called , isomorphic to the of the left ''R''- ''R''. If the ring ''R'' is , that is, its multiplication is commutative, then M(''n'', ''R'') is a unitary noncommutative (unless ''n'' = 1) over ''R''. The of square matrices over a commutative ring ''R'' can still be defined using the ; such a matrix is invertible if and only if its determinant is in ''R'', generalising the situation over a field ''F'', where every nonzero element is invertible. Matrices over s are called . Matrices do not always have all their entries in the same ring– or even in any ring at all. One special but common case is , 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 ; but their sizes must fulfill certain compatibility conditions.
Relationship to linear mapsLinear maps R → R are equivalent to ''m''-by-''n'' matrices, as described . More generally, any linear map between finite- s can be described by a matrix A = (''a''), after choosing v, ..., v of ''V'', and w, ..., w of ''W'' (so ''n'' is the dimension of ''V'' and ''m'' is the dimension of ''W''), which is such that : In other words, column ''j'' of ''A'' expresses the image of v in terms of the basis vectors w 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 . Many of the above concrete notions can be reinterpreted in this light, for example, the transpose matrix A describes the given by A, with respect to the . These properties can be restated more naturally: the of all matrices with entries in a field with multiplication as composition is to the category of finite-dimensional s and linear maps over this field. More generally, the set of ''m''×''n'' matrices can be used to represent the ''R''-linear maps between the free modules ''R'' and ''R'' for an arbitrary ring ''R'' with unity. When ''n''=''m'' composition of these maps is possible, and this gives rise to the of ''n''×''n'' matrices representing the of ''R''.
Matrix groupsA is a mathematical structure consisting of a set of objects together with a , that is, an operation combining any two objects to a third, subject to certain requirements. A group in which the objects are matrices and the group operation is matrix multiplication is called a ''matrix group''. Since a group every element must be invertible, the most general matrix groups are the groups of all invertible matrices of a given size, called the s. 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 of (that is, a smaller group contained in) their general linear group, called a . , determined by the condition :MM = I, form the . Every orthogonal matrix has 1 or −1. Orthogonal matrices with determinant 1 form a subgroup called ''special orthogonal group''. Every is to a matrix group, as one can see by considering the of the . General groups can be studied using matrix groups, which are comparatively well understood, by means of .
Infinite matricesIt is also possible to consider matrices with infinitely many rows and/or columns even if, being infinite objects, one cannot write down such matrices explicitly. 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 ''R'' is any ring with unity, then the ring of endomorphisms of as a right ''R'' module is isomorphic to the ring of column finite matrices whose entries are indexed by , and whose columns each contain only finitely many nonzero entries. The endomorphisms of ''M'' considered as a left ''R'' module result in an analogous object, the row finite matrices whose rows each only have finitely many nonzero entries. 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. For a matrix A to describe a linear map ''f'': ''V''→''W'', bases for both spaces must have been chosen; recall that by definition this means that every vector in the space can be written uniquely as a (finite) linear combination of basis vectors, so that written as a (column) vector''v'' of s, only finitely many entries ''v'' are nonzero. 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. There is no restriction on the rows of ''A'' however: in the product A·''v'' there are only finitely many nonzero coefficients of ''v'' involved, so every one of its entries, even if it is given as an infinite sum of products, involves only finitely many nonzero terms and is therefore well defined. 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 are well defined (provided that the column-index and row-index sets match), are of the same type, and correspond to the composition of linear maps. If ''R'' is a ring, then the condition of row or column finiteness can be relaxed. With the norm in place, 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 , where convergence and 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 can be used instead.
Empty matrixAn ''empty matrix'' is a matrix in which the number of rows or columns (or both) is zero. Empty matrices help dealing with maps involving the . 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 s allow creating and computing with them. The determinant of the 0-by-0 matrix is 1 as follows regarding the occurring in the Leibniz formula for the determinant as 1. This value is also consistent with the fact that the identity map from any finite-dimensional space to itself has determinant1, a fact that is often used as a part of the characterization of determinants.
ApplicationsThere 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 and , the encodes the payoff for two players, depending on which out of a given (finite) set of alternatives the players choose. and automated compilation makes use of such as to track frequencies of certain words in several documents. Complex numbers can be represented by particular real 2-by-2 matrices via : under which addition and multiplication of complex numbers and matrices correspond to each other. For example, 2-by-2 rotation matrices represent the multiplication with some complex number of 1, as . A similar interpretation is possible for s and s in general. Early techniques such as the also used matrices. However, due to the linear nature of matrices, these codes are comparatively easy to break. uses matrices both to represent objects and to calculate transformations of objects using affine to accomplish tasks such as projecting a three-dimensional object onto a two-dimensional screen, corresponding to a theoretical camera observation. Matrices over a are important in the study of . makes use of matrices in various ways, particularly since the use of to discuss and . Examples are the and the used in solving the to obtain the s of the .
Graph theoryThe of a is a basic notion of . It records which vertices of the graph are connected by an edge. Matrices containing just two different values (1 and 0 meaning for example "yes" and "no", respectively) are called . The contains information about distances of the edges. These concepts can be applied to s connected by s or cities connected by roads etc., in which case (unless the connection network is extremely dense) the matrices tend to be , that is, contain few nonzero entries. Therefore, specifically tailored matrix algorithms can be used in .
Analysis and geometryThe of a ''ƒ'': R → R consists of the s of ''ƒ'' with respect to the several coordinate directions, that is, : It encodes information about the local growth behaviour of the function: given a x=(''x'',...,''x''), that is, a point where the first s of ''ƒ'' vanish, the function has a if the Hessian matrix is . can be used to find global minima or maxima of quadratic functions closely related to the ones attached to matrices (see ). Another matrix frequently used in geometrical situations is the of a differentiable map ''f'': R → R. If ''f'', ..., ''f'' denote the components of ''f'', then the Jacobi matrix is defined as : 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 . s can be classified by considering the matrix of coefficients of the highest-order differential operators of the equation. For s this matrix is positive definite, which has a decisive influence on the set of possible solutions of the equation in question. The 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 concerning a sufficiently fine grid, which in turn can be recast as a matrix equation.
Probability theory and statisticsare square matrices whose rows are s, that is, whose entries are non-negative and sum up to one. Stochastic matrices are used to define s with finitely many states. 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 s, 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. is concerned with describing data sets, which can often be represented as , which may then be subjected to techniques. The encodes the mutual of several s. Another technique using matrices are , a method that approximates a finite set of pairs (''x'', ''y''), (''x'', ''y''), ..., (''x'', ''y''), by a linear function :''y'' ≈ ''ax'' + ''b'', ''i'' = 1, ..., ''N'' which can be formulated in terms of matrices, related to the of matrices. are matrices whose entries are random numbers, subject to suitable s, such as . Beyond probability theory, they are applied in domains ranging from to .
Symmetries and transformations in physicsLinear transformations and the associated play a key role in modern physics. For example, s in are classified as representations of the of special relativity and, more specifically, by their behavior under the . Concrete representations involving the and more general are an integral part of the physical description of s, which behave as s. For the three lightest s, there is a group-theoretical representation involving the SU(3); for their calculations, physicists use a convenient matrix representation known as the , which are also used for the SU(3) that forms the basis of the modern description of strong nuclear interactions, . The , in turn, expresses the fact that the basic quark states that are important for s are not the same as, but linearly related to the basic quark states that define particles with specific and distinct es.
Linear combinations of quantum statesThe first model of (, 1925) represented the theory's operators by infinite-dimensional matrices acting on quantum states. This is also referred to as . One particular example is the that characterizes the "mixed" state of a quantum system as a linear combination of elementary, "pure" . 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 s, 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 , which encodes all information about the possible interactions between particles.
Normal modesA general application of matrices in physics is the description of linearly coupled harmonic systems. The of such systems can be described in matrix form, with a mass matrix multiplying a generalized velocity to give the kinetic term, and a matrix multiplying a displacement vector to characterize the interactions. The best way to obtain solutions is to determine the system's s, its s, by diagonalizing the matrix equation. Techniques like this are crucial when it comes to the internal dynamics of : the internal vibrations of systems consisting of mutually bound component atoms. They are also needed for describing mechanical vibrations, and oscillations in electrical circuits.
Geometrical opticsprovides further matrix applications. In this approximative theory, the of light is neglected. The result is a model in which are indeed . If the deflection of light rays by optical elements is small, the action of a 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 : 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. 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.
ElectronicsTraditional and in electronics lead to a system of linear equations that can be described with a matrix. The behaviour of many s can be described using matrices. Let ''A'' be a 2-dimensional vector with the component's input voltage ''v'' and input current ''i'' as its elements, and let ''B'' be a 2-dimensional vector with the component's output voltage ''v'' and output current ''i'' as its elements. Then the behaviour of the electronic component can be described by ''B'' = ''H'' · ''A'', where ''H'' is a 2 x 2 matrix containing one element (''h''), one element (''h''), and two elements (''h'' and ''h''). Calculating a circuit now reduces to multiplying matrices.
HistoryMatrices have a long history of application in solving s but they were known as arrays until the 1800s. The ' written in 10th–2nd century BCE is the first example of the use of array methods to solve , including the concept of s. In 1545 Italian mathematician introduced the method to Europe when he published ''Ars Magna''.''Discrete Mathematics'' 4th Ed. Dossey, Otto, Spense, Vanden Eynden, Published by Addison Wesley, October 10, 2001 , p. 564-565 The used the same array methods to solve simultaneous equations in 1683. The Dutch mathematician'' '' represented transformations using arrays in his 1659 book ''Elements of Curves'' (1659).''Discrete Mathematics'' 4th Ed. Dossey, Otto, Spense, Vanden Eynden, Published by Addison Wesley, October 10, 2001 , p. 564 Between 1700 and 1710 publicized the use of arrays for recording information or solutions and experimented with over 50 different systems of arrays. presented in 1750. The term "matrix" (Latin for "womb", derived from '—mother) was coined by in 1850, who understood a matrix as an object giving rise to several determinants today called , that is to say, determinants of smaller matrices that derive from the original one by removing columns and rows. In an 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. 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. Early matrix theory had limited the use of arrays almost exclusively to determinants and Arthur Cayley's abstract matrix operations were revolutionary. He was instrumental in proposing a matrix concept independent of equation systems. In 1858 published his ''A memoir on the theory of matrices'' in which he proposed and demonstrated the . The English mathematician was the first to use modern bracket notation for matrices in 1913 and he simultaneously demonstrated the first significant use of the notation A = 'a''to represent a matrix where ''a'' refers to the'' i''th row and the ''j''th column. The modern study of determinants sprang from several sources. problems led to relate coefficients of s, that is, expressions such as and s in three dimensions to matrices. further developed these notions, including the remark that, in modern parlance, s are . was the first to prove general statements about determinants, using as definition of the determinant of a matrix A = 'a''the following: replace the powers ''a'' by ''a'' in the :, where Π denotes the of the indicated terms. He also showed, in 1829, that the s of symmetric matrices are real. studied "functional determinants"—later called s by Sylvester—which can be used to describe geometric transformations at a local (or ) level, see ; 's ''Vorlesungen über die Theorie der Determinanten'' and ''Zur Determinantentheorie'', both published in 1903, first treated determinants atically, 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 was proved for 2×2 matrices by Cayley in the aforementioned memoir, and by for 4×4 matrices. , working on s, generalized the theorem to all dimensions (1898). Also at the end of the 19th century, the (generalizing a special case now known as ) was established by . In the early 20th century, matrices attained a central role in linear algebra, partially due to their use in classification of the systems of the previous century. The inception of by , and led to studying matrices with infinitely many rows and columns. Later, carried out the , by further developing notions such as s on s, which, very roughly speaking, correspond to , but with an infinity of .
Other historical usages of the word "matrix" in mathematicsThe word has been used in unusual ways by at least two authors of historical importance. and in their ''Principia Mathematica'' (1910–1913) use the word "matrix" in the context of their . 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 : :"Let us give the name of ''matrix'' to any function, of however many variables, that does not involve any s. Then, any possible function other than a matrix derives from a matrix by means of generalization, that is, by considering the proposition 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". 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, for example, ''y'', by "considering" the function for all possible values of "individuals" ''a'' substituted in place of variable ''x''. And then the resulting collection of functions of the single variable ''y'', that is, , can be reduced to a "matrix" of values by "considering" the function for all possible values of "individuals" ''b'' substituted in place of variable ''y'': : in his 1946 ''Introduction to Logic'' used the word "matrix" synonymously with the notion of as used in mathematical logic.Tarski, Alfred; (1946) ''Introduction to Logic and the Methodology of Deductive Sciences'', Dover Publications, Inc, New York NY, .
See also* * * * * * * * * — A generalization of matrices with any number of indices
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Physics references* * * * * * * * *
Historical references* A. Cayley ''A memoir on the theory of matrices''. Phil. Trans. 148 1858 17-37; Math. Papers II 475-496 * , reprint of the 1907 original edition * * * * * * * *
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