Matrix (mathematics)
In mathematics, a matrix (pl.: matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object.
For example,
Matrices are used to represent
Not all matrices are related to linear algebra. This is, in particular, the case in graph theory, of incidence matrices, and adjacency matrices.[1] This article focuses on matrices related to linear algebra, and, unless otherwise specified, all matrices represent linear maps or may be viewed as such.
In
Matrix theory is the
Definition
A matrix is a rectangular array of numbers (or other mathematical objects), called the entries of the matrix. Matrices are subject to standard operations such as addition and multiplication.[2] Most commonly, a matrix over a field F is a rectangular array of elements of F.[3][4] A real matrix and a complex matrix are matrices whose entries are respectively real numbers or complex numbers. More general types of entries are discussed below. 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.
Size
The 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 rows and columns is called an matrix, or -by- matrix, where and are called its dimensions. For example, matrix above is a matrix.
Matrices with a single row are called
Name | Size | Example | Description | Notation |
---|---|---|---|---|
Row vector
|
1 × n | A matrix with one row, sometimes used to represent a vector | ||
Column vector
|
n × 1 | A matrix with one column, sometimes used to represent a vector | ||
Square matrix | n × n | A matrix with the same number of rows and columns, sometimes used to represent a linear transformation from a vector space to itself, such as reflection, rotation, or shearing. |
Notation
The specifics of symbolic matrix notation vary widely, with some prevailing trends. Matrices are commonly written in
Matrices are usually symbolized using
The entry in the i-th row and j-th column of a matrix A is sometimes referred to as the or entry of the matrix, and commonly denoted by or . Alternative notations for that entry are and . For example, the entry of the following matrix is 5 (also denoted , , or ):
Sometimes, the entries of a matrix can be defined by a formula such as . For example, each of the entries of the following matrix is determined by the formula .
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 or . If matrix size is , the above-mentioned formula is valid for any and any . This can be either specified separately, or indicated using as a subscript. For instance, the matrix above is , and can be defined as or .
Some programming languages utilize doubly subscripted arrays (or arrays of arrays) to represent an m-by-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 .[6] 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, refers to the i-th row of A, while refers to the j-th column.[clarification needed]
The set of all m-by-n real matrices is often denoted or The set of all m-by-n matrices over another field, or over a ring R, is similarly denoted or If m = n, such as in the case of
Basic operations
There are a number of basic operations that can be applied on matrices. Some, such as transposition and submatrix do not depend on the nature of the entries. Others, such as matrix addition, scalar multiplication, matrix multiplication, and row operations involve operations on matrix entries and therefore require that matrix entries are numbers or belong to a field or a ring.[8]
In this section, it is supposed that matrix entries belong to a fixed ring, that is typically a field of numbers.
Addition, scalar multiplication, subtraction and transposition
The sum A+B of two m-by-n matrices A and B is calculated entrywise:
- (A + B)i,j = Ai,j + Bi,j, where 1 ≤ i ≤ m and 1 ≤ j ≤ n.
For example,
The product cA of a number c (also called a scalar in this context) and a matrix A is computed by multiplying every entry of A by c:
- (cA)i,j = c · Ai,j.
This operation is called scalar multiplication, but its result is not named "scalar product" to avoid confusion, since "scalar product" is often used as a synonym for "
- Subtraction
The subtraction of two m×n matrices is defined by composing matrix addition with scalar multiplication by –1:
The transpose of an m-by-n matrix A is the n-by-m matrix AT (also denoted Atr or tA) formed by turning rows into columns and vice versa:
- (AT)i,j = Aj,i.
For example:
Familiar properties of numbers extend to these operations on matrices: for example, addition is
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 dot product of the corresponding row of A and the corresponding column of B:[10]
where 1 ≤ i ≤ m and 1 ≤ j ≤ p.[11] For example, the underlined entry 2340 in the product is calculated as (2 × 1000) + (3 × 100) + (4 × 10) = 2340:
Matrix multiplication satisfies the rules (AB)C = A(BC) (
- AB ≠ BA,
In other words, matrix multiplication is not commutative, in marked contrast to (rational, real, or complex) numbers, whose product is independent of the order of the factors.[10] 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 Hadamard product and the Kronecker product.[13] They arise in solving matrix equations such as the Sylvester equation.
Row operations
There 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
Submatrix
A submatrix of a matrix is a matrix obtained by deleting any collection of rows and/or columns.[14][15][16] 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 minors and cofactors of a matrix are found by computing the determinant of certain submatrices.[16][17]
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.[18][19] 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;[20] this type of submatrix has also been called a leading principal submatrix.[21]
Linear equations
Matrices 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 x1, x2, ..., xn, and b is an m×1-column vector, then the matrix equation
is equivalent to the system of linear equations[22]
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 independent, then this can be done by writing
where A−1 is the
Linear transformations
Matrices 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 Rn → Rm mapping each vector x in Rn to the (matrix) product Ax, which is a vector in Rm. Conversely, each linear transformation f: Rn → Rm arises from a unique m-by-n matrix A: explicitly, the (i, j)-entry of A is the ith coordinate of f(ej), where ej = (0,...,0,1,0,...,0) is the unit vector with 1 in the jth 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 unit square into a parallelogram with vertices at (0, 0), (a, b), (a + c, b + d), and (c, d). 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 R2. The blue original is mapped to the green grid and shapes. The origin (0,0) is marked with a black point.
Horizontal shear with m = 1.25. |
Reflection through the vertical axis | Squeeze mapping with r = 3/2 |
Scaling by a factor of 3/2 |
Rotation by π/6 = 30° |
Under the 1-to-1 correspondence between matrices and linear maps, matrix multiplication corresponds to composition of maps:[23] if a k-by-m matrix B represents another linear map g: Rm → Rk, then the composition g ∘ f 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
Square matrix
A square matrix is a matrix with the same number of rows and columns.[5] 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 aii form the main diagonal 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.
Main types
Name Example with n = 3 Diagonal matrix Lower triangular matrixUpper triangular matrix
Diagonal and triangular matrix
If all entries of A below the main diagonal are zero, A is called an upper triangular matrix. 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 diagonal matrix.
Identity matrix
The identity matrix In of size n is the n-by-n matrix in which all the elements on the main diagonal 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 diagonal matrix. It is called an identity matrix because multiplication with it leaves a matrix unchanged:
- AIn = ImA = 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 matrix
A square matrix A that is equal to its transpose, that is, A = AT, is a symmetric matrix. If instead, A is equal to the negative of its transpose, that is, A = −AT, then A is a skew-symmetric matrix. In complex matrices, symmetry is often replaced by the concept of Hermitian matrices, which satisfy A∗ = A, where the star or asterisk denotes the conjugate transpose of the matrix, that is, the transpose of the complex conjugate of A.
By the
Invertible matrix and its inverse
A square matrix A is called invertible or non-singular if there exists a matrix B such that
where In is the n×n identity matrix with 1s on the main diagonal and 0s elsewhere. If B exists, it is unique and is called the inverse matrix of A, denoted A−1.
Definite matrix
Positive definite matrix |
Indefinite matrix
|
---|---|
Points such that (Ellipse) |
Points such that (Hyperbola) |
A symmetric real matrix A is called
- f (x) = xTA x
has a positive value for every nonzero vector x in Rn. If f (x) only yields negative values then A is
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.[31] The table at the right shows two possibilities for 2-by-2 matrices.
Allowing as input two different vectors instead yields the bilinear form associated to A:[32]
- BA (x, y) = xTAy.
In the case of complex matrices, the same terminology and result apply, with symmetric matrix, quadratic form, bilinear form, and transpose xT replaced respectively by
Orthogonal matrix
An orthogonal matrix is a square matrix with
which entails
where In is the identity matrix of size n.
An orthogonal matrix A is necessarily
The complex analogue of an orthogonal matrix is a unitary matrix.
Main operations
Trace
The
- .
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 cyclic permutations 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(AT).
Determinant
The determinant of a square matrix A (denoted det(A) or |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 R2) or volume (in R3) 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 (rule of Sarrus). The more lengthy Leibniz formula generalises these two formulae to all dimensions.[34]
The determinant of a product of square matrices equals the product of their determinants:
- det(AB) = det(A) · det(B), or using alternate notation:
- |AB| = |A| · |B|.[35]
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.[36] 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 Laplace expansion expresses the determinant in terms of minors, that is, determinants of smaller matrices.[37] 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 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.[38]
Eigenvalues and eigenvectors
A number and a non-zero vector v satisfying
are called an eigenvalue and an eigenvector of A, respectively.[39][40] The number λ is an eigenvalue of an n×n-matrix A if and only if A−λIn is not invertible, which is equivalent to
The polynomial pA in an indeterminate X given by evaluation of the determinant det(XIn−A) is called the characteristic polynomial of A. It is a monic polynomial of degree n. Therefore the polynomial equation pA(λ) = 0 has at most n different solutions, that is, eigenvalues of the matrix.[42] They may be complex even if the entries of A are real. According to the Cayley–Hamilton theorem, pA(A) = 0, that is, the result of substituting the matrix itself into its own characteristic polynomial yields the zero matrix.
Computational aspects
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
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
Determining the complexity of an algorithm means finding
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. There are specifically adapted algorithms for, say, solving linear systems Ax = b for sparse matrices A, such as the conjugate gradient method.[46]
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)
- A−1 = adj(A) / det(A)
may lead to significant rounding errors if the determinant of the matrix is very small. The norm of a matrix can be used to capture the conditioning of linear algebraic problems, such as computing a matrix's inverse.[47]
Most computer
Decomposition
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
The
- An = (VDV−1)n = VDV−1VDV−1...VDV−1 = VDnV−1
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
Abstract algebraic aspects and generalizations
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 is tensors, 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.[56] Matrices, subject to certain requirements tend to form groups known as matrix groups. 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.
Matrices with more general entries
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
More generally, matrices with entries in a
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 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 fulfill certain compatibility conditions.
Relationship to linear maps
Linear maps Rn → Rm are equivalent to m-by-n matrices, as described
In other words, column j of A expresses the image of vj in terms of the basis vectors wi 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.[61] Many of the above concrete notions can be reinterpreted in this light, for example, the transpose matrix AT describes the transpose of the linear map given by A, with respect to the dual bases.[62]
These properties can be restated more naturally: the category of all matrices with entries in a field with multiplication as composition is equivalent to the category of finite-dimensional vector spaces 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 Rm and Rn for an arbitrary ring R with unity. When n = m composition of these maps is possible, and this gives rise to the matrix ring of n×n matrices representing the endomorphism ring of Rn.
Matrix groups
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.[63] A group in which the objects are matrices and the group operation is matrix multiplication is called a matrix group.[64][65] 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 general linear groups.
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.[66] Orthogonal matrices, determined by the condition
- MTM = I,
form the orthogonal group.[67] Every orthogonal matrix has determinant 1 or −1. Orthogonal matrices with determinant 1 form a subgroup called special orthogonal group.
Every
Infinite matrices
It is also possible to consider matrices with infinitely many rows and/or columns[70] even though, 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 coefficients, only finitely many entries vi 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
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,[71] and the abstract and more powerful tools of functional analysis can be used instead.
Empty matrix
An empty matrix is a matrix in which the number of rows or columns (or both) is zero.
Applications
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
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 absolute value 1, as above. A similar interpretation is possible for quaternions[76] 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.[77] Computer graphics uses matrices to represent objects; to calculate transformations of objects using affine rotation matrices to accomplish tasks such as projecting a three-dimensional object onto a two-dimensional screen, corresponding to a theoretical camera observation; and to apply image convolutions such as sharpening, blurring, edge detection, and more.[78] Matrices over a polynomial ring are important in the study of control theory.
Graph theory
The
Analysis and geometry
The Hessian matrix of a differentiable function ƒ: Rn → R consists of the second derivatives of ƒ with respect to the several coordinate directions, that is,[81]
It encodes information about the local growth behaviour of the function: given a critical point x = (x1, ..., xn), that is, a point where the first partial derivatives
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[83]
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.[84]
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 a decisive influence on the set of possible solutions of the equation in question.[85]
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 concerning a sufficiently fine grid, which in turn can be recast as a matrix equation.[86]
Probability theory and statistics
Statistics also makes use of matrices in many different forms.
- yi ≈ axi + b, i = 1, ..., N
which can be formulated in terms of matrices, related to the singular value decomposition of matrices.[91]
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.[92][93]
Symmetries and transformations in physics
Linear transformations and the associated
Linear combinations of quantum states
The first model of
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.[98]
Normal modes
A general application of matrices in physics is the description of linearly coupled harmonic systems. The
Geometrical optics
Electronics
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 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 behaviour 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.
History
Matrices have a long history of application in solving
The term "matrix" (Latin for "womb", "dam" (non-human female animal kept for breeding), "source", "origin", "list", "register", derived from mater—mother[106]) was coined by James Joseph Sylvester in 1850,[107] who understood a matrix as an object giving rise to several 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 1851 paper, Sylvester explains:[108]
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.
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.[103] 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 Cayley published his A memoir on the theory of matrices[109][110] in which he proposed and demonstrated the Cayley–Hamilton theorem.[103]
The English mathematician Cuthbert Edmund Cullis was the first to use modern bracket notation for matrices in 1913 and he simultaneously demonstrated the first significant use of the notation A = [ai,j] to represent a matrix where ai,j refers to the ith row and the jth column.[103]
The modern study of determinants sprang from several sources.
- ,
where denotes the
Many theorems were first established for small matrices only, for example, the
The inception of
Other historical usages of the word "matrix" in mathematics
The word has been used in unusual ways by at 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:[117]
Let us give the name of matrix to any function, of however many variables, that does not involve any
apparent variables. 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" ai substituted in place of variable x. And then the resulting collection of functions of the single variable y, that is, ∀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.[118]
See also
- List of named matrices
- Algebraic multiplicity– Multiplicity of an eigenvalue as a root of the characteristic polynomial
- Geometric multiplicity– Dimension of the eigenspace associated with an eigenvalue
- Gram–Schmidt process – Orthonormalization of a set of vectors
- Irregular matrix
- Matrix calculus – Specialized notation for multivariable calculus
- Matrix function– Function that maps matrices to matrices
- Matrix multiplication algorithm
- Tensor — A generalization of matrices with any number of indices
- Bohemian matrices – Set of matrices
Notes
- ^ However, in the case of adjacency matrices, matrix multiplication or a variant of it allows the simultaneous computation of the number of paths between any two vertices, and of the shortest length of a path between two vertices.
- ^ Lang 2002
- ^ Fraleigh (1976, p. 209)
- ^ Nering (1970, p. 37)
- ^ a b Weisstein, Eric W. "Matrix". mathworld.wolfram.com. Retrieved 2020-08-19.
- ^ Oualline 2003, Ch. 5
- ^ ISBN 978-3-319-54938-5.
- ^ Brown 1991, Definition I.2.1 (addition), Definition I.2.4 (scalar multiplication), and Definition I.2.33 (transpose)
- ^ Brown 1991, Theorem I.2.6
- ^ a b "How to Multiply Matrices". www.mathsisfun.com. Retrieved 2020-08-19.
- ^ Brown 1991, Definition I.2.20
- ^ Brown 1991, Theorem I.2.24
- ^ Horn & Johnson 1985, Ch. 4 and 5
- ^ Bronson (1970, p. 16)
- ^ Kreyszig (1972, p. 220)
- ^ a b Protter & Morrey (1970, p. 869)
- ^ Kreyszig (1972, pp. 241, 244)
- ISBN 978-0-486-13930-2.
- ISBN 978-0-486-66810-9.
- ISBN 978-0-470-45821-1.
- ISBN 978-0-521-83940-2.
- ^ Brown 1991, I.2.21 and 22
- ^ Greub 1975, Section III.2
- ^ Brown 1991, Definition II.3.3
- ^ Greub 1975, Section III.1
- ^ Brown 1991, Theorem II.3.22
- ^ Horn & Johnson 1985, Theorem 2.5.6
- ^ Brown 1991, Definition I.2.28
- ^ Brown 1991, Definition I.5.13
- ^ Horn & Johnson 1985, Chapter 7
- ^ Horn & Johnson 1985, Theorem 7.2.1
- ^ Horn & Johnson 1985, Example 4.0.6, p. 169
- ^ "Matrix | mathematics". Encyclopedia Britannica. Retrieved 2020-08-19.
- ^ Brown 1991, Definition III.2.1
- ^ Brown 1991, Theorem III.2.12
- ^ Brown 1991, Corollary III.2.16
- ^ Mirsky 1990, Theorem 1.4.1
- ^ Brown 1991, Theorem III.3.18
- ^ Eigen means "own" in German and in Dutch.
- ^ Brown 1991, Definition III.4.1
- ^ Brown 1991, Definition III.4.9
- ^ Brown 1991, Corollary III.4.10
- ^ Householder 1975, Ch. 7
- ^ Bau III & Trefethen 1997
- ^ Golub & Van Loan 1996, Algorithm 1.3.1
- ^ Golub & Van Loan 1996, Chapters 9 and 10, esp. section 10.2
- ^ Golub & Van Loan 1996, Chapter 2.3
- ISSN 0036-1445.
- Mathematica, see Wolfram 2003, Ch. 3.7
- ^ Press, Flannery & Teukolsky et al. 1992
- ^ Stoer & Bulirsch 2002, Section 4.1
- ^ Horn & Johnson 1985, Theorem 2.5.4
- ^ Horn & Johnson 1985, Ch. 3.1, 3.2
- ^ Arnold & Cooke 1992, Sections 14.5, 7, 8
- ^ Bronson 1989, Ch. 15
- ^ Coburn 1955, Ch. V
- ^ Lang 2002, Chapter XIII
- ^ Lang 2002, XVII.1, p. 643
- ^ Lang 2002, Proposition XIII.4.16
- ^ Reichl 2004, Section L.2
- ^ Greub 1975, Section III.3
- ^ Greub 1975, Section III.3.13
- ^ See any standard reference in a group.
- closedin the general linear group.
- ^ Baker 2003, Def. 1.30
- ^ Baker 2003, Theorem 1.2
- ^ Artin 1991, Chapter 4.5
- ^ Rowen 2008, Example 19.2, p. 198
- ^ See any reference in representation theory or group representation.
- ^ See the item "Matrix" in Itõ, ed. 1987
- ^ "Not much of matrix theory carries over to infinite-dimensional spaces, and what does is not so useful, but it sometimes helps." Halmos 1982, p. 23, Chapter 5
- ^ "Empty Matrix: A matrix is empty if either its row or column dimension is zero", Glossary Archived 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 Archived 2009-12-28 at the Wayback Machine
- ^ Fudenberg & Tirole 1983, Section 1.1.1
- ^ Manning 1999, Section 15.3.4
- ^ Ward 1997, Ch. 2.8
- ^ Stinson 2005, Ch. 1.1.5 and 1.2.4
- ^ Association for Computing Machinery 1979, Ch. 7
- ^ Godsil & Royle 2004, Ch. 8.1
- ^ Punnen 2002
- ^ Lang 1987a, Ch. XVI.6
- ^ Nocedal 2006, Ch. 16
- ^ Lang 1987a, Ch. XVI.1
- ^ Lang 1987a, Ch. XVI.5. For a more advanced, and more general statement see Lang 1969, Ch. VI.2
- ^ Gilbarg & Trudinger 2001
- stiffness method.
- ^ Latouche & Ramaswami 1999
- ^ Mehata & Srinivasan 1978, Ch. 2.8
- ISBN 978-0-19-850702-4
- ^ Krzanowski 1988, Ch. 2.2., p. 60
- ^ Krzanowski 1988, Ch. 4.1
- ^ Conrey 2007
- ^ Zabrodin, Brezin & Kazakov et al. 2006
- ^ Itzykson & Zuber 1980, Ch. 2
- ^ see Burgess & Moore 2007, section 1.6.3. (SU(3)), section 2.4.3.2. (Kobayashi–Maskawa matrix)
- ^ Schiff 1968, Ch. 6
- ^ Bohm 2001, sections II.4 and II.8
- ^ Weinberg 1995, Ch. 3
- ^ Wherrett 1987, part II
- ^ Riley, Hobson & Bence 1997, 7.17
- ^ Guenther 1990, Ch. 5
- ^ Shen, Crossley & Lun 1999 cited by Bretscher 2005, p. 1
- ^ ISBN 978-0-321-07912-1, p. 564-565
- ISBN 978-0-521-05801-8.
- ISBN 978-0-321-07912-1, p. 564
- ^ Merriam-Webster dictionary, Merriam-Webster, retrieved April 20, 2009
- ^ Although many sources state that J. J. Sylvester coined the mathematical term "matrix" in 1848, Sylvester published nothing in 1848. (For proof that Sylvester published nothing in 1848, see: J. J. Sylvester with H. F. Baker, ed., The Collected Mathematical Papers of James Joseph Sylvester (Cambridge, England: Cambridge University Press, 1904), vol. 1.) His earliest use of the term "matrix" occurs in 1850 in J. J. Sylvester (1850) "Additions to the articles in the September number of this journal, "On a new class of theorems," and on Pascal's theorem," The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 37: 363-370. From page 369: "For this purpose, we must commence, not with a square, but with an oblong arrangement of terms consisting, suppose, of m lines and n columns. This does not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants ... "
- ^ The Collected Mathematical Papers of James Joseph Sylvester: 1837–1853, Paper 37, p. 247
- ^ Phil.Trans. 1858, vol.148, pp.17-37 Math. Papers II 475-496
- ^ Dieudonné, ed. 1978, Vol. 1, Ch. III, p. 96
- ^ Knobloch 1994
- ^ Hawkins 1975
- ^ Kronecker 1897
- ^ Weierstrass 1915, pp. 271–286
- ^ Bôcher 2004
- ^ Mehra & Rechenberg 1987
- ^ Whitehead, Alfred North; and Russell, Bertrand (1913) Principia Mathematica to *56, Cambridge at the University Press, Cambridge UK (republished 1962) cf page 162ff.
- ISBN 0-486-28462-X.
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{{citation}}
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Physics references
- Bohm, Arno (2001), Quantum Mechanics: Foundations and Applications, Springer, ISBN 0-387-95330-2
- Burgess, Cliff; Moore, Guy (2007), The Standard Model. A Primer, Cambridge University Press, ISBN 978-0-521-86036-9
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Historical references
- A. Cayley A memoir on the theory of matrices. Phil. Trans. 148 1858 17–37; Math. Papers II 475–496
- ISBN 978-0-486-49570-5, reprint of the 1907 original edition
- Cayley, Arthur (1889), The collected mathematical papers of Arthur Cayley, vol. I (1841–1853), Cambridge University Press, pp. 123–126
- Dieudonné, Jean, ed. (1978), Abrégé d'histoire des mathématiques 1700-1900, Paris, FR: Hermann
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- MR 1308079
- Kronecker, Leopold (1897), Hensel, Kurt (ed.), Leopold Kronecker's Werke, Teubner
- ISBN 978-0-387-96284-9
- Shen, Kangshen; Crossley, John N.; Lun, Anthony Wah-Cheung (1999), Nine Chapters of the Mathematical Art, Companion and Commentary (2nd ed.), ISBN 978-0-19-853936-0
- Weierstrass, Karl (1915), Collected works, vol. 3
Further reading
- "Matrix", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- Kaw, Autar K. (September 2008), Introduction to Matrix Algebra, Lulu.com, ISBN 978-0-615-25126-4
- The Matrix Cookbook (PDF), retrieved 24 March 2014
- Brookes, Mike (2005), The Matrix Reference Manual, London: Imperial College, retrieved 10 Dec 2008
External links
- MacTutor: Matrices and determinants
- Matrices and Linear Algebra on the Earliest Uses Pages
- Earliest Uses of Symbols for Matrices and Vectors