Matrix (mathematics)

For the square matrix section, see square matrix.
In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, of elements of a ringlike algebraic structure. In this article, the entries of a matrix are real or complex numbers unless otherwise noted.
Matrices are useful to record data that depend on two categories, and to keep track of the coefficients of systems of linear equations and linear transformations.
For the development and applications of matrices, see matrix theory.
Contents 
Definitions and notations
The horizontal lines in a matrix are called rows and the vertical lines are called columns. A matrix with m rows and n columns is called an mbyn matrix (or m×n matrix) and m and n are called its dimensions.
The entry of a matrix A that lies in the i th row and the jth column is called the i,j entry or (i,j)th entry of A.
This is written as A[i,j] or A_{i,j} (or in notation of the C programming language, A[i][j]
).
We often write <math>A:=(a_{i,j})_{m \times n}<math> to define an m × n matrix A with each entry in the matrix A[i,j] called a_{ij} for all 1 ≤ i < m and 1 ≤ j < n.
Example
The matrix
 <math>\begin{bmatrix}
1 & 2 & 3 \\ 1 & 2 & 7 \\ 4&9&2 \\ 6&1&5\end{bmatrix}<math>
is a 4×3 matrix. The element A[2,3] or a_{2,3} is 7.
Adding and multiplying matrices
Sum
Main article: Matrix addition
If two mbyn matrices A and B are given, we may define their sum A + B as the mbyn matrix computed by adding corresponding elements, i.e., (A + B)[i, j] = A[i, j] + B[i, j]. For example
 <math>
\begin{bmatrix} 1 & 3 & 2 \\ 1 & 0 & 0 \\ 1 & 2 & 2 \end{bmatrix}
+
\begin{bmatrix} 0 & 0 & 5 \\ 7 & 5 & 0 \\ 2 & 1 & 1 \end{bmatrix}
=
\begin{bmatrix} 1+0 & 3+0 & 2+5 \\ 1+7 & 0+5 & 0+0 \\ 1+2 & 2+1 & 2+1 \end{bmatrix}
=
\begin{bmatrix} 1 & 3 & 7 \\ 8 & 5 & 0 \\ 3 & 3 & 3 \end{bmatrix}
<math>
Another, much less often used notion of matrix addition is the direct sum.
Scalar multiplication
If a matrix A and a number c are given, we may define the scalar multiplication cA by (cA)[i, j] = cA[i, j]. For example
 <math>2
\begin{bmatrix} 1 & 8 & 3 \\ 4 & 2 & 5 \end{bmatrix}
=
\begin{bmatrix} 2\times 1 & 2\times 8 & 2\times 3 \\ 2\times 4 & 2\times 2 & 2\times 5 \end{bmatrix}
=
\begin{bmatrix} 2 & 16 & 6 \\ 8 & 4 & 10 \end{bmatrix}
<math>
These two operations turn the set M(m, n, R) of all mbyn matrices with real entries into a real vector space of dimension mn.
Multiplication
Main article: Matrix multiplication
Multiplication of two matrices is welldefined only if the number of columns of the first matrix is the same as the number of rows of the second matrix. If A is an mbyn matrix (m rows, n columns) and B is an nbyp matrix (n rows, p columns), then their product AB is the mbyp matrix (m rows, p columns) given by
 (AB)[i, j] = A[i, 1] * B[1, j] + A[i, 2] * B[2, j] + ... + A[i, n] * B[n, j] for each pair i and j.
For instance
 <math>
\begin{bmatrix} 1 & 0 & 2 \\ 1 & 3 & 1 \\ \end{bmatrix}
\times
\begin{bmatrix} 3 & 1 \\ 2 & 1 \\ 1 & 0 \end{bmatrix}
=
\begin{bmatrix} (1 \times 3 + 0 \times 2 + 2 \times 1) & (1 \times 1 + 0 \times 1 + 2 \times 0) \\ (1 \times 3 + 3 \times 2 + 1 \times 1) & (1 \times 1 + 3 \times 1 + 1 \times 0) \\ \end{bmatrix}
=
\begin{bmatrix} 5 & 1 \\ 4 & 2 \\ \end{bmatrix}
<math>
This multiplication has the following properties:
 (AB)C = A(BC) for all kbym matrices A, mbyn matrices B and nbyp matrices C ("associativity").
 (A + B)C = AC + BC for all mbyn matrices A and B and nbyk matrices C ("distributivity").
 C(A + B) = CA + CB for all mbyn matrices A and B and kbym matrices C ("distributivity").
It is important to note that commutativity does not generally hold; that is, given matrices A and B and their product defined, then generally AB ≠ BA.
Matrices are said to anticommute if AB = BA. Such matrices are very important in representations of Lie algebras and in Representations of Clifford algebras
Linear transformations, ranks and transpose
Matrices can conveniently represent linear transformations because matrix multiplication neatly corresponds to the composition of maps, as will be described next.
Here and in the sequel we identify R^{n} with the set of "rows" or nby1 matrices. For every linear map f : R^{n} > R^{m} there exists a unique mbyn matrix A such that f(x) = Ax for all x in R^{n}. We say that the matrix A "represents" the linear map f. Now if the kbym matrix B represents another linear map g : R^{m} > R^{k}, then the linear map g o f is represented by BA. This follows from the abovementioned associativity of matrix multiplication.
The rank of a matrix A is the dimension of the image of the linear map represented by A; this is the same as the dimension of the space generated by the rows of A, and also the same as the dimension of the space generated by the columns of A.
The transpose of an mbyn matrix A is the nbym matrix A^{tr} (also sometimes written as A^{T} or ^{t}A) gotten by turning rows into columns and columns into rows, i.e. A^{tr}[i, j] = A[j, i] for all indices i and j. If A describes a linear map with respect to two bases, then the matrix A^{tr} describes the transpose of the linear map with respect to the dual bases, see dual space.
We have (A + B)^{tr} = A^{tr} + B^{tr} and (AB)^{tr} = B^{tr} * A^{tr}.
Square matrices and related definitions
A square matrix is a matrix which has the same number of rows as columns. The set of all square nbyn matrices, together with matrix addition and matrix multiplication is a ring. Unless n = 1, this ring is not commutative.
M(n, R), the ring of real square matrices, is a real unitary associative algebra. M(n, C), the ring of complex square matrices, is a complex associative algebra.
The unit matrix or identity matrix I_{n}, with elements on the main diagonal set to 1 and all other elements set to 0, satisfies MI_{n}=M and I_{n}N=N for any mbyn matrix M and nbyk matrix N. For example, if n = 3:
 <math>
I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}
<math> The identity matrix is the identity element in the ring of square matrices.
Invertible elements in this ring are called invertible matrices or nonsingular matrices. An n by n matrix A is invertible if and only if there exists a matrix B such that
 AB = I_{n} ( = BA).
In this case, B is the inverse matrix of A, denoted by A^{−1}. The set of all invertible nbyn matrices forms a group (specifically a Lie group) under matrix multiplication, the general linear group.
If λ is a number and v is a nonzero vector such that Av = λv, then we call v an eigenvector of A and λ the associated eigenvalue. (Eigen means "own" in German.) The number λ is an eigenvalue of A if and only if A−λI_{n} is not invertible, which happens if and only if p_{A}(λ) = 0. Here p_{A}(x) is the characteristic polynomial of A. This is a polynomial of degree n and has therefore n complex roots (counting multiple roots according to their multiplicity). In this sense, every square matrix has n complex eigenvalues.
The determinant of a square matrix A is the product of its n eigenvalues, but it can also be defined by the Leibniz formula. Invertible matrices are precisely those matrices with nonzero determinant.
The GaussJordan elimination algorithm is of central importance: it can be used to compute determinants, ranks and inverses of matrices and to solve systems of linear equations.
The trace of a square matrix is the sum of its diagonal entries, which equals the sum of its n eigenvalues.
Every orthogonal matrix is a square matrix.
Special types of matrices
In many areas in mathematics, matrices with certain structure arise. A few important examples are
 Symmetric matrices are such that elements symmetric to the main diagonal (from the upper left to the lower right) are equal, that is, a_{i,j}=a_{j,i}.
 Skewsymmetric matrices are such that elements symmetric to the main diagonal are the negative of each other, that is, a_{i,j}=  a_{j,i}. In a skewsymmetric matrix, all diagonal elements are zero, that is, a_{i,i}=0.
 Hermitian (or selfadjoint) matrices are such that elements symmetric to the diagonal are each others complex conjugates, that is, a_{i,j}=a^{*}_{j,i}, where the superscript '*' signifies complex conjugation.
 Toeplitz matrices have common elements on their diagonals, that is, a_{i,j}=a_{i+1,j+1}.
 Stochastic matrices are square matrices whose columns are probability vectors; they are used to define Markov chains.
For a more extensive list see list of matrices.
Matrices in abstract algebra
If we start with a ring R, we can consider the set M(m,n, R) of all m by n matrices with entries in R. Addition and multiplication of these matrices can be defined as in the case of real or complex matrices (see below). The set M(n, R) of all square n by n matrices over R is a ring in its own right, isomorphic to the endomorphism ring of the left Rmodule R^{n}.
Similarly, if the entries are taken from a semiring S, matrix addition and multiplication can still be defined as usual. The set of all square n×n matrices over S is itself a semiring. Note that fast matrix multiplication algorithms such as the Strassen algorithm generally only apply to matrices over rings and will not work for matrices over semirings that are not rings.
If R is a commutative ring, then M(n, R) is a unitary associative algebra over R. It is then also meaningful to define the determinant of square matrices using the Leibniz formula; a matrix is invertible if and only if its determinant is invertible in R.
All statements mentioned in this articles for real or complex matrices remain correct for matrices over an arbitrary field.
Matrices over a polynomial ring are important in the study of control theory.
History
The study of matrices is quite old. Latin squares and magic squares have been studied since prehistoric times.
Matrices have a long history of application in solving linear equations. Leibniz, one of the two founders of calculus, developed the theory of determinants in 1693. Cramer developed the theory further, presenting Cramer's rule in 1750. Carl Friedrich Gauss and Wilhelm Jordan developed GaussJordan elimination in the 1800s.
The term "matrix" was first coined in 1848 by J. J. Sylvester. Cayley, Hamilton, Grassmann, Frobenius and von Neumann are among the famous mathematicians who have worked on matrix theory.
Olga Taussky Todd (19061995) started to use matrix theory when investigating an aerodynamic phenomenon called flutter, during WWII.
Further reading
A more advanced article on matrices is matrix theory.
External links
 Matrix name and history: very brief overview (http://www.ualr.edu/~lasmoller/matrices.html)
 WIMS Matrix Calculator (http://wims.unice.fr/wims/wims.cgi?module=tool/linear/matrix.en) computes determinant, rank, inverse etc. online.bg:Матрица
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