Matrix mathematics

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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 ring-like 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.

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 m-by-n 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 j-th column is called the i,j entry or (i,j)-th entry of A. This is written as Ai,j or A[i,j].

We often write Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A:=(a_{i,j})_{m \times n}} to define an m × n matrix A with each entry in the matrix A[i,j] called aij for all 1 ≤ im and 1 ≤ jn.

Example

The matrix

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} 1 & 2 & 3 \\ 1 & 2 & 7 \\ 4&9&2 \\ 6&0&5\end{bmatrix}}

is a 4×3 matrix. The element A[2,3] or a2,3 is 7.

Adding, subtracting, and multiplying matrices

Sum

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If two m-by-n matrices A and B are given, we may define their sum A + B as the m-by-n matrix computed by adding corresponding elements, i.e., (A + B)[i, j] = A[i, j] + B[i, j]. For example

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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} }

Another, much less often used notion of matrix addition is the direct sum.

Difference

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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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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} }

These two operations turn the set M(m, n, R) of all m-by-n matrices with real entries into a real vector space of dimension mn.

Multiplication

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Multiplication of two matrices is well-defined 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 m-by-n matrix (m rows, n columns) and B is an n-by-p matrix (n rows, p columns), then their product AB is the m-by-p 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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} }

This multiplication has the following properties:

  • (AB)C = A(BC) for all k-by-m matrices A, m-by-n matrices B and n-by-p matrices C ("associativity").
  • (A + B)C = AC + BC for all m-by-n matrices A and B and n-by-k matrices C ("right distributivity").
  • C(A + B) = CA + CB for all m-by-n matrices A and B and k-by-m matrices C ("left 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. This same property makes them powerful data structures in high-level programming languages.

Here and in the sequel we identify Rn with the set of "rows" or n-by-1 matrices. For every linear map f : Rn -> Rm there exists a unique m-by-n matrix A such that f(x) = Ax for all x in Rn. We say that the matrix A "represents" the linear map f. Now if the k-by-m matrix B represents another linear map g : Rm -> Rk, then the linear map g o f is represented by BA. This follows from the above-mentioned associativity of matrix multiplication.

More generally, a linear map from an n-dimensional vector space to an m-dimensional vector space is represented by an m-by-n matrix, provided that bases have been chosen for each.

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 m-by-n matrix A is the n-by-m matrix Atr (also sometimes written as AT or tA) formed by turning rows into columns and columns into rows, i.e. Atr[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 Atr describes the transpose of the linear map with respect to the dual bases, see dual space.

We have (A + B)tr = Atr + Btr and (AB)tr = Btr * Atr.

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 n-by-n 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 In, with elements on the main diagonal set to 1 and all other elements set to 0, satisfies MIn=M and InN=N for any m-by-n matrix M and n-by-k matrix N. For example, if n = 3:

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The identity matrix is the identity element in the ring of square matrices.

Invertible elements in this ring are called invertible matrices or non-singular matrices. An n by n matrix A is invertible if and only if there exists a matrix B such that

AB = In ( = BA).

In this case, B is the inverse matrix of A, denoted by A−1. The set of all invertible n-by-n matrices forms a group (specifically a Lie group) under matrix multiplication, the general linear group.

If λ is a number and v is a non-zero 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−λIn is not invertible, which happens if and only if pA(λ) = 0. Here pA(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 Gauss-Jordan 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.

Matrix exponential is defined for square matrices, using power series.

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, ai,j=aj,i.
  • Skew-symmetric matrices are such that elements symmetric to the main diagonal are the negative of each other, that is, ai,j= - aj,i. In a skew-symmetric matrix, all diagonal elements are zero, that is, ai,i=0.
  • Hermitian (or self-adjoint) matrices are such that elements symmetric to the diagonal are each others complex conjugates, that is, ai,j=a*j,i, where the superscript '*' signifies complex conjugation.
  • Toeplitz matrices have common elements on their diagonals, that is, ai,j=ai+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 R-module Rn.

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 Gauss-Jordan 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 (1906-1995) 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

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