Linear Algebra

Matrices

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 depends 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 mþn 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 A[i,j] or A_i,j, or in notation of the C programming language, A[i][j].

The notation A = (a_ij) means that A[i,j] = a_ij for all indices i and j.

Examples

The matrix

$\begin{bmatrix} 1 & 2 & 3 \\ 1 & 2 & 7 \\ 4&9&2 \\ 6&1&5\end{bmatrix}$

is a 4þ3 matrix. The element A[2,3] or a_2,3 is 7.

Adding and multiplying matrices

Sum

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

$\begin{bmatrix} 1 & 3 & 2 \\ 1 & 0 & 0 \\ 1 & 2 & 2 \end{bmatrix} + \begin{bmatrix} 0 & 0 & 5 \\ 7 & 5 & 0 \\ 2 & 1 & 1 \end{bmatrix} = \begin{bmatrix} 1+0 & 3+0 & 2+5 \\ 1+7 & 0+5 & 0+0 \\ 1+2 & 2+1 & 2+1 \end{bmatrix} = \begin{bmatrix} 1 & 3 & 7 \\ 8 & 5 & 0 \\ 3 & 3 & 3 \end{bmatrix}$

Another, much less often used notion of matrix addition can be found at Direct sum (Matrix).

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

$2 \begin{bmatrix} 1 & 8 & -3 \\ 4 & -2 & 5 \end{bmatrix} = \begin{bmatrix} 2\times 1 & 2\times 8 & 2\times -3 \\ 2\times 4 & 2\times -2 & 2\times 5 \end{bmatrix} = \begin{bmatrix} 2 & 16 & -6 \\ 8 & -4 & 10 \end{bmatrix}$

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

Main article: Matrix multiplication

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

$\begin{bmatrix} 1 & 0 & 2 \\ -1 & 3 & 1 \\ \end{bmatrix} \times \begin{bmatrix} 3 & 1 \\ 2 & 1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} (1 \times 3 + 0 \times 2 + 2 \times 1) & (1 \times 1 + 0 \times 1 + 2 \times 0) \\ (-1 \times 3 + 3 \times 2 + 1 \times 1) & (-1 \times 1 + 3 \times 1 + 1 \times 0) \\ \end{bmatrix} = \begin{bmatrix} 5 & 1 \\ 4 & 2 \\ \end{bmatrix}$

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 ("distributivity").
C(A + B) = CA + CB for all m-by-n matrices A and B and k-by-m 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 Rⁿ with the set of "rows" or n-by-1 matrices. For every linear map f : Rⁿ -> R^m 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 : R^m -> R^k, then the linear map g o f is represented by BA. This follows from the above-mentioned 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 m-by-n matrix A is the n-by-m matrix A^tr (also sometimes written as A^T or ^tA) 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 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 I_n, with elements on the main diagonal set to 1 and all other elements set to 0, satisfies MI_n=M and I_nN=N for any m-by-n matrix M and n-by-k matrix N. For example, if n = 3:

$I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

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 = I_n ( = 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−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 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.

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.
Hermitian (or self-adjoint) 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.

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