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. For example the matrix below is a 4-by-3 matrix:

<math>\begin{bmatrix}

The entry of a matrix A that lies in the ith 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]. In the example above, A[2,3]=7.

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

Adding and multiplying matrices

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

<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}

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

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}

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 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.

<math>

  \begin{bmatrix}
    1 & 0 & 2 \\
    -1 & 3 & 1 \\
  \end{bmatrix}

  \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").

For other, less commonly encountered ways to multiply matrices, see matrix multiplication.

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 isn't 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:

<math>

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

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).

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 γ the associated eigenvalue. The number λ is an eigenvalue of A if and only if A-λI_n isn't 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.

Partitioning Matrices

A Partitioned Matrix or Block Matrix is a matrix of matricies. For example, take a matrix P:

<math>P = \begin{bmatrix}

We could partition it into a 2-by-2 partitioned matrix like this:

<math>P_{11} = \begin{bmatrix}

<math>P_{partitioned} = \begin{bmatrix}

This technique is used to cut down calculations of matricies, column-row expansions, and many computer science applications, including VLSI chip design.

Classes of real and complex matrices

Certain special matrices are so important that they are given special names, as listed in list of matrices. Below are some examples:

• 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.

Matrices with entries in arbitrary rings

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 above, and it has the same properties. 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.

If R is commutative, 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 above 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.

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