Matrix

A matrix (plural matrices) is a rectangular table of numbers. Matrices are useful to record data that depend on two categories, such as the sales in three branches of a store in each of the four quarters of a year, or to keep track of the coefficients of linear expressions such as linear transformations and systems of linear equations. The field of mathematics that studies matrices is called matrix theory, a branch of linear algebra. Closely related terms from computing are "two-dimensional array" and "spreadsheet".

Horizontal lines in a matrix are termed rows while vertical lines are termed columns. If a matrix has m rows and n columns, then it is called an m-by-n matrix.

The position of an entry or element in a matrix is usually indicated with two indices as shown for this 4-by-3 matrix:

  / a₁₁  a₁₂  a₁₃ \
  | a₂₁  a₂₂  a₂₃ |
  | a₃₁  a₃₂  a₃₃ |
  \ a₄₁  a₄₂  a₄₃ /

As you see, in an expression a_ij the first index always denotes the row and the second index denotes the column of the element. If the matrix is called A, then the element in row i and column j is also written as A[i, j], or in C notation, A[i][j].

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] + B[j] for all i and j. Furthermore, given any real number or complex number c, we define the matrix cA by multiplying every element of A by c, i.e., (cA)[i, j] = c A[i, j] for all i and j. 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.

Matrices may also be multiplied, but the operation is not as simple as matrix addition. If A is an m-by-n matrix (i.e. m rows and n columns) and B is an n-by-p matrix (n rows, p columns), then their product AB is an 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

/ 1  0  2 \   / 3  1 \   / 1*3+0*2+2*1  1*1+0*1+2*0 \   / 5  1 \
\-1  3  1 / * | 2  1 | = \-1*3+3*2+1*1 -1*1+3*1+1*0 / = \ 4  2 /    
              \ 1  0 /

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").

Matrices can conveniently represent linear transformations because matrix multiplication neatly corresponds to the composition of maps, as will be described next.

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. Here and in the sequel we identify Rⁿ with the set of "rows" or n-by-1 matrices. 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 associativity.

The set M(n, R) of all square n-by-n matrices with real entries, together with matrix addition and matrix multiplication is a ring, in fact a real unitary associative algebra. Unless n = 1, this ring is not commutative. The unit matrix I_n, with all elements on the main diagonal set to 1 and all other elements set to 0, is the unit element of this ring. For example, if n = 3:

       / 1  0  0 \
I  =   | 0  1  0 |
 3     \ 0  0  1 /

Invertible elements in this ring are called invertible matrices or non-singular matrices. Most square matrices are invertible, namely all the ones with non-zero determinant. To compute the inverse of a matrix, use Gauss-Jordan elimination. The set Gl(n, R) of all invertible n-by-n matrices forms a group (specifically a Lie group) under matrix multiplication, the general linear group.

For non-invertible (and even non-square) matrices, the concept of rank is useful: in a sense, it measures "how close" the matrix is to being invertible.

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

In the applications, for a given square matrix A it is often important to get a handle on those non-zero vectors x and numbers λ such that A x = λ x. Such a vector is called an eigenvector of A and λ is called an eigenvalue of A.

The sum of all the diagonal entries of a square matrix is called the trace of the matrix.

There are several classes of matrixes, that is, matrixes with special properties. Such classes are:

symmetric matrices are such that elements symmetric to the diagonal 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.