Matrix Definitions

A matrix  is a rectangular array of numbers. If the array has nrows  and m columns  then it is said to be an n x mmatrix. The element in the i^th row and j^th column of the matrix A is denoted by A_ij. The notation A = (A_ij) is common.

$$\begin{pmatrix}A_{11}&A_{12}&A_{13}&\cdots&A_{1m}\\ A_{21}&A_{22}... ...dots& \vdots& \ddots& \vdots\\ A_{n1}&A_{n2}&A_{n3}&\cdots&A_{nm} \end{pmatrix}$$

A matrix with the same number of rows as columns is called a square  matrix. A matrix with one column is called a column vector  and a matrix with one row is called a row vector .

The diagonal  of the square matrix A is the top-left to bottom-right diagonal A₁₁, A₂₂,...,A_nn.

A square matrix is said to be diagonal  if all the elements off the diagonal are zero. That is to say A_ij = 0 if i$\ne$j.

A square matrix is said to be upper triangular  if all elements below the diagonal are zero. Similarly for lower triangular .

A square matrix is said to be symmetric  if its elements are symmetric about the diagonal. That is to say A_ij = A_ji for all i and j.

A square matrix is said to be skew-symmetric  or anti-symmetric  if A_ij = - A_ji for all i and j. In particular this implies that the diagonal elements must be zero (since A_ii = - A_ii).

$$\begin{pmatrix}1&2&3\\ 2&5&7\\ 3&7&2 \end{pmatrix} \begin{pmatrix}0&2&3\\ -2&0&1\\ -3&-1&0 \end{pmatrix}$$

The n x n Identity Matrix  (or Unit Matrix) is the diagonal matrix I_n with 1's down its diagonal. It has to be a square matrix. If the size of the matrix is obvious from context we usually just write I for the Identity matrix.

The n x m Zero Matrix  0 is the matrix of that size which has all its elements zero.

The Transpose  of the n x m matrix A is the m x nmatrix A^T given by A^T_ij = A_ji.