Linear Algebra, Matrix

Matrix

This seems like a good time to define a matrix. A matrix is a two dimensional array, or grid, of elements from a ring R, usually real or complex numbers. Sometimes we will present the grid in two dimensions, especially if it is large, but smaller matricies will be given in line, with pipes acting as row delimiters. Here is a 3 by 3 matrix presented both ways.

[7,3,9|6,2,1|4,8,-3]

7	3	9
6	2	1
4	8	-3

If you program in C, Java, Perl, and most computer languages, the element in the i^th row and j^th column of a matrix is denoted M_i,j, where rows and columns are numbered starting at 0. Mathematicians also use row-order notation, but they often number rows and columns starting at 1. Thus the upper left entry is M_1,1. Fortran also starts numbering at 1, but it runs in column order, not row order.

A matrix is m by n, or m×n, if it has m rows and n columns. A matrix is square if m = n.

The main diagonal of a matrix, usually a square matrix, is the set of elements M_i,i, from upper left to lower right.

The identity matrix has ones down the main diagonal and zeros elsewhere.

The transpose of a matrix is its reflection through the main diagonal. The transpose of M is written M^T. A matrix is symmetric if it equals its transpose. A symmetric matrix is necessarily square.

If R is commutative, (ab)^T = b^Ta^T.

The conjugate of a matrix of complex numbers is produced by conjugating all the entries. The tranjugate of a matrix is the conjugate of the transpose, or the transpose of the conjugate.