D.1 Matrices and Determinants

A rectangular array of real or complex numbers of the form

is a matrix. Matrix A is square if M = N . A horizontal line of numbers is called a row or row vector and a vertical line is called a column or column vector. The M N matrix has elements a _ij in which the first subscript denotes the row and the second subscript denotes the column.

The transpose A ^T of matrix A is obtained by interchanging the rows and columns.

A real square matrix is symmetric if it is equal to its transpose, so that A = A ^T. A real square matrix is skew symmetric if A = ? A ^T, in which case the elements a _ij = ? a _ji and a _ii = 0.

Multiplication of an M N matrix A with an R P matrix B is only defined when R = N . The resulting M P matrix C consists of elements that is the (inner) product of the j-th row vector of the matrix A and the k-th column vector of the matrix B. Matrix multiplication is associative and distributive

but is not, in general, commutative. Hence, in general

Further, AB = 0 does not require A = 0 or B =...