C.1 Unitary Vector Space

This section recalls the basic definitions and the basic formulae of linear algebra. For a more detailed treatment, see for example [1 3].

A matrix is a rectangular array of scalars (e.g., integer, real, or complex numbers or integer-, real-, or complex-valued functions). The rectangular array

(C.1)

is called a matrix of m rows and n columns. We denote matrices with boldface letters. We may use the index ? m n ? to indicate the numbers of rows and columns. A matrix of type ? m n ? is said to be an m by n matrix. A matrix of type ? n n ? is called a square matrix of order n. A matrix of type ? m 1 ? is a column vector, and a matrix of type ?1 n ? is a row vector,

(C.2a)

(C.2b)

For two matrices A and B of the same size ? m n ? a matrix sum

(C.3)

is defined such that

(C.4)

for each i and j. The product of a matrix A and B with a scalar b

(C.5)

is defined by

(C.6)

For two matrices A and B of the type ? m l ? and ? l n ?, a product

(C.7)

is...