Vector Space of Matrices (VSM)

The vector space [image] is the set of all mxn matrices with entries from the set of complex numbers.

Matrix Equality (ME)

The mxn matrices A and B are equal, written A = B, provided [image] for all [image].

Matrix Addition (MA)

Given the mxn matrices A and B, define the sum of A and B as an mxn matrix, with A + B, according to

[image]

Matrix Scalar Multiplication (MSM)

Given the mxn matrix A and the scalar [image], the scalar multiple of A is an mxn matrix with [image] defined according to

[image]

Transpose of a Matrix (TM)

Given an mxn matrix A, the transpose is the nxm matrix [image] given by

[image]

Symmetric Matrix (SYM)

The matrix A is symmetric if [image]

Complex Conjugate of a Matrix (CCM)

Suppose that A is an mxn matrix. Then the conjugate of A, written [image], is an mxn matrix defined by

[image]

[Each entry is conjugated]

Adjoint (A)

If A is a matrix, then the adjoint is [image]

([image] may also be written as [image])

[The adjoint is the conjugate and transposed or transposed and conjugated version of A].

Note: There are two unrelated meanings for 'adjoint' in linear algebra; need to check meaning when reading texts.

Suppose that A is an m x n matrice with columns [image] and u is a vector of size n. Then the matrix-vector product of Au is the linear combination

[image]

The result will be a vector of size m.

Suppose A is an m x n matrix and [image] are the columns in an n x p matrix, B. Then the matrix-product of A with B is the m x p matrix whter column i is the matrix-vector product [image]. Symbollically,

[image]

The square matrix A is Hermitian (or self-adjoint) if [image].

The set of real number matrices that are Hermitian is exactly the set of symmetric matrices.

Suppose A and B are square matrices of size n such that [image]. Then A is invertible and B is the inverse of A. In this situation we write [image].

Note that we can just as easily say A is the inverse of B.

Suppose that U is a square matrix of size n such that [image]. Then we say U is unitary.

Suppose that A is an m x n matrix with columns [image]. Then the column space of A, written C(A), is the subset of [image] containing all linear combinations of the columns of A

[image]

Another, popular, defintion is

[image]

Suppose A is an m x n matrix. Then the row space of A, R(A), is the column space of [image] i.e. R(A) = C([image]).