Let

be a general
matrix and let

denote a
general

matrix. Denote the matrix product by

. Then
matrix multiplication is carried out by computing
the
inner product of every row of

with every column of

. Let the

th row of

be denoted by

,

, and the

th column of

by

,

. Then the matrix product

is
defined as

This definition can be extended to
complex matrices by using a
definition of inner product which does not conjugate its second
argument.
H.2
Examples:
An

matrix

can be multiplied on the
right by an

matrix, where

is any positive integer. An

matrix

can be multiplied on the
left by a

matrix, where

is any positive integer. Thus, the number of columns in
the matrix on the left must equal the number of rows in the matrix on the
right.
Matrix multiplication is
non-commutative, in general. That is,
normally

even when both products are defined (such as when the
matrices are square.)
The
transpose of a matrix product is the product of the
transposes in
reverse order:
The
identity matrix is denoted by

and is defined as
Identity matrices are always
square. The

identity
matrix

, sometimes denoted as

, satisfies

for every

matrix

. Similarly,

, for every

matrix

.
As a special case, a matrix

times a vector

produces a new vector

which consists of the inner product of every row of

with
A matrix

times a vector

defines a
linear transformation
of

. In fact, every linear function of a vector

can be
expressed as a matrix multiply. In particular, every linear
filtering operation can be expressed as a matrix multiply applied to the
input
signal. As a special case, every linear, time-invariant (
LTI)
filtering operation can be expressed as a matrix multiply in which the
matrix is
Toeplitz,
i.e.,
![$ \mathbf{A}^{\!\hbox{\tiny T}}[i,j] = \mathbf{A}^{\!\hbox{\tiny T}}[i-j]$](http://www.dsprelated.com/josimages_new/mdft/img2098.png)
(constant along
diagonals).
As a further special case, a row vector on the left may be multiplied by a
column vector on the right to form a
single inner product:
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