Let be a general matrix and let denote a general matrix. Denote the matrix product by or . Then matrix multiplication is carried out by computing the inner product of every row of with every column of . Let theth row of be denoted by , , and theth column of by , . Then the matrix product
is defined as
This definition can be extended to complexmatrices by using a
definition of inner product which does not conjugate its second
argument.7.4
Examples:
An matrix can only be multiplied on the right by an matrix, where is any positive integer. An matrix can only 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 transformationof . 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., (constant along alldiagonals).
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:
where the alternate transpose notation '''' is defined to
include complex conjugation so that the above result holds also for
complex vectors. Using this result, we may rewrite the general matrix
multiply as
be a general
matrix and let
denote a general
matrix. Denote the matrix product by
or
. Then matrix multiplication is carried out by computing the inner product of every row of
th row of
,
, and the
th column of
,
. Then the matrix product
can only be multiplied on the right by an
is any positive integer. An
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.
even when both products are defined (such as when the matrices are square.)
and is defined as
identity matrix
, satisfies
for every
matrix
, for every
produces a new vector
which consists of the inner product of every row of
(constant along alldiagonals).
'' is defined to include complex conjugation so that the above result holds also for complex vectors. Using this result, we may rewrite the general matrix multiply as
