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

**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

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

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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University