# Matrices

A *matrix* is defined as a rectangular array of numbers, *e.g.*,

*rows*, and is the number of

*columns*of the matrix. For example, the general matrix is

^{H.1}of a matrix may be denoted by , , or . For example, in the above two examples. The rows and columns of matrices are normally numbered from instead of from 0; thus, and . When , the matrix is said to be

*square*.

The *transpose* of a real matrix
is denoted by
and is defined by

A *complex matrix*
, is simply a
matrix containing complex numbers. The
*transpose* of a complex matrix is normally defined to
include *conjugation*. The conjugating transpose operation is called the
*Hermitian transpose*. To avoid confusion, in this tutorial,
and the word ``transpose'' will always denote transposition
*without* conjugation, while conjugating transposition will be
denoted by and be called the ``Hermitian transpose'' or the
``conjugate transpose.'' Thus,

## Matrix Multiplication

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

*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

*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

*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.*, (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*:

## Solving Linear Equations Using Matrices

Consider the linear system of equations

in matrix form:

*determinant*of the matrix ) is nonzero. For larger matrices, numerical algorithms are used to invert matrices, such as used by Matlab based on LINPACK [25]. An initial introduction to matrices and linear algebra can be found in [47].

**Next Section:**

Matlab/Octave Examples

**Previous Section:**

Number Systems for Digital Audio