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

which is a

(``two by two'') matrix. A general matrix may
be

, where

is the number of
rows,
and

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

matrix is
Either square brackets or large parentheses may be used to delimit the
matrix. The

th element
H.1 of a matrix

may be denoted by
![$ \mathbf{A}[i,j]$](http://www.dsprelated.com/josimages_new/mdft/img2053.png)
,

, or

. For example,
![$ \mathbf{A}[1,2]=b$](http://www.dsprelated.com/josimages_new/mdft/img2056.png)
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
While

is

, its transpose is

. We may
say that the ``rows and columns are interchanged'' by the transpose
operation, and transposition can be visualized as ``flipping'' the
matrix about its main diagonal. For example,
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,
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:
Consider the
linear system of equations
in
matrix form:
This can be written in higher level form as
where

denotes the two-by-two matrix above, and

and

denote the two-by-one vectors. The solution to this equation
is then
The general two-by-two matrix inverse is given by
and the inverse exists whenever

(which is called the
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].
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