"Matrix" is the Latin word for womb. The origin of mathematical matrices has a long history. The term "matrix" in combinatorics was introduced in 1850 by the British mathematician James Joseph Sylvester (1814--1897), who also coined many mathematical terms or used them in "new or unusual ways" mathematically, such as graphs, discriminants, annihilators, canonical forms, minor, nullity, and many others.

When defining a whole matrix, vectors can be combined: try for example

` [1:2:9; 2:2:10] `

` zeros(5,2)`

**I**) of size

*m*can defined by typing eye(m) . The following command generates a \( 4 \times 4 \) identity matrix:

` eye(4)`

` [eye(4) zeros(4)]`

` [zeros(3); 3*eye(3)]`

` A=[1 2; 3 4]`

` A(2)`

` twenty=[1:10;11:20]`

` first_row=twenty(1,:)`

` third_column=twenty(:,3)`

chooses the third column. matlab can also easily calculate things like the trace(A), which is sum of the diagonal elements:

` trace(A)`

The norm of a matrix may be thought of as its size because it is a nonnegative number. Matrix norms are directly related to vector norms.
The definitions are summarized below for an \( m \times n \) matrix **A**.

1-norm (is commonly known as the maximum column sum norm) of a matrix **A** may be computed as

**A**may be computed as

In the special case of

*p*= 2 we get the Euclidean norm (which is equal to the largest singular value of a matrix)

Some properties of the matrix norms are presented in the following

Theorem: Let **A** and **B** be \( m \times n \) matrices
and let \( k \) be a scalar.

- \( \| {\bf A} \| \ge 0 \) for any square matrix
**A**. - \( \| {\bf A} \| =0 \) if and only if the matrix
**A**is zero: \( {\bf A} = {\bf 0}. \) - \( \| k\,{\bf A} \| = |k| \, \| {\bf A} \| \) for any scalar \( k. \)
- \( \| {\bf A} + {\bf B}\| \le \| {\bf A} \| + \| {\bf B} \| .\)
- \( \| {\bf A} \, {\bf B}\| \le \| {\bf A} \| \, \| {\bf B} \| \)

matlab has special commands for evaluating norms:

norm(A,1) for evaluating 1-norm of the matrix **A**;

norm(A,inf) for evaluating \( infty - \)norm of the matrix **A**;

norm(A,2) for evaluating 2-norm (Euclidean) of the matrix **A**;

**Example**. Evaluate the norms of the matrix
\( {\bf A} = \left[ \begin{array}{cc} 1 & -7 \\ -2 & -3 \end{array} \right] . \)

The absolute column sums of **A** are \( 1 + | -2 | =3 \) and \( |-7| + | -3 | =10 . \)
The larger of these is 10 and therefore \( \| {\bf A} \|_1 = 10 . \)

The absolute row sums of **A** are \( 1 + | -7 | =8 \) and
\( | -2 | + |-3| = 5 , \) therefore, \( \| {\bf A} \|_{\infty} = 8 . \)

The Euclidean norm of **A**
is

This matrix\( {\bf A}\,{\bf A}^{\ast} \) has two eigenvalues
\( \frac{1}{2} \left( 63 \pm \sqrt{2813} \right) . \) Hence, the Euclidean norm of the matrix **A** is
\( \sqrt{\frac{1}{2} \left( 63 + \sqrt{2813} \right)} . \)

The Frobenius norm: