# Matrices and Vectors

**3 Matrices and Vectors**

This Section is a very concise introduction to the algebra of matrices and
vectors. This will let us

study mathematical models of the types we saw in the preceding illustrative
examples in a more

general, unified fashion.

**3.1 Matrices
**

A (real) matrix of size m *n is an array of mn real numbers arranged in m rows and n columns:

The n*m matrix A^{T} obtained by exchanging rows
and columns of A is called the transpose of

A. A matrix A is said to be symmetric if A = A^{T} .

The sum of two matrices of equal size is the matrix of the entry-by-entry sums,
and the scalar

product of a real number a and an m *n matrix A is the m* n matrix of all the
entries of A, each

multiplied by a . The difference of two matrices of equal size A and B is

A - B = A + (-1)B :

The product of an m *p matrix A and a p* n matrix B is an m* n matrix C with
entries

**Examples**

Let

Then,

and

The product AB is not defined, because A and B have incompatible sizes. Furthermore,

and

**3.2 Vectors**

A (real) n-dimensional vector is an n-tuple of real numbers

v = (v_{1},..., v_{n}) .

There is a natural, one-to-one correspondence between n-dimensional vectors and
n *1 matrices:

The matrix on the right is called the column vector
corresponding to the vector on the left.

There is also a natural, one-to-one correspondence between n-dimensional vectors
and 1* n

matrices:

The matrix on the right is called the row vector corresponding to the vector on
the left.

If a is a vector, then the symbol a also denotes the corresponding column
vector, so that the

corresponding row vector is a^{T} .

All algebraic operations on vectors are inherited from the corresponding matrix
operations,

when defined. In addition , the inner product of two n-dimensional vectors

a = (a_{1},..., a_{n}) and b = (b_{1},..., b_{n})

is the real number equal to the matrix product a^{T}b. It is easy to
verify that this is also equal to

b^{T} a. Two vectors that have a zero inner product are said to be
orthogonal.

The norm of a vector a is

obviously a nonnegative number . A unit vector is a vector
with norm one.

Examples

The vector a = (2, -1, 0) corresponds to row vector a^{T} = [2, -1, 0]
and to column vector

3.2 Vectors

The inner product of a and b = (1, 0, -1) is

and the norm of a is

The vector

has unit norm:

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