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 = (v₁,..., 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 = (a₁,..., a_n) and b = (b₁,..., b_n)

is the real number equal to the matrix product a^Tb. 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: