Exercise 13 Solve the system
using the result of Exercise 12(b).
Finding the Inverse of a Square Matrix
Given 
, we must find
such that AB = I (the
identity matrix). Therefore, the first column of B must satisfy
(this vector
is the 1st column of I). Similarly, for the other columns of B. For example, the
jth column of B
satisfies 
(the jth column of I). So in order
to get the inverse of an n × n
matrix, we must solve n linear systems . However, the same steps of the
Gauss -Jordan elimination
procedure are needed for all of these systems. So we solve them all at once,
using the matrix form.
Example: Find the inverse of 
We need to solve the following matrix equation
We divide the first row by 3 to introduce a 1 in the top
left corner.
Then we add four times the first row to the second row to
introduce a 0 in the first column.
Multiply the second row by 3.
Add 
the second row to
the first. (All this is classical Gauss-Jordan elimination.)
As IB = B, we get
It is important to note that, in addition to the two
elementary row operations introduced earlier
in the context of the Gauss-Jordan elimination procedure, a third elementary row
operation may
sometimes be needed here, namely permuting two rows.
Example: Find the inverse of 
.
Because the top left entry of A is 0, we need to permute
rows 1 and 2 first.
Now we divide the first row by 2.
Next we add -
the
second row to the first
and we are done, since the matrix in front of B is the
identity.
Exercise 14 Find the inverse of the matrix
Exercise 15 Find the inverse of the matrix
1.5 Determinants
To each square matrix, we associate a number , called its determinant, defined as
follows:
If 
, then
If 
, then
.
For a square matrix A of dimensions n × n, the de terminant can be obtained as
follows. First,
define 
as the matrix of dimensions (n - 1) × (n - 1) obtained from A by
deleting row 1 and
column j. Then
Note that, in this formula, the signs alternate between + and -.
For example, if 
, then
Determinants have several interesting properties . For example, the fol lowing
statements are
equivalent for a square matrix A:
•
det(A) = 0,
•
A has no inverse, i.e. A is singular,
•
the columns of A form a set of linearly dependent vectors,
•
the rows of A form a set of linearly dependent vectors.
For our purpose, however, determinants will be needed mainly in our discussion
of classical opti-
mization, in conjunction with the material from the following section.
Exercise 16 Compute the determinant of 
.
1.6 Positive Definite Matrices
When we study functions of several variables (see Chapter 3!), we will need the
following matrix
notions.
A square matrix A is positive definite if xTAx > 0 for all non zero column vectors
x. It is
negative definite if xTAx < 0 for all nonzero x. It is positive semidefinite if
xTAx ≥ 0 and negative
semidefinite if xTAx ≤ 0 for all x. These definitions are hard to check directly and
you might as
well forget them for all practical purposes.
More useful in practice are the following properties, which hold when the matrix
A is symmetric
(that will be the case of interest to us), and which are easier to check.
The ith principal minor of A is the matrix 
formed by the
first i rows and
columns of A. So,
the first principal minor of A is the matrix 
, the second principal
minor is the matrix
, and so on.
•The matrix A is positive definite if all its principal
minors 
An have strictly positive
determinants.
• f these determinants are nonzero and alternate in signs, starting with
, then the
matrix A is negative definite.
•
If the determinants are all nonnegative, then the matrix is positive semidefinite,
If the determinant alternate in signs, starting with
, then the matrix
is negative
semidefinite.
To x ideas, consider a 2 × 2 symmetic matrix 
.
It is positive definite if:
and negative definite if:
It is positive semidefinite if:
and negative semidefinite if:
Exercise 17 Check whether the following matrices are positive definite, negative
definite, positive
semidefinite, negative semidefinite or n one of the above .
