5.8 Diagonalizable Matrices
So far most of the examples we have considered were matrices that had as many distinct
eigenvalues as the matrix had rows and columns. This implies these eigenvalues have
multiplicity one. In this section and the next we consid
5.4 Differential Equations
In a differential equation the unknowns are one or more functions, e.g. x(t) and y(t). In
applications we are often observing some system that is changing with time and x and y
represent physical quantities that change with time
5.3 Difference Equations
In a difference equation the unknowns are one or more sequences of numbers, e.g.
x0, x1, x2, x3, , xn,
y0, y1, y2, y3, , yn,
In applications we are often observing some system that is changing with time and x and
y represent physi
5.2 Diagonalization and Matrix Powers
If A is a square matrix then An = AAA is A multiplied by itself n times.
Example 1. If A = , then
A2 =
=
A3 = A2A =
=
=
=
There is a nice formula for An in terms of the eigenvalues and eigenvectors of A. This
formula
5 Eigenvalues
Most of the material in the previous chapters has been related to solving linear equations.
In this chapter we look at eigenvalues and eigenvectors of matrices which are involved in
applications of matrices to problems in dynamics and optimi
5.5 Complex Eigenvalues
Let's look an example of a matrix whose eigenvalues are complex numbers.
Example 1. Find the eigenvalues and eigenvectors of A = .
For the eigenvalues one has
A - I =
0 = det( A - I ) =
= (- 1 - )(- 3 - ) (1)(- 26)
= 2 + 4 + 3 + 26
5.6 Differential Equations with Complex Eigenvalues
If the coefficient matrix in a system of differential equations has complex eigenvalues,
then the same solution method discussed in section 5.4 still works. However, there are
some steps that can be shor
6 Quadratic Functions
So far we have been concerned with linear functions. In this chapter we apply some of
the results on symmetric matrices to real valued quadratic functions. In particular, we are
interested in whether such a function assumes a maximum
6.2 Quadratic Forms
In the previous section we saw that a general quadratic function w = xTAx + bx + c could
be written as w = yTAy + d where y = x + A-1b and d = c - bTA-1b. In this way questions
about the maximum and minimum of a general quadratic funct
5.7 Symmetric Matrices
One interesting aspect of matrix algebra is drawing conclusions about the eigenvalues of
a matrix A from properties of the matrix itself. Some examples of this for a matrix with
real entries.
(1)
A symmetric
Eigenvalues real
(2)
A s
6.3 Positive Definite Quadratic Forms
In the previous section we saw that a quadratic function w = xTAx + bx + c had a
minimum at y = x + A-1b if all the eigenvalues of A are positive and a maximum at this
point if all the eigenvalues of A are positive. T
5.9 Generalized Eigenvectors
In the previous section we looked at the case where each eigenvalue of a square matrix A
has as many linearly independent eigenvectors as its multiplicity. In that case we could
diagonalize A and use this to solve differential