Engineering Analysis 4
Workshop questions  week of 11/09/08
1. Give an example of a matrix which has repeated eigenvalues, but which is not defective.
Answer 
Defective matrices are characterized by having repeated eigenvalues, but not enough
eigenvectors for each of the eigenvalues. A nondefective matrix can have a repeated eigen
value, but it must have two linearly independent eigenvectors to go along with the eigenvalue.
A possible matrix is
±
4
0
0
4
²
,
which has 4 as a repeated eigenvalue. It is easy to see that this matrix has two linearly inde
pendent eigenvectors, for example,
~
v
1
=
±
1
0
²
,
~
v
2
=
±
0
1
²
.
2. Consider the forced system
~x
0
=
A~x
+
te
2
t
~
f ,
(1)
where
~
f
a constant vector and
A
is a matrix.
(a) Set up the equations to Fnd a particular solution using undetermined coefFcient vectors.
Answer 
You have to take as trial functions the forcing term and all of its derivatives.
Thus, we try
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 TAFLOVE
 Linear Algebra, Equations, Vector Space, linearly independent eigenvectors

Click to edit the document details