# EP220 Lecture notes - prepared by ```EP220 Lecture notes
- prepared by-
Prof. Dr. Eser OLĞAR
3.3 Power of a Matrix
Example:
18
2014-Fall
EP220 Lecture notes
- prepared by-
Prof. Dr. Eser OLĞAR
Example:
3.4 Application of eigenvalues to solving systems of Differential Equations
In this section we shall apply the matrix analysis, eigenvalues, and eigenvectors to solve
systems of first order ordinary differential equations. Differential equations and their
systems arise in many areas of mathematics and engineering, for examples in control theory
and in analysis of electrical circuits. The unknows in this equations are functions.
A number of techniques have been developed to solva such systems of equations. The
Laplace transform is one example of this techniques. We shall study anather analytical
technique based on eigenvalues and eigenvectors.
18
2014-Fall
EP220 Lecture notes
- prepared by-
Prof. Dr. Eser OLĞAR
Example:
18
2014-Fall
EP220 Lecture notes
- prepared by-
Prof. Dr. Eser OLĞAR
2014-Fall
Example: Solve the homogeneous linear system using eigenvalues and eigenvectors.
18
EP220 Lecture notes
- prepared by-
Prof. Dr. Eser OLĞAR
2014-Fall
2. Show that the matrix is not digonalizable.
3. Show that the matrix is digonalizable.
4. Show that the matrix is digonalizable with only one eigenvalue.
5. Find P and D such that P-1AP =D where
6. Find P and D such that P-1AP =D
7. Solve the following differential equations using eigenvalues
Exercises
1. Produce a matrix that diagonalizes the given matrix, or show that this matrix
is not diagonalizable.
0 −1
5 3
1 0
−5 3
a)
b)
c)
d)
4 3
−4 1
1 3
0 9
0 0 0
−2 0 0
2 0 0
5 0 0
e) 1 0 3
f) 1 0 2 g) 0
h) 0 2 1
2 1
0 1 3
0 −1 2
0 −1 2
0 0 −2
18
```