Sec7'3: Eigenvalue Method

1
Sec7.3 Eigenvalue Method
Our Goal
Solve the Homog linear system
Solution 1) Find n linearly independent
solutions 2) The general
solution is their linear combination
Example
Solve
Method
Find all eigenvalues of the matrix A
1
Distinct real Eigenvalues
repeated real Eigenvalues
complex Eigenvalues
2
Sec7.3 Eigenvalue Method
Method
Find all eigenvalues of the matrix A
1
Distinct real Eigenvalues
Find all eigenvectors of the matrix A
2
N-lin. Independent solutions are
3
The general solution is
4
3
Sec7.3 Eigenvalue Method
1
Solve
2
3
4
The general solution
4
Sec7.3 Eigenvalue Method
1
2
3
4
Solve
The general solution
5
Sec7.3 Eigenvalue Method
Method
Find all eigenvalues of the matrix A
1
Complex conjugate eigenvalues
Find an eigenvector for
2
Complex vector
solution is
3
Two-lin independent solutions are
4
Two-lin independent solutions are
5
6
Sec7.3 Eigenvalue Method
Solve
The general solution
7
Sec7.3 Eigenvalue Method
Method
Method
Find all eigenvalues of the matrix A
1
Find all eigenvalues of the matrix A
1
Complex conjugate eigenvalues
Complex conjugate eigenvalues
Find an eigenvector for
2
Find an eigenvector for
2
Complex vector
Two-lin independent solutions are
solution is
3
3
Two-lin independent solutions are
4
Two-lin independent solutions are
4
Two-lin independent solutions are
5
8
Sec7.3 Eigenvalue Method
Solve