Finding a Fundamental Matrix for the System

[sndoyle1]

Homework Statement

Find a fundamental matrix for the system x'(t)=Ax(t); where
5 -3 -2
A = 8 -5 -4
-4 3 3

The second part of the question is to find e^At

Homework Equations

I know that you have to find the 3 eigenvectors and then the 'general solution' without putting the constants into the fundamental matrix. I'm having a super hard time finding the last eigen vector.

The Attempt at a Solution

I found the eigenvalues to be (λ-1)³ and when I completed for the first λ=1 I found that v1=[1 4 0] and v2= [1 0 2]. I tried to generalize to find v3 by using (A-λI)v3=v2 but I keep getting an invalid system.

Any help would be great.

 

[Mark44]

Homework Statement

Find a fundamental matrix for the system x'(t)=Ax(t); where
5 -3 -2
A = 8 -5 -4
-4 3 3

The second part of the question is to find e^At

Homework Equations

I know that you have to find the 3 eigenvectors and then the 'general solution' without putting the constants into the fundamental matrix. I'm having a super hard time finding the last eigen vector.

The Attempt at a Solution

I found the eigenvalues to be (λ-1)³

No, the eigenvalue is λ = 1.

and when I completed for the first λ=1 I found that v1=[1 4 0] and v2= [1 0 2].

You made a mistake in v₁ - it isn't an eigenvector. You can check this by verifying that Av₁ ≠ 1v₁.

Your other eigenvector is correct.

I tried to generalize to find v3 by using (A-λI)v3=v2 but I keep getting an invalid system.

Any help would be great.

 

[Dick]

And your matrix doesn't have three linearly independent eigenvectors. It only has two. If it had three it would be the identity matrix since it would be diagonalizable. I'm not sure what you mean by the 'fundamental matrix'. Can you define it?