# Linear Control Theory: How to solve this matrix

1. Oct 20, 2011

### aznkid310

1. The problem statement, all variables and given/known data
So the first part of the problem asks us to solve for the state transition matrix, which I found to be:

Phi(t) = [2 0 0; (2-e^-t) e^-t 0; 0 0 e^-2t];
I need to find the matrix A, which I assume can be done with the following relation:

Phit(t) = e^(At)

where A is also a matrix

2. Relevant equations
Need to solve for A. Not sure how to solve this and what properties hold when matrices are concerned.

3. The attempt at a solution

I initially thought to diagonlize Phi(t) but didnt really get anywhere.

Last edited: Oct 20, 2011
2. Oct 22, 2011

### Staff: Mentor

If Phi(t) = eAt, then ln(Phi(t)) = At.

See this wikipedia article on the logarithm of a matrix.

3. Oct 22, 2011

### aznkid310

Thanks for the reply Mark44. How do we deal with the ln(0) terms?

4. Oct 22, 2011

### Staff: Mentor

The log of a matrix is not just the log of the entries in the matrix. Is that what you're trying to do?

The link I provided shows some examples of finding the log of a matrix.

5. Oct 23, 2011

### aznkid310

Yes thats what I thought. Reading a little furthur, it seems that I should find the eigenvectors of phi(t), then do: ln(phi(t)) = V*ln(inv(V)*phi(t)*V)*inv(V), where V is the matrix of eigenvectors. But phi(t) is not diagonalizable, so I dont know how to proceed from here.

Last edited: Oct 23, 2011