# Get matrix exponential

1. Feb 11, 2009

### clayy

1. The problem statement, all variables and given/known data
I have a exam and i dont know how get matrix exponential:

| 2*t t|
| 3*t -t|
it is a 2x2 matrix.
where 't' is not a constant ,it is a variable

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Feb 11, 2009

### HallsofIvy

If A is a diagonal matrix with only 0 off the diagonal, aii on the diagonal, then eA is a diagonal matrix with eaii on the diagonal. If A is a "diagonalizable" matrix, that is, if there exist a matrix P such that PAP-1= D, a diagonal matrix, then $e^A= P^{-1}e^DP$.

If A is not diagonalizable, then you can still put it in "Jordan Normal Form", which is "near diagonal", but the exponential is more complicated.

Fortunately for you, this particular matrix has two distinct eigenvalues for every t (except 0 in which case the matrix is identically 0 and so its exponential is the identity matrix) and so is diagonalizable. Find the eigenvalues and corresponding eigenvectors. The exponential will be P-1DP where P is the matrix having the eigenvectors as columns and D is the diagonal matrix with $e^{\lambda}$ on the diagonal with $\lambda$ being the eigenvalues.

3. Feb 11, 2009