# Eigenvalues and vectors - finding original matrix

1. Dec 11, 2007

### Caeder

How do I determine what the original matrix was that yielded these two eigenvalues with the corresponding eigenvectors:

$$\lambda_1 = -3$$ Eigenvector: [0,1]

$$\lambda_2 = 2$$ Eigenvector: [1,0]

I've played around with det(A-lambda I) but can't find the matrix! I even just did some trial and error matrices in Maple trying to figure it out. If anyone can find me the matrix I'd be very impressed.

2. Dec 11, 2007

### rock.freak667

Well if you have the eigenvectors of a matrix A then A can be represented as:
A=PDP$^{-1}$
where D is a diagonal matrix with the diagonal elements as $\lambda_1 \ and \ \lambda_2$

and P is the eigenvectors of the eigenvalues....

D would be the matrix:
[-3 0]
[0 2]

and the first column for P would be the eigenvector for -3 and the 2nd column would be the eigenvector for 2...so you now have P..find P$^{-1}$ and multiply out

3. Dec 11, 2007

### Office_Shredder

Staff Emeritus
What does the matrix
a11 a12
a21 a22

map (1,0) and (0,1) to?