# Thread: Determining Matrix P

1. ## Determining Matrix P

Hey there,
So I've been given this question, determine a matrix P so that P^-1AP=B?
The questions relate to the matrices
A = [1 2 3]
[0 -1 2]
[0 0 3]

B= [-1 -2 3]
[0 1 -5]
[0 0 3]

The eigenvalues for the matrices are the same: lamda equals 3, 1 and -1.
And the eigenvectors for lamda equals 1 and -1 are the same for both matrices: for 1: 1, 0, 0 and for -1: -1, 1, 0.
The eigenvectors for lamda equals 3 in A: 2, 1/2, 1.
The eigenvectors for lamda equals 3 in B: 2, -5/2, 1.

So I've done a lot of working out so far but now I'm stuck! Can anyone let me know how to go about finding matrix P.
I thought I was just meant to put the eigenvectors of A into a matrix:
e.g. [1 -1 2]
[0 1 1/2]
[0 0 1]
However, that didn't work so I'm really unsure now!
Thanks heaps in advance.

2. ## Re: Determining Matrix P

Hey grooverandshaker.

If you have the eigen-vectors then P is just made up by using the eigenvectors corresponding to a particular eigen-value while D has the eigenvalues in the diagonal (with zeroes everywhere else).

Try putting the eigenvectors on the appropriate rows of P, calculate P inverse and then do the multiplication and see if you get B back.

If you ever want to double check your work use something like Octave which does a lot of MATLAB does only its free and open source. (GUIOctave is the GUI front end for it and is a separate download).

3. ## Re: Determining Matrix P

You don't need to calculate eigenvectors and eigenvalues. Note that B can be derived from A by "row operations".

1) Multiply row 1 by -1.
2) Multiply row 2 by -1.
3) Add -1 times row 3 to row 2.

Each of those corresponds to an "elementary matrix", the matrix you get by applying the same row operation to the identity matrix, and their product gives "P".

4. ## Re: Determining Matrix P

Thanks so much for your help. I don't really understand what the row operation for row 3 is though? Can anyone explain?