
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.

Re: Determining Matrix P
Hey grooverandshaker.
If you have the eigenvectors then P is just made up by using the eigenvectors corresponding to a particular eigenvalue 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).

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".

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?