# Similar matrices

• May 10th 2008, 09:13 AM
lute
Similar matrices
Hi

Are these 2 matrices similar to each other?

A= [0 5 0]
.....[0 0 5]
.....[0 0 0]

B = [0 10 0 ]
......[0 0 10]
......[0 0 0]

They have the same eigenvalues of 0,0,0 and the same rank, but there is no P or P^-1 that you can find to give you a diagonal matrix, being that the P you would get is entirely singular. Does that entail then that A and B here aren't similar at all? Thanks
• May 10th 2008, 01:16 PM
Opalg
Quote:

Originally Posted by lute
Hi

Are these 2 matrices similar to each other?

A= [0 5 0]
.....[0 0 5]
.....[0 0 0]

B = [0 10 0 ]
......[0 0 10]
......[0 0 0]

They have the same eigenvalues of 0,0,0 and the same rank, but there is no P or P^-1 that you can find to give you a diagonal matrix, being that the P you would get is entirely singular. Does that entail then that A and B here aren't similar at all? Thanks

Try taking P to be a diagonal matrix, say $\displaystyle P = \begin{bmatrix}x&0&0\\ 0&y&0\\ 0&0&z\end{bmatrix}$. Calculate the product $\displaystyle P^{-1}AP$ and see if you can make it equal to B, for suitable values of x,y and z.
• May 10th 2008, 05:17 PM
lute
Thanks! I was indeed able to find a P and a P^-1 that was able to get A equal to B, which means that they are similar matrices afterall.