# Are Similar Matrices Necessarily Congruent?

• Jan 17th 2008, 01:02 AM
kevin_chn
Are Similar Matrices Necessarily Congruent?
As two similar matrices have a same set of eigenvalues, are they necessarily congruent?

Thank you~
• Jan 17th 2008, 01:24 AM
kevin_chn
Now I understand the answer should be no, but who can give me a brief conclusion?
• Jan 17th 2008, 07:18 PM
Jhevon
Quote:

Originally Posted by kevin_chn
Now I understand the answer should be no, but who can give me a brief conclusion?

go by the definitions:

an nxn matrix $A$ is similar to an nxn matrix $B$ if there exists an (invertible) nxn matrix $P$ such that:

$P^{-1}AP = B$

an nxn matrix $A$ is congruent to an nxn matrix $B$ if there exists an (invertible) nxn matrix $P$ such that:

$P^TAP = B$

where $P^T$ is the transpose of the matrix $P$

for similar matrices to be congruent as well, therefore, we must have $P^{-1} = P^T$, which will not always be true. you can find a counter-example to show this

see here for a counter-example