# Thread: Are Similar Matrices Necessarily Congruent?

1. ## Are Similar Matrices Necessarily Congruent?

As two similar matrices have a same set of eigenvalues, are they necessarily congruent?

Thank you~

2. Now I understand the answer should be no, but who can give me a brief conclusion?

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

$\displaystyle P^{-1}AP = B$

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

$\displaystyle P^TAP = B$

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

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

see here for a counter-example