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