As two similar matrices have a same set of eigenvalues, are they necessarily congruent?
an nxn matrix is similar to an nxn matrix if there exists an (invertible) nxn matrix such that:
an nxn matrix is congruent to an nxn matrix if there exists an (invertible) nxn matrix such that:
where is the transpose of the matrix
for similar matrices to be congruent as well, therefore, we must have , which will not always be true. you can find a counter-example to show this
see here for a counter-example