# Congruency and eigen values

$If$ $two$ $matrices$ $are$ $congruent$ $then$ $they$ $have$ $the$ $same$ $eigen$ $values$ $??$

I have tried solving it using definition of congruent matrices taking eigen value but didn't come up with a proper solution ... how to solve it ???

• The closest to a true statement similar to this may be Sylvester's inertia theorem ... – Hagen von Eitzen May 7 '16 at 22:13
• Like we can prove Aand B are similar matrices then they have the same eigen values | B-kl | = | A-kI | where k is their eigen value .. is there any similr proof to prove | B-kl | is not equal to | A-kI – Shona May 7 '16 at 22:22
• What is Sylvester's inertia theorem ? – Shona May 8 '16 at 6:35

It is wrong $\pmatrix{0&1\cr1&0\cr}$ and $\pmatrix{2&1\cr0&-1/2\cr}$ are congruent but the first one has $1$ and $-1$ as eigen values and the second one has $2$ and $-1/2$ as eigen values.
• Even simpler(?) $I$ and $(2I)^TI(2I)=4I$ are congruent and have different eigenvalues – Hagen von Eitzen May 7 '16 at 22:12
• well it is a counter example, so your statement is false. And imagine that you find a matrix such as $A$ and $D$ are congruent thanks to the orthogonal matrix $P$ and have the same eigen values and $D$ is diagonal. If you multiply $D$ by $\lambda >0$ then you always have $\lambda D$ and $A$ congruent, so they almost never have the same eigen values. – Jennifer May 7 '16 at 22:18