Two real symmetric matrices are congruent if and only if they have the same rank and signature.

Question

So I saw this statement in an exercise :

Two real $n \times n$ matrices are congruent if and only if they have the same rank and the same signature.

But I was wondering why do we need to state the fact that they must have the same rank. If two real $n \times n$ symmetric matrices have the same signature, doesn't they necessarily have the same rank ? So shouldn't it be :

Two real $n \times n$ matrices are congruent if and only if they have the same signature.

Answer 1

You are correct by any standard interpretation of the term "signature". It's difficult to know what the author is going for here.