# Show that congruence of matrices is an equivalence relation.

How would one solve a question of this nature?

We know that given an arbitrary square matrix, A, that a matrix B is said to be congruent to A if there exists a nonsingular (invertible) matrix P such that B = (P^T)AP.

Also, a matrix B is equivalent to a matrix A if B can be obtained from A by a sequence of elementary row and column operations. Alternatively, B is equivalent to A if there exist nonsingular matrices P and Q such that B = PAQ. Just like row and column equivalence, equivalence of matrices is an equivalence relation.

• Which of the requirements for an equivalence relation is causing you headaches? – Marc van Leeuwen Apr 18 '18 at 16:05
• B = PAQ in the second paragraph above. – HR 8938 Cephei Apr 18 '18 at 16:12
• That is not a requirement for an equivalence relation. There are exactly three requirements for an relation to be an equivalence relation (but unfortunately, being called "equivalence" is not among them): reflexivity, symmetry, and transitivity. By the way, I think your title should say "equivalence" where it says "congruence". – Marc van Leeuwen Apr 18 '18 at 16:55
• I see. It seems as though my book doesn't discuss those requirements for an equivalence relation. It does discuss symmetric matrices but that's the closest topic to those requirements. My book only covers elementary linear algebra so perhaps that is why. The title is the same as the exercise out of the book. It could be a typo. – HR 8938 Cephei Apr 18 '18 at 17:14
• Could you cite your book's definition of an equivalence relation? – Marc van Leeuwen Apr 18 '18 at 17:47