A Pythagorean triple is a list of positive integers (a, b, c) such that a2 + b2 = c2. Euclid knew how to find all Pythagorean triples: pick two positive integers m and n with m > n and let
a = m2 – n2, b = 2mn, c = m2 + n2.
Now what if we look at matrices with integer entries such that A2 + B2 = C2? Are there any? Yes, here’s an example:
How can you find matrix-valued Pythagorean triples? One way is to create diagonal matrices with Pythagorean triple integers. Take a list of n Pythagorean triples (ai, bi, ci) and create diagonal matrices A, B, and C with the ai along the diagonal of A, bi along the diagonal of B, and ci along the diagonal of C.
But that seems cheap. We’re really not using any properties of matrix multiplication other than the fact that you can square a diagonal matrix by squaring its elements. One way to create more interesting matrix Pythagorean triples is to start with the diagonal matrices described above and conjugate them by some matrix Q with integer entries such that Q-1 also has integer entries. Here’s why that works:
The example above was created by starting with the Pythagorean triples (3, 4, 5) and (5, 12, 13), creating diagonal matrices, and conjugating by
This gives a way to create lots of matrix Pythagorean triples, but there may triples that this procedure leaves out.