It looks like you've answered your own question. As for real life applications, function orthogonality is used all over the place in differential equations, where, say, the orthonormal basis (that's a set of functions) of an Hermitian operator is used to write down the solution to the DE. Matrix orthogonality, in a like manner, might be used in a discretization of the same problem, I suppose, though I don't know for sure.