What does it mean for two $2 \times 2$ matrices to be orthogonal to each other?

Question

This is just a quick question about definitions. What exactly does it mean for two matrices to be orthogonal to each other? Thank you.

Answer 1

A matrix can BE orthogonal (as described here) but I have never heard of matrices being orthogonal to each other.

Answer 2

This is an inner product on the space of all $2\times 2$ matrices:

$$\begin{pmatrix}a_1&b_1\\c_1&d_1\end{pmatrix}.\begin{pmatrix}a_2&b_2\\c_2&d_2\end{pmatrix}=a_1a_2+2b_1b_2+c_1c_2+2d_1d_2.$$

As soon as we have an inner product, we have that the matrices are orthogonal if their inner product is zero.

Of course, there may be other inner products and the definition of orthogonality will change accordingly.