I have two $a \times b$ matrices of complex numbers, representing acoustic fields at some planes, and I want to measure their similarity.
One way I know is the cosine similarity, but the cosine similarity between complex matrices seems not perfectly reflects the similarity.
The way I implemented the cosine similarity is (suppose matrices P and Q):
$$c = \frac{PQ^H}{\left\|P\right\|\left\|Q\right\|}$$ where $Q^H$ denotes the Hermitian transpose of the complex matrix $Q$.
Am I going to the right direction? Is there any other matrices that measures them well?