# How to compress a unitary matrix?

I have a set of unitary matrices (four-dimensional, but would like to get an answer in the general case). I get it from singular value decomposition (svd).

There are a lot of matrices, and I began to think about how to store my data in a compressed form. A unitary matrix is a generalization of an orthogonal matrix, which consists of orthonormal bases and describes rotation or reflection. This makes me think that I can store my matrices as a set of angles in complex space, and then restore them fairly accurately.

But when I tried to generalize the formulas for real rotation matrices to the complex case, I got stuck with the concepts of rotation, dot product, and others in the complex case. Therefore, here is my question: is it possible to describe a unitary matrix with a smaller number of complex numbers (it seems to me that it should be 4 complex numbers for a 4x4 matrix), and then restore the original version accurately enough?

• A unitary matrix $U$ satisfies $U^HU = I$ where $U^H$ is obtained by transposing the complex conjugate of $U$. In the case of $n=4$ this is equivalent to 10 conditions on the columns of $U$. It strikes me as unlikely that we will always be able to satisfy 10 equations with only 4 parameters. Would you edit your question and develop you reasoning on this issue? You are implying that you do not need an exact representation, but that an approximation might do. What level of accuracy are you looking for? Commented Jun 2, 2023 at 11:48
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