I have a question about what I would call, for want of a better name, matrix decomposition. However, my question does not concern standard decompositions like eigenvalue or Cholesky decomposition.

The problem:

Assume given two real and square matrices C and D. C is symmetric, while D is antisymmetric. Find two real and square matrices A and B, such that:

C = A*A - B*B

and

D = A*B + B*A

Here * denotes standard matrix multiplication.

Does anybody know a suitable algorithm for this or a similar problem? Most likely several solutions A and B exist.

This problem arises when trying to describe radar Doppler measurements of hydrometeors (cloud and rain drops).

Any help will be appreciated.