# Matrix decomposition

#### schutgens

Hello,

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.

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#### fresh_42

Mentor
2018 Award
There is probably a typo in the condition for $C$.

The way is usually from the other direction: Given any square matrix $A$, then $A+A^\tau$ is symmetric and $A-A^\tau$ antisymmetric, better: skew symmetric. The decomposition is $A=\frac{1}{2}\cdot \left((A+A^\tau)+(A-A^\tau) \right)\,.$

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