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I want to write a real matrix $M$ as a sum of unitary (or symmetric) matrices.

$M$ can be broken down into a diagonal and skew-symmetric matrix:

$$M = D + A$$

$$D = \begin{bmatrix} D_1 & 0 & 0 & ... \\ 0 & D_2 & 0 & ... \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & D_N \end{bmatrix}, \quad A = \begin{bmatrix} 0 & A_{12} & ... & A_{1N} \\ -A_{12} & 0 & ... & A_{2N} \\ \vdots & \vdots & \ddots & \vdots \\ -A_{1N} & -A_{2N} & ... & 0 \end{bmatrix}$$

I tried using the procedure given in this post. But the answer is trivial because $M$ doesn't contain any imaginary values. (Using the notation in the post, if I let $M=B+iC$ and calculate $U_{B,C}$, $V_{B,C}$ then the solution comes out to $M=B$.)

How can the procedure be adapted to write any real matrix as a sum of unitary or symmetric matrices?

Medulla Oblongata
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    The sum of symmetric matrices is symmetric. Therefore, not every real square matrix can be written as a sum of symmetric matrices. On the other hand, every real matrix is a complex matrix. So, the decompositions outlined in the answers to the linked question are still valid. – user1551 Jun 06 '23 at 17:52
  • @Medulla What exactly do you mean by "the answer is trivial"? – Ben Grossmann Jun 06 '23 at 21:43

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