Show that if $X$ is a orthogonal matrix, then for each symmetric matrix $S$, there exist at least one matrix H such that $XH^{T}+HX^{T}=S$.

Question

This is a exercise of Elon Lima's book.

Let be $E=M_{n}(\mathbb{R})$, consider the application $f:E\rightarrow E$, $X \mapsto XX^{T}$.

a) Describe the derivative $f'(X): E\rightarrow E$.

b) Show that if $X$ is a orthogonal matrix, then for each symmetric matrix $S$, there exist at least one matrix $H$ such that $f'(X)H=S$.

Here $X^{T}$ means the transpose of matrix $X$ and $M_{n}(\mathbb{R}) $ the vector space of $n\times n$ matrix with real entries.

I found that the derivative of $f'(X)$ is $H\mapsto XH^{T}+HX^{T}=S$. But my problem was the item b).

I would like some help. Thanks!!

Answer 1

An "obvious" solution is $H_0=1/2SX$. The general solution is $H=H_0+KX$ where $K$ is an arbitrary skew-symmetric matrix.