If I have a block matrix of complex matrices
$$ \begin{bmatrix} P &Q\\ Q^T & P \end{bmatrix} $$ while Q being skew symmetric, the decomposition is $$ \begin{bmatrix} I & -iI\\ . & I \end{bmatrix} \begin{bmatrix} P-iQ & .\\ -Q & I \end{bmatrix} \begin{bmatrix} I & iI\\ . & P+iQ \end{bmatrix} $$ These are results from (R.A Wooding 1956). Can anyone please explain what kind of complex decomposition is this or please provide any reference explaing this above decomposition. I have looked into Block LU decomposition but this does not explain how the above stated decomposition is working.
Appreciate your suggestions !
Hint: the matrix valued function of a scalar $a$ : $$m(a) = \left[\begin{array}{cc}1&0\\a&1\end{array}\right]$$ has the property that $$m(a)\cdot m(b) = m(a+b)$$ and this property carries over to block form with $\bf I$ corresponding to $1$ and $\bf 0$ corresponding to $0$: $$m({\bf A}) = \left[\begin{array}{cc}\bf I&\bf 0\\{\bf A}&{\bf I}\end{array}\right]$$ so that: $$m({\bf A})m({\bf B}) = m({\bf A+B})$$
Hopefully this can help you gain some understanding.