I was working around with the decomposition of the multidimensional linear canonical transform (which is not even continuous w.r.t. the parameters) into a few fractional Fourier transforms (and other related operators that are real analytic w.r.t. the parameters); this requires decomposition of the LCT matrix, and long story short, it is necessary to be able to decompose an arbitrary real matrix $B$ into a product $D¹P¹SP²D²$, with $D¹$ and $D²$ being diagonal, $P¹$ and $P²$ possessing at least one real logarithm each, $S$ being symmetric, and all being real.
So far I could not find any combination of known mutliplicative matrix decomposition methods that arrives at this result; I need help.
Note: Here $D¹$ and $D²$ are completely independent, the $¹$ and $²$ having absolutely no meanings, and so on.
UPDATE: After workup, I found out that it is sufficient to prove the following statement: given any two real symmetric matrices $F¹$ and $F²$, there exist real diagonal matrices $D¹$ and $D²$, s.t. the polar decomposition $D¹F¹=P¹U¹$, $P¹$ being positive semidefinite and $U¹$ orthogonal, and so on, with the product of $U¹$ with the transpose of $U²$ being symmetric.
UPDATE 2: After a bit more workup, I found out that it is sufficient to prove that, for any real symmetric matrices $F, G$, there exist real diagonal matrices $H, I$ s.t. $HFGI = √(HF^2H)Q√(IG^2I)$ with $Q$ being symmetric. A direct symbolic computation, leaving the diagonals of $H, I$ to be determined, would solve the trick- except that the symbolic computation of the square root becomes demanding.
UPDATE 3: Using the scale-invariant SVD, one only needs to prove that, given a real symmetric positive semi-definite matrix $S¹$ and orthogonal real matrix $Z$, there exist real $H¹$, $H¹$ and symmetric $Q$ such that $S¹Z = exp(H¹)Qexp(H²)$.
