In the Answer to a different Question, a curious matrix relation came up:
M is symmetric and non-singular, G is non-symmetric and singular.
Theorem: When $M$ is positive/negative definite, or more generally, if $G$ and $(G^\dagger G) M (G^\dagger G)$ have the same ranks then $$G^{T}\left(GMG^{T}\right)^{\dagger}G=\left((G^\dagger G)M(G^\dagger G)\right)^\dagger$$
I have no idea how to prove this or why the rank relationships asserted would be true.
An example:
$M=\left( \begin{array}{ccc} 1. & 2. & 3. \\ 2. & 4. & 5. \\ 3. & 5. & 6. \\ \end{array} \right)$ $G=\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 0 \\ \end{array} \right)$ $GMG^T=\left( \begin{array}{ccc} 1. & 4. & 0. \\ 4. & 16. & 0. \\ 0. & 0. & 0. \\ \end{array} \right)$
$GMG^T$ has a rank of 1 and the relation does not hold. But if we add a small number times the Identity Matrix to M, $GMG^T$ will have rank 2 and the relation will hold.
What is going on there? Can anyone help with an explanation or proof? And under exactly what circumstances is this true or not true?