Approximate Change of Basis Matrix/ Solution to Algebraic Riccatti System

Question

Let $B,Q,P$ be compatible matrices. Is there a closed-form solution to $$ \|PBB^TP^T -M\|_{F}=0; $$ where $F$ is the Frobenius norm.

Ideas:

• Solve the simplified algebraic Riccati equation $$ PBB^TP^T =M? $$
• Proximal Mapping Approach.

However, I've found no solution to the "closest change of matrix P" making B into $M$...

Answer 1

I assume, per the comments, that $M$ is (symmetric and) positive semidefinite.

By Sylvester's law of intertia, an exact solution to $P(BB^T)P^T = M$ will exist if and only if the rank of $M$ is less than or equal to the rank of $BB^T$, which must in turn be equal to the rank of $B$. We can characterize every solution to this equation as follows:

• Let $LL^T = M$ be a Cholesky decomposition.
• For some orthogonal matrix $Q$ (we could simply take $Q = I$ for instance), $P$ must solve the linear equation $PB = LQ$.

If the rank of $M$ is greater than that of $B$, then I we can find a minimizer to $\|PBB^TP^T - M\|_F$ as follows: Let $r$ denote the rank of $B$. Find the best rank-$r$ approximation to $M$ (using a truncated SVD or truncated spectral decomposition), call this low-rank matrix $\tilde M$. Then, a $P$ solving $P(BB^T)P^T = \tilde M$ will minimize our objective function.