Maximising det(AB) with respect to B [closed]

I have a $$y \times u$$ matrix $$A$$ and a $$u \times y$$ sized matrix $$B$$, where $$y \leq u$$.

Given $$A$$, I want to choose $$B$$ such that $$\det(AB)$$ is maximised, subject to the constraint that the columns of $$B$$ have magnitude 1.

Is there a closed form solution to this?

Presumably $$A$$ has full row rank (or else $$\det(AB)$$ is always zero). Therefore $$A$$ admits a polar decomposition $$PU^T$$, where $$P$$ is positive definite and $$U^T$$ has orthonormal rows (i.e., $$U$$ has orthonormal columns). Then $$\det(AB)=\det(P)\det(U^TB)$$. Since $$U^T$$ has orthonormal rows and the columns of $$B$$ are unit vectors, each column of $$U^TB$$ has norm $$\le1$$. It follows from Hadamard’s determinant theorem that $$\det(U^TB)\le1$$. Consequently, $$\det(AB)\le\det(P)$$. The optimal solutions are the matrices $$B$$ that make equality hold. In particular, $$B=U=A^T(AA^T)^{-1/2}$$ is an optimal solution.
• @Jabby Yes. If $A=USV^T$ is a SVD, you may also take $B=VU^T$ or $B=V\operatorname{diag}(1,\ldots,1,\det(U))$. The former choice is actually the one in my answer. In the latter choice, the purpose of the diagonal matrix is to make sure that $\det(AB)$ is positive. Commented Jun 24 at 22:47
• I'm slightly confused. $V$ here would be $u \times u$ and $U$ would be $y \times y$, so I don't understand how $B = V U^T$ is something you can compute. $B$ needs to be $u \times y$. Commented Jun 24 at 22:57
• @Jabby Oh, I mean a compact/economic SVD, not a full SVD. That is, if $A=U\pmatrix{S&0}W^T$ is a full SVD and $V^T$ is the top $y$ rows of $W^T$ (or $V$ is the leftmost $y$ columns of $W$), then $USV^T$ is the compact SVD of $A$, with $V^T$ having the same size as $A$. Commented Jun 24 at 23:02
• Ah ok. For my sake, are you able to spell out the proof that $𝐵=𝑉𝑈^𝑇$ in this SVD decomposition case? Can you also comment on the pros and cons of computing B via $𝐵=𝑉𝑈^𝑇$ (SVD decomposition) vs. $𝐵=A^T(AA^T)^{-1/2}$ (polar decomposition) (e.g. in terms of computational efficiency)? Commented Jun 24 at 23:16