2
$\begingroup$

I want to solve the folowing problem B*M=V, where B is the unknown of size 3x3, M of size 3xN and V of size 3xN. The difficulty is, that B has to be unitary.

N is in the range of 500. All matrices are real.

Solving the problem by multiplying from right with the pseudoinverse of M gives a solution that is close to being unitary.

Does somebody have a hint how to solve it? Best would be a solution by aid of Matlab or a C++ library like Eigen.

$\endgroup$
1
$\begingroup$

For this to work, the singular value decompositions of $V$ and $M$ can be written in the form $$ V = U_1 \Sigma U_2^*, \ M = U_3 \Sigma U_2^*$$ with the same $\Sigma$ and $U_2$, and you have $B = U_1 U_3^*$.

$\endgroup$
  • $\begingroup$ Thank you, that worked! It's a really elegant solution! But I still have some questions =) Since for me, the equality sign in $BM=V$ does not hold (but the error has to be minimized), I have $$V = U_1 \Sigma_1 U_2^*, M = U_3\Sigma_2 U_4^* $$ I have reordered $\Sigma_2$ and $U_4^*$ in a way that they are closest to $\Sigma_1$ and $U_2^*$. Of course $U_3$ was reordered accordingly to that. Now the question is: Is the resulting $B$ the best possible unitary solution to $BM=V$ or is it just one close solution? And if so: why? $\endgroup$ – yar Feb 21 '17 at 2:02

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.