# Numerically finding matrix approximation by lower-dimensional “pseudo-similar” matrix

Consider an $$N\times N$$ (real or complex) matrix $$A$$, and some $$n. Is there a good numerical algorithm that finds the set consisting of an $$n\times n$$ matrix $$B$$, an $$n\times N$$ matrix $$I$$, and an $$N\times n$$ matrix $$I^-$$ such that $$I^-I=\mathbb{1}$$, which minimizes $$\|A-IBI^-\|^2\;,$$ where $$\|X\|^2 = \mathop{Tr}(XX^\dagger)$$ is the (squared) Hilbert-Schmidt norm?

If yes, I'd like to know the same for a finite collection $$A^j$$, $$0\leq j. That is, find $$B^j$$, $$I$$ and $$I^-$$ minimizing $$\sum_{0\leq j

If it helps, I'd be fine with the following modifications to the problem:

• I'm not super insistant about the precise choice of norm. $$A^j-IB^jI^-$$ can be considered a $$3$$-index array in the vector space $$\mathbb{C}^{NNJ}$$, and if you have an answer for other norms in that vector space than the $$2$$-norm I'd be very happy, too. In the use-cases I have in mind, the underlying assumption is that the minimum is very close to $$0$$ such that the precise choice of norm is not super important.
• It should be fine to restrict $$I$$ to be an isometry.
• One might consider minimizing something like $$\|A-IBI^-\|^2\ + \|I^-I-\mathbb{1}\|^2$$ instead of demanding $$I^-I=\mathbb{1}$$.
• The problem in your first paragraph is a classical one; I suggest you start from reviewing the theory on that one. – Federico Poloni Feb 5 at 21:29
• And don't use $I$ or anything similar to denote a matrix other than the identity. – David Handelman Feb 5 at 23:59