I'm a physics grad student working, for my research, in a matrix vector space over the complex numbers. That is, both operators and the vectors they act upon are $n\times n$ complex matrices living in the same space. When it comes to the eigenvalue equation I've been using
where $A$ and $B$ are $n\times n$ complex matrices. I can "justify" this choice based on the Heisenberg picture of Quantum Mechanics, but I would like to give it a more solid mathematical background. So in this regard I have several questions:
Is the choice above sensible? what other choices could I try? and last what is the relevant literature dealing with matrix vector spaces? I mean, every semi-decent book mentions the space of matrices as a vector space, but that's just about it. I've found some valuable pieces of information scattered througout the problems in the Hoffman linear algebra text, but again, that's just about it. I have yet to find a text that treats matrix vector spaces in any depth. Any help would be really appreciated.