Regarding eigenvalue equation in a matrix vector space 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
$AB-BA=\lambda B$,
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.
 A: The linear operator $\text{ad}_A : B \mapsto AB - BA = [A, B]$ defines the adjoint representation of the Lie algebra $\mathfrak{gl}_n$, and what you are studying are its eigenvalues. You don't want a linear algebra book; you want a book on Lie theory. The classic reference (although it is not particularly directed towards physicists) is Fulton and Harris's Representation Theory: a First Course.
The important structure here is not the vector space but the action on it, which is why books on linear algebra won't say anything specific about "matrix vector spaces" because they are, after all, still vector spaces. 
What other choices you have depends, of course, on what it is you're doing. 
A: Here's something that might help.  Using Krnoecker products, we can write
$$AX-XA = 0$$
as 
$$[(I \otimes A) - (A^T \otimes I)] \vec X = 0.$$
This is taken directly from Horn and Johnson's Topics in Matrix Analysis, pg. 257.  
A: it is good to see you here. First let me start why your problem is not correctly stated. This also explains why my MO answer was correct. One can see an equation by using different glasses, but no one considers $3x=\lambda x$ as an eigenvalue value problem or a PDE! To have an eigenvalue problem you should have  a correctly stated linear operator $T$  whose action on a vector space is a priori known. Solving an eigenvalue problem means determining the eigenvalues and the corresponding eigenspaces of that operator. 
The main reason why not every matrix equation containing a $\lambda$ is an eigenvalue equation is that not every matrix corresponds to a well defined operator. (remember that a matrix is nothing more than a rectangular representaion of a linear map)
Now, answers to your questions: First of all, I still don't see a correctly stated problem. Is $\lambda$ given and are you looking for $A+B?$ Or, is $A$ given and are you looking for $\lambda$ or $B?$ And most importantly, what do you want to obtain at the end of the day?         
