I have a set of matrix equation as described below:
1-AP=PX
2-b=P*y
In above equations, A and b are known matrices of size n by n and n by 1 respectively. P, X and y are unknown matrices of size n by r, r by r and r by 1 of appropriate sizes. Also we know that r<<n (r is much smaller than n).
I also know that first equation has an answer iff Im(AP) is subset of Im(P) and likewise, second equation has answer iff b is a subset of Im(P). Now, the hope is that we can select (design) P such that above conditions are satisfied; then finding X and y is possible by using conventional algorithms.
Just as a hint, suppose that equation (1) was the only constraint; by choosing columns of P as eigen vectors of A we can easily compute the exact answer; however, adding the equation (2), if b cannot be expressed as linear combination of eigen vectors of A equality cannot be satisfied (best answer for equation (2) would be Least Squares output which doesn't result in exact equality necessarily).
So, in brief, I'm looking for a systematic way to design P (with respect to above conditions) such that I know that X and y exist.