Consider the matrix equation $$ A^TUA + C = U, $$ in the variable $U$, with $A, C$ given. Above, $A, U, C$ are real matrices of compatible dimensions.
What are necessary and sufficient conditions to ensure that a solution to the above equation exists?
Some observations/facts:
- if the spectral radius $\rho(A) < 1$, then via Gelfand's formula, one verifies the existence of a solution.
- in one dimension, the equation is $c = (1 - a^2) u$. Hence, the equation always has a solution if and only if either $c = 0$ or $|a| \neq 1$.
Can we generalize the second condition? What should it be?
Suppose $A$ has a full set of left eigenvectors, then with $L(U) = A^TU A -U$, we have $L(u_i u_j^T) = (\lambda_i \lambda_j -1) u_i u_j^T$, so we see that $L$ has eigenvalues $\lambda_i \lambda_j -1$.
The general result follows from continuity.
So, to (partially) answer your question, $L(U)=-C$ has a unique solution for all $C$ iff $\lambda_i\lambda_j \neq 1$ for all $i,j$.
More generally, a (not necessarily unique) solution exists iff $C$ is in the range of $L$.