Let $K, M\in \mathbb{C}^{n\times n}$ be positive definite matrices and consider the following linear system: $$(M-K)\vec{u}=\vec{b}\quad (1)$$
There are three cases of existence and uniqueness of solutions of this system:
Case (1): $M-K$ is positive or negative definite: $M-K$ is then nonsingular a unique solution exists.
Case (2): $M-K$ is positive or negative semidefinite:
$M-K$ is then singular and
- infinitely many solutions exist;
- no solutions exist (inconsistent system)
Case (3): $M-K$ is indefinite:
- $M-K$ is nonsingular and a unique solution exists;
- $M-K$ is singular and infinitely many solutions exist;
- $M-K$ is singular no solutions exist (inconsistent system)
So, all in all, system (1) can possibly take any of the possibilities of existence and uniqueness of solutions.
I'd appreciate if you could comment on this: is this all true? Thanks.