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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.

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  • $\begingroup$ As far as I can see it, yes. However the problem of existence and uniqueness of solutions has basically nothing to do with positive definiteness: it is a matter of rank of the matrix of the system and rank of the augmented matrix. Furthermore it is valid over any field (even thos without any order relation). $\endgroup$ – Bernard Apr 8 at 22:43
  • $\begingroup$ @Bernard Can you please clarify about rank? If a matrix is positive definite then it is nonsingular so it necessarily has a unique solution. $\endgroup$ – sequence Apr 8 at 23:10
  • $\begingroup$ Yes, but the real criterion is just that the matrix has full rank. $\endgroup$ – Bernard Apr 8 at 23:12
  • $\begingroup$ If it is positive definite then it has full rank, since no eigenvalue is $0$, as far as I understand it. $\endgroup$ – sequence Apr 8 at 23:13
  • $\begingroup$ Absolutely, but the problem makes sense even in fields for which positive definiteness can't even be defined. $\endgroup$ – Bernard Apr 8 at 23:14

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