A matrix with real entries has *positive characteristic polynomial* if for all real . For example,

has this property: . For brevity, let’s say that is a PCP matrix.

Clearly, any PCP matrix must be of even size. Incidentally, this implies that the algebraists’ characteristic polynomial and the analysts’ characteristic polynomial coincide.

In general, there is no reason for the PCP property to be preserved under either addition or multiplication of matrices. But there are some natural rings of PCP matrices, such as

- complex numbers
- quaternions

A matrix is PCP if and only if its eigenvalues are either complex or repeated. This is equivalent to . In general, a matrix of even size is PCP if and only if it has no real eigenvalues of odd algebraic multiplicity.

Is this post merely an excuse for 1995 flashback: Ангельская Пыль by Ария?