## Angel dust of matrices

A matrix ${A}$ with real entries has positive characteristic polynomial if ${\det(tI-A)\ge 0}$ for all real ${t}$. For example, $\displaystyle A=\begin{pmatrix} 1 & -5 \\ 3 & -2 \end{pmatrix}$

has this property: ${\det (tI-A)=(t-1)(t+2)+15=t^2+t+13}$. For brevity, let’s say that ${A}$ is a PCP matrix.

Clearly, any PCP matrix must be of even size. Incidentally, this implies that the algebraists’ characteristic polynomial ${\det (tI-A)}$ and the analysts’ characteristic polynomial ${\det (A-tI)}$ 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 ${\begin{pmatrix} a & b \\ -b & a \end{pmatrix} }$
• quaternions ${\begin{pmatrix} a & b & c & d \\ -b & a & -d & c \\ -c & d & a & -b \\ -d & -c & b & a \end{pmatrix}}$

A ${2\times 2}$ matrix is PCP if and only if its eigenvalues are either complex or repeated. This is equivalent to ${(\mathrm{tr}\, A)^2 \le 4\det A}$. 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?