# 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?

# Integer sets without singular matrices

Let’s say that a set $A\subset \mathbb N$ contains an $n\times n$ singular matrix if there exists an $n\times n$ matrix with determinant zero whose entries are distinct elements of $A$. For example, the set of prime numbers does not contain any $2\times 2$ singular matrices; however, for every $n\ge 3$ it contains infinitely many $n\times n$ singular matrices.

I don’t know of an elementary proof of the latter fact. By a 1939 theorem of van der Corput, the set of primes contains infinitely many progressions of length 3. (Much more recently, Green and Tao replaced 3 with an arbitrary $k$.) If every row of a matrix begins with a 3-term arithmetic progression, the matrix is singular.

In the opposite direction, one may want to see an example of an infinite set $A\subset \mathbb N$ that contains no singular matrices of any size. Here is one: