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?

It might not be.

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:
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