On the dimension of the locus of determinantal hypersurfaces

TL;DR

This paper investigates the structure of determinantal hypersurfaces generated by matrix tuples, showing that for general position matrices with at least three matrices, the set of such matrices sharing the same characteristic polynomial is finite up to conjugacy, and the locus of these hypersurfaces has a specific irreducible dimension.

Contribution

It establishes the irreducibility and precise dimension of the locus of determinantal hypersurfaces for matrix tuples in general position when r ≥ 3.

Findings

Finiteness of matrix tuples with same characteristic polynomial up to conjugacy for r ≥ 3.

The locus of determinantal hypersurfaces is irreducible.

Dimension of the locus is (r-1)n^2 + 1.

Abstract

The characteristic polynomial of an $r$-tuple $(A_1,..., A_r)$ of $n \times n$ matrices is the determinant $\det(x_0 I + x_1 A_1 + ... + x_r A_r)$. We show that if $r$ is at least 3 and $A = (A_1,..., A_r)$ is an $r$-tuple of matrices in general position, then up to conjugacy there are only finitely many $r$-tuples of matrices with the same characteristic polynomial as $A$. Equivalently, the locus of determinantal hypersurfaces of degree $n$ in $P^r$ is irreducible of dimension $(r-1)n^2 + 1$.