Principal Minor Ideals and Rank Restrictions on their Vanishing Sets

TL;DR

This paper investigates the algebraic structure of matrices with fixed rank by analyzing the ideals generated by principal minors and computing the dimension of specific rank-restricted subsets.

Contribution

It provides a detailed computation of the dimension of matrices of rank n-2 with vanishing principal minors, advancing understanding of algebraic conditions on matrix ranks.

Findings

Dimension of rank n-2 matrices with vanishing principal minors is n^2 - n - 4.

Characterization of principal minor ideals in relation to matrix rank restrictions.

Insights into the algebraic geometry of matrix varieties with fixed rank.

Abstract

All matrices we consider have entries in a fixed algebraically closed field $K$. A minor of a square matrix is principal means it is defined by the same row and column indices. We study the ideal generated by size $t$ principal minors of a generic matrix, and restrict our attention to locally closed subsets of its vanishing set, given by matrices of a fixed rank. The main result is a computation of the dimension of the locally closed set of $n\times n$ rank $n-2$ matrices whose size $n-2$ principal minors vanish; this set has dimension $n^2-n-4$.