# The order-$n$ minors of certain $(n+k) \times n$ matrices

## TL;DR

This paper establishes conditions for the non-vanishing of order-$n$ minors in certain rectangular matrices and constructs high-dimensional subspaces with specific Schmidt rank properties.

## Contribution

It provides new criteria for minors' non-vanishing and explicit formulas for special matrix classes, advancing understanding of matrix minors and tensor subspace structures.

## Key findings

- All order-$n$ minors are nonzero under certain conditions.
- Explicit formulas for minors of special $(n+1) imes n$ matrices.
- Constructed subspaces of $ ext{C}^m imes ext{C}^n$ with maximal dimension and prescribed Schmidt ranks.

## Abstract

We determine sufficient conditions for certain classes of $(n+k) \times n$ matrices $E$ to have all order-$n$ minors to be nonzero. For a special class of $(n+1) \times n$ matrices $E,$ we give the formula for the order-$n$ minors. As an application we construct subspaces of $\C^m \otimes \C^n$ of maximal dimension, which does not contain any vector of Schmidt rank less than $k$ and which has a basis of Schmidt rank $k$ for $k=2,3,4$.