ABSTRACT

The commuting variety of matrices over a given field is a well-studied object in linear algebra and algebraic geometry. As a set, it consists of all pairs of square matrices with entries in that field that commute with one another. In this paper, we generalize the commuting variety by using the commuting distance of matrices. We show that over an algebraically closed field, each of our sets does indeed form a variety. We compute the dimension of the distance-2 commuting variety and characterize its irreducible components. We also work over other fields, showing that the distance-2 commuting set is a variety but that the higher distance commuting sets may or may not be varieties, depending on the field and on the size of the matrices.

摘要

给定域上的矩阵的交换多样性是线性代数和代数几何中一个经过充分研究的对象。作为一个集合，它由所有具有该字段中的条目的方矩阵对组成，这些条目相互通勤。在本文中，我们利用矩阵的通勤距离来概括通勤多样性。我们表明，在代数闭域上，我们的每个集合确实形成了一个变体。我们计算距离为 2 的通勤品种的维数并表征其不可约成分。我们还研究了其他领域，表明距离为 2 的通勤集是一个变体，但更高距离的通勤集可能是变体，也可能不是变体，这取决于领域和矩阵的大小。