Geometry of Varieties of Mutually Orthogonal Matrices

For the ring of square matrices Mat_n(K) of order n over a field K, one can construct the orthogonality graph O(Mat_n(K)), whose vertices are the zero divisors of the ring Mat_n(K). Two vertices A and B are connected by an edge if AB = BA = 0. The notion of the distance between two elements of the ring naturally implies that one can consider the set Ond of pairs of elements lying within distance at most d. It is proved that such sets form affine algebraic varieties; a decomposition of these varieties into irreducible components is provided, and their dimensions are calculated. Also the sets that are similarly defined for the ring of upper triangular matrices are described, and generalizations of these results to arbitrary finite-dimensional algebras are suggested.