Some Properties of Some Special Matrices

The terminology and notation used in this paper have been introduced in the following articles: [8], [3], [11], [12], [1], [10], [9], [6], [2], [4], [5], [13], and [7]. For simplicity, we adopt the following convention: n denotes a natural number, K denotes a field, a denotes an element of K, and M , M1, M2, M3, M4 denote matrices over K of dimension n. Let n be a natural number, let K be a field, and let M1, M2 be matrices over K of dimension n. We say that M1 is permutable with M2 if and only if: (Def. 1) M1 · M2 = M2 · M1. Let us note that the predicate M1 is permutable with M2 is symmetric. Let n be a natural number, let K be a field, and let M1, M2 be matrices over K of dimension n. We say that M1 is reverse of M2 if and only if: