Schur form of a complex matrix relative to consimilarity and its calculation (Q1803449)

Let \(A\) be a nondegenerate complex \(n \times n\)-matrix and \(\Gamma_ A\) be the real algebra of all matrix \(B\) from \(\mathbb{C}^{n \times n}\) satisfying the pseudocommutative property \(AB=\overline BA\). The author proves the following main results. 1. Let \(n \times n\)-matrix \(C\) has block triangular structure and pseudospectra of its diagonal blocks \(C_{ii}\) pairwise disjoint. Then each matrix \(D\) pseudocommutative with \(C\) has analogous block triangular form, and the order of the block \(D_{ii}\) is equal to that of the block \(C_{ii}\). 2. Suppose that \(H_ 1=Q_ 1^*A \overline Q_ 1\), \(H_ 2=Q_ 2^*A \overline Q_ 2\) are upper Hessenberg forms of a given matrix \(A\), where \(Q_ 1\) and \(Q_ 2\) are unitary matrices. Suppose also that at least one of these Hessenberg matrices is indecomposable. If the first columns of the matrices \(Q_ 1\) and \(Q_ 2\) coincide, then \(Q_ 2=Q_ 1U\), \(H_ 2=U^*H_ 1U\), where \(U\) is unitary diagonal matrix.