Upper triangular similarity of upper triangular matrices
The author considers the following equivalence relation in the set of all complex upper triangular \(n\times n\) matrices: \(A\) and \(B\) are \(U\)-similar if there exists an invertible upper triangular matrix \(S\) such that \(A=S^{-1} BS\). If \(A\) and \(B\) are \(U\)-similar, then they must have the same diagonal and the same Jordan form. It is known that for \(n\geq 6\) there are infinitely many pairwise non-\(U\)-similar nilpotent upper triangular matrices with the same Jordan form. The author generalizes the concept of Jordan block (called irreducible) and proves that an upper triangular matrix is \(U\)-similar to a ``generalized'' direct sum of irreducible blocks, where the location and order of the blocks is fixed and each block is uniquely determined up to \(U\)-similarity.
scientific article; zbMATH DE number 1040491