Upper triangular similarity of upper triangular matrices

AbstractWe consider the following equivalence relation in the set of all complex upper triangular n × n matrices: A and B are called U-similar if there exists an invertible upper triangular matrix S such that A = S−1BS. If A, B are U-similar, then they must have the same diagonal and the same Jordan form. It is known that for n ⩾ 6 there are infinitely many mutually non-U-similar nilpotent upper triangular matrices with the same Jordan form. We introduce an appropriate generalization of the Jordan block (called an irreducible matrix), and we prove that each upper triangular matrix is U-similar to a “generalized” direct sum of irreducible blocks, where the location and the order of the blocks is fixed and each block is determined uniquely up to U-similarity