Let A and B be matrices of order n that are direct sums of nilpotent Jordan blocks. Suppose that A and B are not just different arrangements of the same blocks but, rather, they differ in the sizes of the blocks. It is shown that, in this case, A and B cannot be congruent. This result can be regarded as a new proof of the uniqueness of the singular part in the Horn–Sergeichuk canonical form of a singular matrix.
Keywords:
direct sum, Jordan block, congruence transformation, span of a system of vectors
