QUADRATICALLY NORMAL AND CONGRUENCE-NORMAL MATRICES

摘要

A matrix A ∈C~(n×n) is unitarily quasidiagonalizable if A can be brought by a unitary similarity transformation to a block diagonal form with 1×1 and 2×2 diagonal blocks. In particular, the square roots of normal matrices, i.e., the so-called quadratically normal matrices are unitarily quasidiagonalizable. A matrix A ∈C~(n×n) is congruence-normal if B = AA is a conventional normal matrix. We show that every congruence-normal matrix A can be brought by a unitary congruence transformation to a block diagonal form with 1×1 and 2×2 diagonal blocks. Our proof emphasizes and exploits a likeness between the equations X~2 = B and XX = B for a normal matrix B.