Quadratically normal and congruence-normal matrices
A matrix A ∈ Cn×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 ∈ Cn×n is congruence-normal if \( B = A\overline A \) 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 andexploitsalikenessbetween theequations X2 = B and \( X\overline X = B \) for a normal matrix B. Bibliography: 13 titles.
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