Quadratically normal matrices of type 1 and unitary similarities between them

Verification of the unitary similarity between matrices having quadratic minimal polynomials is known to be much cheaper than the use of the general Specht-Pearcy criterion. Such an economy is possible due to the following fact: n x n matrices A and B with quadratic minimal polynomials are unitarily similar if and only if A and B have the same eigenvalues and the same singular values. It is shown that this fact is also valid for a subclass of matrices with cubic minimal polynomials, namely, quadratically normal matrices of type 1.