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Hermitian and unitary matrices are two representatives of the class of normalmatrices whose full eigenvalue decomposition can be stably computed inquadratic computing com plexity. Recently, fast and reliable eigensolversdealing with low rank perturbations of unitary and Hermitian matrices wereproposed. These structured eigenvalue problems appear naturally when computingroots, via confederate linearizations, of polynomials expressed in, e.g., themonomial or Chebyshev basis. Often, however, it is not known beforehand whetheror not a matrix can be written as the sum of an Hermitian or unitary matrixplus a low rank perturbation. We propose necessary and sufficient conditionscharacterizing the class of Hermitian or unitary plus low rank matrices. Thenumber of singular values deviating from 1 determines the rank of aperturbation to bring a matrix to unitary form. A similar condition holds forHermitian matrices; the eigenvalues of the skew-Hermitian part differing from 0dictate the rank of the perturbation. We prove that these relations are linkedvia the Cayley transform. Based on these conditions we are able to identify theclosest Hermitian and unitary plus low rank matrix in Frobenius and spectralnorm and a practical Lanczos iteration to detect the low rank perturbation ispresented. Numerical tests prove that this straightforward algorithm is robustwith respect to noise.

When is a matrix unitary or Hermitian plus low rank?