Geodesic
A Note on Normal Matrices
Canadian mathematical bulletin, Tome 4 (1961) no. 1, pp. 23-27

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In 1954 A.J. Hoffman and O. Taussky [1] showed that if A is an n-square complex matrix with eigenvalues λ = (λ1, ..., λn ) and P is a permutation matrix for which αA + βA* has eigenvalues for some αβ ≠ 0 then A is normal. Here is the conjugate vector of λ. As a companion result they also proved that if the eigenvalues of AA* are , i = 1, ..., n then A is normal. 
Marcus, Marvin; Khan, Nisar. A Note on Normal Matrices. Canadian mathematical bulletin, Tome 4 (1961) no. 1, pp. 23-27. doi: 10.4153/CMB-1961-004-8
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