Geodesic
Classification of squared normal operators on unitary and Euclidean spaces
Fundamentalʹnaâ i prikladnaâ matematika, Tome 13 (2007) no. 4, pp. 225-232

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We give a canonical form for a complex matrix whose square is normal under transformations of unitary similarity as well as a canonical form for a real matrix whose square is normal under transformations of orthogonal similarity. 
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     author = {V. M. Futornyi and R. A. Horn and V. V. Sergeichuk},
     title = {Classification of squared normal operators on unitary and {Euclidean} spaces},
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V. M. Futornyi; R. A. Horn; V. V. Sergeichuk. Classification of squared normal operators on unitary and Euclidean spaces. Fundamentalʹnaâ i prikladnaâ matematika, Tome 13 (2007) no. 4, pp. 225-232. http://geodesic.mathdoc.fr/item/FPM_2007_13_4_a10/