Geodesic
Unitary similarity of algebras generated by pairs of orthoprojectors
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XVIII, Tome 323 (2005), pp. 5-14

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It is shown that the unitary similarity of two matrix algebras generated by pairs of orthoprojectors $\{P_1,Q_1\}$ and $\{P_2,Q_2\}$ can be verified by comparing the traces of $P_1$, $Q_1$, and $(P_1Q_1)^i$, $i=1,2,\dots,n$, with those of $P_2$, $Q_2$, and $(P_2Q_2)^i$. The conditions of the unitary similarity of two matrices with quadratic minimal polynomials presented in [A. George and Kh. D. Ikramov, Unitary similarity of matrices with quadratic minimal polynomials. – Linear Algebra Appl., 349 (2002), 11–16] are refined. 
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     title = {Unitary similarity of algebras generated by pairs of orthoprojectors},
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Yu. A. Alpin; Kh. D. Ikramov. Unitary similarity of algebras generated by pairs of orthoprojectors. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XVIII, Tome 323 (2005), pp. 5-14. http://geodesic.mathdoc.fr/item/ZNSL_2005_323_a0/