The unitary orbit of an complex matrix A is the set consisting of matrices unitarily similar to A. In this note we offer an alternative proof for a recent result, due to Li, Poon and Sze, on the possible ranks of the difference of matrices taken from the unitary orbits of two given orthogonal projections.
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[3] Horn, A. “Doubly Stochastic matrices and the diagonal of a rotation matrix,” Amer. J. Math., Vol. 76, pp. 620-630 (1954).
[4] Li, C. K. and Poon, Y. T., “Principal submatrices of a Hermitian matrix,” Linear and Multilinear Algebra, Vol. 51, no. 21, pp. 199-208 (2003).
[5] Thompson, R. C., “Principal submatrices of normal and Hermitian matrices,” Illinois J. Math., Vol. 10, pp. 296-308 (1966).
[6] Choi, M. D., Li, C. K. and Poon, Y. T., “Some convexity features associated with unitary orbits,” Canad. J. Math., Vol. 55, pp. 91-111 (2003).
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