Geodesic
Some Spectral Properties of Polar Decompositions
Canadian journal of mathematics, Tome 36 (1984) no. 6, pp. 973-985

Voir la notice de l'article provenant de la source Cambridge University Press

The results in this paper respond to two rather natural questions about a polar decomposition A = UP, where U is a unitary matrix and P is positive semidefinite. Let λ 1, ..., λn be the eigenvalues of A. The questions are: (A) When will |λ1|, ..., |λn| be the eigenvalues of P? (B) When will λ1/|λ1|, ..., λn/|λn| be the eigenvalues of U? The complete answer to (A) is “if and only if U and P commute.” In an important special case the answer to (B) is “if and only if U 2 and P commute.“Since these matters are best couched in terms of two different inertias, we begin with a unifying definition of inertia which views all inertias from a single perspective.For each square complex matrix A and each complex number z let m(A, z) denote the multiplicity of z as a root of the characteristic polynomial 
Cain, Bryan E. Some Spectral Properties of Polar Decompositions. Canadian journal of mathematics, Tome 36 (1984) no. 6, pp. 973-985. doi: 10.4153/CJM-1984-055-4
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