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Invariant measure on sums of symmetric 3\(\times 3\) matrices with specified eigenvalues. (English) Zbl 0733.15005

Let A and B be real symmetric \(3\times 3\) matrices. If the eigenvalues of A and B are specified, then the possible eigenvalues of \(A+B\) are subject to certain inequalities. This problem and related questions arise in the context of the computation of the quadrupole moment of two rigid rotors.
In this paper these questions are studied by looking at the equivalent ones for a sum \(A+B^{\sigma}\), where A and B are diagonal matrices and \(\sigma\) is an element of SO(3). In fact \(\sigma\) is related to the relative orientations of the rotors. Of particular interest is the case when B has a repeated eigenvalue. Then, by translating and multiplying by a scalar, it is possible to assume B has the special form Diag(1,0,0). In this case, the SO(3) Haar measure of the set of eigenvalues of the sum matrix is found.

MSC:

15A22 Matrix pencils
15A18 Eigenvalues, singular values, and eigenvectors
28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures
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