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Characterization and computation of matrices of maximal trace over rotations. (English) Zbl 1429.15006

Summary: Given a \(d\times d\) matrix \(M\), it is well known that finding a \(d\times d\) rotation matrix \(U\) that maximizes the trace of \(UM\), i.e., that makes \(UM\) a matrix of maximal trace over rotation matrices, can be achieved with a method based on the computation of the singular value decomposition (SVD) of \(M\). We characterize \(d\times d\) matrices of maximal trace over rotation matrices in terms of their eigenvalues, and for \(d=2,3\), we identify alternative ways, other than the SVD, of computing \(U\) so that \(UM\) is of maximal trace over rotation matrices.

MSC:

15A15 Determinants, permanents, traces, other special matrix functions
15A18 Eigenvalues, singular values, and eigenvectors
15A42 Inequalities involving eigenvalues and eigenvectors
15B10 Orthogonal matrices
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