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Full-Text Articles in Physical Sciences and Mathematics
Principal Angles And Approximation For Quaternionic Projections [Dataset], Terry A. Loring
Principal Angles And Approximation For Quaternionic Projections [Dataset], Terry A. Loring
Math and Statistics Datasets
We extend Jordan's notion of principal angles to work for two subspaces of quaternionic space, and so have a method to analyze two orthogonal projections in the matrices over the real, complex or quaternionic field (or skew field). From this we derive an algorithm to turn almost commuting projections into commuting projections that minimizes the sum of the displacements of the two projections. We quickly prove what we need using the universal real C*-algebra generated by two projections.
Estimating Norms Of Commutators [Dataset], Terry A. Loring, Freddy Vides
Estimating Norms Of Commutators [Dataset], Terry A. Loring, Freddy Vides
Math and Statistics Datasets
We find estimates on the norm of a commutator of the form [f(x),y] in terms of the norm of [x,y], assuming that x and y are bounded linear operators on Hilbert space, with x normal and with spectrum within the domain of f. In particular we discuss |[x^2,y]| and |[x^{1/2},y]| for 0leq x leq 1. For larger values of delta = |[x,y]| we can rigorous calculate the best possible upper bound |[f(x),y]| leq eta_f(delta) for many f. In other cases we have conducted numerical experiments that strongly suggest that we have in many cases found the correct formula for the …
Computing A Logarithm Of A Unitary Matrix With General Spectrum [Dataset], Terry A. Loring
Computing A Logarithm Of A Unitary Matrix With General Spectrum [Dataset], Terry A. Loring
Math and Statistics Datasets
We analyze an algorithm for computing a skew-Hermitian logarithm of a unitary matrix, and also skew-Hermitian approximate logarithms for nearly unitary matrices. This algorithm is very easy to implement using standard software and it works well even for unitary matrices with no spectral conditions assumed. Certain examples, with many eigenvalues near -1, lead to very non-Hermitian output for other basic methods of calculating matrix logarithms. Altering the output of these algorithms to force an Hermitian output creates accuracy issues which are avoided by the considered algorithm. A modification is introduced to deal properly with the J-skew symmetric unitary matrices. Applications …