1,099 research outputs found
Diagonality Measures of Hermitian Positive-Definite Matrices with Application to the Approximate Joint Diagonalization Problem
In this paper, we introduce properly-invariant diagonality measures of
Hermitian positive-definite matrices. These diagonality measures are defined as
distances or divergences between a given positive-definite matrix and its
diagonal part. We then give closed-form expressions of these diagonality
measures and discuss their invariance properties. The diagonality measure based
on the log-determinant -divergence is general enough as it includes a
diagonality criterion used by the signal processing community as a special
case. These diagonality measures are then used to formulate minimization
problems for finding the approximate joint diagonalizer of a given set of
Hermitian positive-definite matrices. Numerical computations based on a
modified Newton method are presented and commented
Novel Modifications of Parallel Jacobi Algorithms
We describe two main classes of one-sided trigonometric and hyperbolic
Jacobi-type algorithms for computing eigenvalues and eigenvectors of Hermitian
matrices. These types of algorithms exhibit significant advantages over many
other eigenvalue algorithms. If the matrices permit, both types of algorithms
compute the eigenvalues and eigenvectors with high relative accuracy.
We present novel parallelization techniques for both trigonometric and
hyperbolic classes of algorithms, as well as some new ideas on how pivoting in
each cycle of the algorithm can improve the speed of the parallel one-sided
algorithms. These parallelization approaches are applicable to both
distributed-memory and shared-memory machines.
The numerical testing performed indicates that the hyperbolic algorithms may
be superior to the trigonometric ones, although, in theory, the latter seem
more natural.Comment: Accepted for publication in Numerical Algorithm
Small-Deviation Inequalities for Sums of Random Matrices
Random matrices have played an important role in many fields including
machine learning, quantum information theory and optimization. One of the main
research focuses is on the deviation inequalities for eigenvalues of random
matrices. Although there are intensive studies on the large-deviation
inequalities for random matrices, only a few of works discuss the
small-deviation behavior of random matrices. In this paper, we present the
small-deviation inequalities for the largest eigenvalues of sums of random
matrices. Since the resulting inequalities are independent of the matrix
dimension, they are applicable to the high-dimensional and even the
infinite-dimensional cases
Computing a logarithm of a unitary matrix with general spectrum
We analyze an algorithm for computing a skew-Hermitian logarithm of a unitary
matrix. 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 in the considered algorithm.
A modification is introduced to deal properly with the -skew symmetric
unitary matrices. Applications to numerical studies of topological insulators
in two symmetry classes are discussed.Comment: Added discussion of Floquet Hamiltonian
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