75 research outputs found
Change of the congruence canonical form of 2-by-2 and 3-by-3 matrices under perturbations and bundles of matrices under congruence
We construct the Hasse diagrams and for the closure ordering on
the sets of congruence classes of and complex matrices.
In other words, we construct two directed graphs whose vertices are
or, respectively, canonical matrices under congruence and there is
a directed path from to if and only if can be transformed by an
arbitrarily small perturbation to a matrix that is congruent to .
A bundle of matrices under congruence is defined as a set of square matrices
for which the pencils belong to the same bundle under
strict equivalence. In support of this definition, we show that all matrices in
a congruence bundle of or matrices have the same
properties with respect to perturbations. We construct the Hasse diagrams
and for the closure ordering on the sets of
congruence bundles of and, respectively, matrices. We
find the isometry groups of and congruence canonical
matrices.Comment: 34 page
Computing Eigenspaces with Specified Eigenvalues of a Regular Matrix Pair (A, B) and Condition Estimation: Theory, Algorithms and Software
Theory, algorithms and LAPACK-style software for computing a pair of de ating subspaces with specified eigenvalues of a regular matrix pair (A, B) and error bounds for computed quantities (eigenvalues and eigenspaces) are presented. The reordering of specified eigenvalues is performed with a direct orthogonal transformation method with guaranteed numerical stability. Each swap of two adjacent diagonal blocks in the real generalized Schur form, where at least one of them corresponds to a complex conjugate pair of eigenvalues, involves solving a generalized Sylvester equation and the construction of two orthogonal transformation matrices from certain eigenspaces associated with the diagonal blocks. The swapping of two 1 x 1 blocks is performed using orthogonal (unitary) Givens rotations. The error bounds are based on estimates of condition numbers for eigenvalues and eigenspaces. The software computes reciprocal values of a condition number for an individual eigenvalue (or a cluster of eigenvalues), a condition number for an eigenvector (or eigenspace), and spectral projectors onto a selected cluster. By computing reciprocal values we avoid over ow. Changes in eigenvectors and eigenspaces are measured by their change in angle. The condition numbers yield both asymptotic and global error bounds. The asymptotic bounds are only accurate for small perturbations (E, F) of(A, B), while the global bounds work for all ||(E, F)|| up to a certain bound, whose size is determined by the conditioning of the problem. It is also shown how these upper bounds can be estimated. Fortran 77 software that implements our algorithms for reordering eigenvalues, computing (left and right) deflating subspaces with specified eigenvalues and condition number estimation are presented. Computational experiments that illustrate the accuracy, efficiency and reliability of our software are also described
Parallel Scientific Computing and Optimization
Introduces the developments in the construction, analysis, and implementation of parallel computing algorithms. This book includes construction and analysis of parallel algorithms for linear algebra and optimization problems. It covers different aspects of parallel architectures, including distributed memory computers with multicore processors
Orbit closure hierarchies of skew-symmetric matrix pencils
We study how small perturbations of a skew-symmetric matrix pencil may change its canonical form under congruence. This problem is also known as the stratification problem of skew-symmetric matrix pencil orbits and bundles. In other words, we investigate when the closure of the congruence orbit (or bundle) of a skew-symmetric matrix pencil contains the congruence orbit (or bundle) of another skew-symmetric matrix pencil. This theory relies on our main theorem stating that a skew-symmetric matrix pencil A-λB can be approximated by pencils strictly equivalent to a skew-symmetric matrix pencil C-λD if and only if A-λB can be approximated by pencils congruent to C-λD
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