63 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
Change of the *congruence canonical form of 2-by-2 matrices under perturbations
We study how small perturbations of a 2-by-2 complex matrix can change its
canonical form for *congruence. We construct the Hasse diagram for the closure
ordering on the set of *congruence classes of 2-by-2 matrices.Comment: 8 pages. arXiv admin note: substantial text overlap with
arXiv:1105.216
Generic symmetric matrix pencils with bounded rank
We show that the set of n × n complex symmetric matrix pencils of rank at most r is the union of the closures of [r/2] + 1 sets of matrix pencils with some, explicitly described,complete eigenstructures. As a consequence, these are the generic complete eigenstructures of n × n complex symmetric matrix pencils of rank at most r. We also show that these closures correspondto the irreducible components of the set of n × n symmetric matrix pencils with rank at most r when considered as an algebraic set
Safety neighbourhoods for the Kronecker canonical form
Abstract We give safety neighbourhoods for the necessary conditions in the change of the Kronecker canonical form of a matrix pencil under small perturbations
Studies in matrix perturbation and robust statistics
Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1999.Includes bibliographical references (p. 124-132).by Yanyuan Ma.Ph.D
Structure-Preserving Model Reduction of Physical Network Systems
This paper considers physical network systems where the energy storage is naturally associated to the nodes of the graph, while the edges of the graph correspond to static couplings. The first sections deal with the linear case, covering examples such as mass-damper and hydraulic systems, which have a structure that is similar to symmetric consensus dynamics. The last section is concerned with a specific class of nonlinear physical network systems; namely detailed-balanced chemical reaction networks governed by mass action kinetics. In both cases, linear and nonlinear, the structure of the dynamics is similar, and is based on a weighted Laplacian matrix, together with an energy function capturing the energy storage at the nodes. We discuss two methods for structure-preserving model reduction. The first one is clustering; aggregating the nodes of the underlying graph to obtain a reduced graph. The second approach is based on neglecting the energy storage at some of the nodes, and subsequently eliminating those nodes (called Kron reduction).</p
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