1,932 research outputs found
Miniversal deformations of matrices of bilinear forms
V.I. Arnold [Russian Math. Surveys 26 (2) (1971) 29-43] constructed a
miniversal deformation of matrices under similarity; that is, a simple normal
form to which not only a given square matrix A but all matrices B close to it
can be reduced by similarity transformations that smoothly depend on the
entries of B. We construct a miniversal deformation of matrices under
congruence.Comment: 39 pages. The first version of this paper was published as Preprint
RT-MAT 2007-04, Universidade de Sao Paulo, 2007, 34 p. The work was done
while the second author was visiting the University of Sao Paulo supported by
the Fapesp grants (05/59407-6 and 2010/07278-6). arXiv admin note:
substantial text overlap with arXiv:1105.216
Miniversal deformations of pairs of symmetric matrices under congruence
For each pair of complex symmetric matrices we provide a normal form
with a minimal number of independent parameters, to which all pairs of complex
symmetric matrices , close to can be
reduced by congruence transformation that smoothly depends on the entries of
and . Such a normal form is called a miniversal
deformation of under congruence. A number of independent parameters in
the miniversal deformation of a symmetric matrix pencil is equal to the
codimension of the congruence orbit of this symmetric matrix pencil and is
computed too. We also provide an upper bound on the distance from to
its miniversal deformation.Comment: arXiv admin note: text overlap with arXiv:1104.249
On the condensed density of the generalized eigenvalues of pencils of Hankel Gaussian random matrices and applications
Pencils of Hankel matrices whose elements have a joint Gaussian distribution
with nonzero mean and not identical covariance are considered. An approximation
to the distribution of the squared modulus of their determinant is computed
which allows to get a closed form approximation of the condensed density of the
generalized eigenvalues of the pencils. Implications of this result for solving
several moments problems are discussed and some numerical examples are
provided.Comment: 30 pages, 16 figures, better approximations provide
The linear pencil approach to rational interpolation
It is possible to generalize the fruitful interaction between (real or
complex) Jacobi matrices, orthogonal polynomials and Pade approximants at
infinity by considering rational interpolants, (bi-)orthogonal rational
functions and linear pencils zB-A of two tridiagonal matrices A, B, following
Spiridonov and Zhedanov.
In the present paper, beside revisiting the underlying generalized Favard
theorem, we suggest a new criterion for the resolvent set of this linear pencil
in terms of the underlying associated rational functions. This enables us to
generalize several convergence results for Pade approximants in terms of
complex Jacobi matrices to the more general case of convergence of rational
interpolants in terms of the linear pencil. We also study generalizations of
the Darboux transformations and the link to biorthogonal rational functions.
Finally, for a Markov function and for pairwise conjugate interpolation points
tending to infinity, we compute explicitly the spectrum and the numerical range
of the underlying linear pencil.Comment: 22 page
Structured backward errors for eigenvalues of linear port-Hamiltonian descriptor systems
When computing the eigenstructure of matrix pencils associated with the
passivity analysis of perturbed port-Hamiltonian descriptor system using a
structured generalized eigenvalue method, one should make sure that the
computed spectrum satisfies the symmetries that corresponds to this structure
and the underlying physical system. We perform a backward error analysis and
show that for matrix pencils associated with port-Hamiltonian descriptor
systems and a given computed eigenstructure with the correct symmetry structure
there always exists a nearby port-Hamiltonian descriptor system with exactly
that eigenstructure. We also derive bounds for how near this system is and show
that the stability radius of the system plays a role in that bound
Simplest miniversal deformations of matrices, matrix pencils, and contragredient matrix pencils
V. I. Arnold [Russian Math. Surveys 26 (2) (1971) 29-43] constructed a simple
normal form for a family of complex n-by-n matrices that smoothly depend on
parameters with respect to similarity transformations that smoothly depend on
the same parameters. We construct analogous normal forms for a family of real
matrices and a family of matrix pencils that smoothly depend on parameters,
simplifying their normal forms by D. M. Galin [Uspehi Mat. Nauk 27 (1) (1972)
241-242] and by A. Edelman, E. Elmroth, B. Kagstrom [Siam J. Matrix Anal. Appl.
18 (3) (1997) 653-692].Comment: 20 page
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