اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدReduction of a pair of skew-symmetric matrices to its canonical form under congruence
69
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Let $(A,B)$ be a pair of skew-symmetric matrices over a field of characteristic not 2. Its regularization decomposition is a direct sum [ (underline{underline A},underline{underline B})oplus (A_1,B_1)oplusdotsoplus(A_t,B_t) ] that is congruent to $(A,B)$, in which $(underline{underline A},underline{underline B})$ is a pair of nonsingular matrices and $(A_1,B_1),$ $dots,$ $(A_t,B_t)$ are singular indecomposable canonical pairs of skew-symmetric matrices under congruence. We give an algorithm that constructs a regularization decomposition. We also give a constructive proof of the known canonical form of $(A,B)$ under congruence over an algebraically closed field of characteristic not 2.
قيم البحث
اقرأ أيضاً
We consider a family of pairs of m-by-p and m-by-q matrices, in which some entries are required to be zero and the others are arbitrary, with respect to transformations (A,B)--> (SAR,SBL) with nonsingular S, R, L. We prove that almost all of these pa
irs reduce to the same pair (C, D) from this family, except for pairs whose arbitrary entries are zeros of a certain polynomial. The polynomial and the pair (C D) are constructed by a combinatorial method based on properties of a certain graph.
In a thorough paper Kuchar has examined the canonical reduction of the most general action functional describing the geometrodynamics of the maximally extended Schwarzschild geometry. This reduction yields the true degrees of freedom for (vacuum) sph
erically symmetric general relativity. The essential technical ingredient in Kuchars analysis is a canonical transformation to a certain chart on the gravitational phase space which features the Schwarzschild mass parameter $M_{S}$, expressed in terms of what are essentially Arnowitt-Deser-Misner variables, as a canonical coordinate. In this paper we discuss the geometric interpretation of Kuchars canonical transformation in terms of the theory of quasilocal energy-momentum in general relativity given by Brown and York. We find Kuchars transformation to be a ``sphere-dependent boost to the rest frame, where the ``rest frame is defined by vanishing quasilocal momentum. Furthermore, our formalism is general enough to cover the case of (vacuum) two-dimensional dilaton gravity. Therefore, besides reviewing Kuchav{r}s original work for Schwarzschild black holes from the framework of hyperbolic geometry, we present new results concerning the canonical reduction of Witten-black-hole geometrodynamics.
We provide a solution to the problem of simultaneous $diagonalization$ $via$ $congruence$ of a given set of $m$ complex symmetric $ntimes n$ matrices ${A_{1},ldots,A_{m}}$, by showing that it can be reduced to a possibly lower-dimensional problem whe
re the question is rephrased in terms of the classical problem of simultaneous $diagonalization$ $via$ $similarity$ of a new related set of matrices. We provide a procedure to determine in a finite number of steps whether or not a set of matrices is simultaneously diagonalizable by congruence. This solves a long standing problem in the complex case.
We present fast numerical methods for computing the Hessenberg reduction of a unitary plus low-rank matrix $A=G+U V^H$, where $Gin mathbb C^{ntimes n}$ is a unitary matrix represented in some compressed format using $O(nk)$ parameters and $U$ and $V$
are $ntimes k$ matrices with $k< n$. At the core of these methods is a certain structured decomposition, referred to as a LFR decomposition, of $A$ as product of three possibly perturbed unitary $k$ Hessenberg matrices of size $n$. It is shown that in most interesting cases an initial LFR decomposition of $A$ can be computed very cheaply. Then we prove structural properties of LFR decompositions by giving conditions under which the LFR decomposition of $A$ implies its Hessenberg shape. Finally, we describe a bulge chasing scheme for converting the initial LFR decomposition of $A$ into the LFR decomposition of a Hessenberg matrix by means of unitary transformations. The reduction can be performed at the overall computational cost of $O(n^2 k)$ arithmetic operations using $O(nk)$ storage. The computed LFR decomposition of the Hessenberg reduction of $A$ can be processed by the fast QR algorithm presented in [8] in order to compute the eigenvalues of $A$ within the same costs.
In this paper we establish Springer correspondence for the symmetric pair $(mathrm{SL}(N),mathrm{SO}(N))$ using Fourier transform, parabolic induction functor, and a nearby cycle sheaves construction due to Grinberg. As applications, we obtain result
s on cohomology of Hessenberg varieties and geometric constructions of irreducible representations of Hecke algebras of symmetric groups at $q=-1$.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
