Some Theorems on the Inertia of General Matrices

Ostrowski, Alexander; Schneider, Hans

doi:10.1007/978-3-0348-9355-8_37

Cited by 19 publications

(42 citation statements)

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“…While giving an equivalent formulation of the inertia theorem of Ostrowski and Schneider [18], Wimmer [24] proves the following result on Hermitian matrices: Suppose A is a n × n complex Hermitian matrix given in the block form by…”

Section: Theorem 42mentioning

confidence: 99%

The Cauchy interlacing theorem in simple Euclidean Jordan algebras and some consequences

Gowda

Tao

2011

Linear and Multilinear Algebra

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Section: Theorem 42mentioning

confidence: 99%

The Cauchy interlacing theorem in simple Euclidean Jordan algebras and some consequences

Gowda

Tao

2011

Linear and Multilinear Algebra

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“…However, the proofs there are based on different ideas, related to the classical Lyapunov's asymptotic stability theorem (see [10,11] This proposition implies, in particular, the change of stability properties of the fixed point x = 0 when the quadratic form (2) loses the minimum property. Indeed, the number of eigenvalues of A lying outside (and inside) the unit disk is i − .…”

Section: Theorem 1 Suppose That the Following Nondegeneracy Conditiomentioning

confidence: 99%

Spectral properties of operators with polynomial invariants in real finite-dimensional spaces

Козлов

2010

Proc. Steklov Inst. Math.

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Abstract-We consider linear operators lying in the orthogonal group of a quadratic form and study those spectral properties of such operators that can be expressed in terms of the signature of this form. We show that in the typical case these transformations are symplectic. Some of the results can be extended to the general case when the operator admits a homogeneous form of degree ≥ 3. DOI: 10.1134/S0081543810010128Let E be an n-dimensional vector space over the field of real numbers and A : E → E be a linear operator that admits an invariant homogeneous form F of degree m,A meaningful example is given by linear differential equations with periodic coefficients in the case when such an equation admits a homogeneous first integral F . Then A is a monodromy operator. However, in this situation A is homotopic to the identity operator I, but we will use this property only partially in Sections 2 and 3.The case of m = 1 is trivial: the mappinghas an invariant linear form F = 0 if and only if λ = 0 is an eigenvalue of the operator A (det A = 0). Therefore, we consider the case of m ≥ 2. For m = 2 the transformations (1) constitute the orthogonal group of the form F . In the real case, this group is determined only by the signature of F and is often denoted as O(i + , i − ), where i ± are the indices of inertia of F . The group O(1, n − 1) is called the Lorentz group.Our goal is to express the spectral properties of typical operators in the group O(i + , i − ) in terms of the signature of an invariant quadratic form.By typical operators we mean operators without eigenvalues +1 and −1. In the theory of differential equations, such monodromy operators correspond to nondegenerate periodic solutions. General results of this kind are presented in Section 1. Note one of their corollaries: The spectrum of a typical Lorentz transformation contains a real pair λ and λ −1 , λ = ±1, the other points of the spectrum lying on a unit circle and having simple elementary divisors.The following fact looks somewhat unexpected: a typical transformation (1) is symplectic. In particular, n is even and the characteristic polynomial of the operator A is self-reciprocal. For linear operators with invariant forms of degree m > 2, this is not so.Some of the results presented in Section 1 can be interpreted as the properties of the fixed point set of some auxiliary continuous automorphism (Section 2). This trick allows us to find topological

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“…В 1960-е годы было подмечено следующее важное обстоятель-ство: течения идеальной несжимаемой жидкости -геодезические линии метрики (26). Таким образом, идеальная несжимаемая жидкость -это бесконечномерный "волчок Эйлера" с правоин-вариантной метрикой на группе SDiff Q [20]- [22].…”

Section: это утверждение обобщает свойствоunclassified

“…Этот случай отвечает си-стемам с диссипацией энергии. Если диссипация полная (квадра-тичная форма (32) отрицательно определена), то u = i − (теорема Островского-Шнейдера [26]). …”

Section: энергетические критерии устойчивостиunclassified