Invariant neutral subspaces for Hamiltonian matrices

Let G be a group of linear transformations on a finite dimensional real or complex vector space X. Assume X is completely reducible as a G-module. Let S be the ring of all complex-valued polynomials on X, regarded as a G-module in the obvious way, and let J ? S be the sub-ring of all G-invariant polynomials on X.

Similarity vs unitary similarity and perturbation analysis of sign characteristics: Complex and real indefinite inner products

Linear algebra is considered an essential mathematical theory that has many engineering applications. While many theorem provers support linear spaces, they only consider finite dimensional spaces. In addition, available libraries only deal with real vectors, whereas complex vectors are extremely useful in many fields of engineering. In this paper, we propose a new linear space formalization which covers both finite and infinite dimensional complex vector spaces, implemented in HOL-Light. We give the definition of a linear space and prove many properties about its operations, e.g., addition and scalar multiplication. We also formalize a number of related fundamental concepts such as linearity, hermitian operation, self-adjoint, and inner product space. Using the developed linear algebra library, we were able to implement basic definitions about quantum mechanics and use them to verify a quantum beam splitter, an optical device that has many applications in quantum computing.

Let V be a finite dimensional complex vector space, and for each positive integer n let denote the vector space whose elements are the complex valued symmetric n-linear functions on (n-copies). If , that is, f is a linear functional on V, then we define the symmetric n-linear function by for all . It is well known that the elements , known as polar elements, span . A basis for consisting entirely of polar elements is a polar basis. We present a surprisingly simple recursive algorithm for the generation of polar bases. Explicit stand alone formulas for polar bases are also presented. One such explicit basis is so natural that it should probably be called the canonical polar basis for . We introduce the notion of subspace filter and show that each subspace filter of order n provides polar bases for all of the spaces simultaneously, and we extend this result to the entire symmetric algebra .

On the operation Alg Lat in finite dimensions

This paper is an expanded and updated version of the preprint arXiv:math/0406499. It includes a more detailed description of the basics of the theory of Cherednik and Hecke algebras of varieties started in arXiv:math/0406499, as well as a new Section 4, which reviews the developments in this theory since 2004 with references to the relevant literature. Let $G$ be a finite group of linear transformations of a finite dimensional complex vector space $V$. To this data one can attach a family of algebras $H_{t,c}(V,G)$, parametrized by complex numbers $t$ and conjugation invariant functions $c$ on the set of complex reflections in $G$, which are called rational Cherednik algebras. These algebras have been studied for over 15 years and revealed a rich structure and deep connections with algebraic geometry, representation theory, and combinatorics. In this paper, we define global analogs of Cherednik algebras, attached to any smooth algebraic or analytic variety $X$ with a finite group $G$ of automorphisms. We show that many interesting properties of Cherednik algebras (such as the PBW theorem, universal deformation property, relation to Calogero-Moser spaces, action on quasiinvariants) still hold in the global case, and give several interesting examples. Then we define the KZ functor for global Cherednik algebras, and use it to define (in the case $\pi_2(X)\otimes \Bbb Q=0$) a flat deformation of the orbifold fundamental group of the orbifold $X/G$, which we call the Hecke algebra of $X/G$. This includes usual, affine, and double affine Hecke algebras for Weyl groups, Hecke algebras of complex reflection groups, as well as many new examples.

We consider the space $[\Gamma, rr_{0}, v^{r}, \varrho]$ of all vector-valued holomorphic modular forms $f\colon \mathbb{H}_{n} \to \mathcal{Z}$ of transformation type \begin{equation*} f(MZ)=v^{r}(M)\det(CZ+D)^{r_{0}r}\varrho(CZ+D)f(Z). \end{equation*} $\varrho\colon \mathrm{GL}(n, \mathbb{C}) \to \mathrm{GL}(\mathcal{Z})$ is a rational representation on a finite dimensional complex vector space $\mathcal{Z}$. These spaces can be collected in a graded $A(\Gamma)$-module \begin{equation*} \mathcal{M} = \mathcal{M}_{\Gamma}(r_{0}, v, \varrho) := \bigoplus_{r \in \mathbb{Z}}[\Gamma, rr_{0}, v^{r}, \varrho]. \end{equation*} We treat in this paper some special cases in genus 2. The first one is essentially due to Wieber. Here the starting weight is $1/2$, the starting multiplier system is the multiplier system $v_{\Theta}$ and for $\varrho$ we take the second symmetric power of standard representation. Thus we consider a variant of this case and a new example. In this final case the starting weight is $1/2$, the starting multiplier system is the theta multiplier system $v_{\vartheta}$ and for $\varrho$ we take the standard representation. In all these cases we will determine the structure of $\mathcal{M}$.

If f: X → Y is a holomorphic map between complex spaces X and Y, the direct images f (q)(S) of a coherent O X -sheaf S in general are not coherent O Y -sheaves. For example if Y is just a single (reduced) point, coherence of the q-th image sheaf f(q)(S) means that f (q) (S)=Hq(X, S) is a finite dimensional complex vector space. There are many examples of non-compact spaces X where this is not the case.KeywordsManifoldCoherenceSteinResoThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper we give complete analytic invariants for the set of germs of holomorphic foliations in$(\mathbb{C}^{2},0)$that become regular after a single blow-up. Some of the invariants describe the holonomy pseudogroup of the germ and are called transverse invariants. The other invariants lie in a finite dimensional complex vector space. Such singularities admit separatrices tangentially to any direction at the origin. When enough separatrices are leaves of a radial foliaton (a condition that can always be attained if the multiplicity of the germ at the origin is at most four) we are able to describe and realize all the analytical invariants geometrically and provide analytic normal forms. As a consequence, we prove that any two such germs sharing the same transverse invariants are conjugated by a very particular type of birational transformation. We also provide explicit examples of universal equisingular unfoldings of foliations that cannot be produced by unfolding functions. With these at hand we are able to explicitly parametrize families of analytically distinct foliations that share the same transverse invariants.

A characterization of positive operators on finite dimensional complex vector spaces, developed from the Routh-Hurwitz criterion with review of some basic concepts and results.

Sum and intersection of linear subspaces in a vector space over a field are fundamental operations in linear algebra. The purpose of this survey paper is to give a comprehensive approach to the sums and intersections of two linear subspaces and their orthogonal complements in the finite-dimensional complex vector space. We shall establish a variety of closed-form formulas for representing the direct sum decompositions of them-dimensional complex column vector space 𝔺mwith respect to a pair of given linear subspaces 𝒨 and 𝒩 and their operations, and use them to derive a huge amount of decomposition identities for matrix expressions composed by a pair of orthogonal projectors onto the linear subspaces. As applications, we give matrix representation for the orthogonal projectors onto the intersections of a pair of linear subspaces using various matrix decomposition identities and Moore–Penrose inverses; necessary and su˚cient conditions for two linear subspaces to be in generic position; characterization of the commutativity of a pair of orthogonal projectors; necessary and su˚cient conditions for equalities and inequalities for a pair of subspaces to hold; equalities and inequalities for norms of a pair of orthogonal projectors and their operations; as well as a collection of characterizations of EP-matrix.

A linear transformation X of the n -dimensional real or complex vector space E can be thought of as a vector field: X associates to each point p in E , the vector X (p ). As an aid to one’s imagination one may think of a fluid in motion, so that the velocity of the fluid particles passing through the point p is always X (p ); the vector field is then the current of the flow and the paths of the fluid particles are the trajectories .

On the normalizer of the reflexive cover of a unital algebra of linear transformations

Let Ik denote the identity matrix of order k and set **-Let SrX&,) denote the group of all complex Ik X k matrices which satisfy the equation gskg' sk.Let be the linear space of all n X 2k complex matrices with k > n, and let S(E*) denote the symmetric algebra of all complex-valued polynomial functions on E. The study of the action of Sp(At,C), which is obtained by right translation on S(E*), leads to a concrete and simple realization of all irreducible holomorphic representations of Sp(k,C).In connection with this realization, a theory of symplectic Stiefel harmonics is also established.This notion may be thought of as a generalization of the spherical harmonics for the symplectic Stiefel manifold. Introduction. Let Ik denote the unit matrix of order k and setLet Sp(zt,C) denote the group of all complex 2k X 2k matrices g satisfying gskg' = sk.If G is a complex Lie group and F is a finite-dimensional complex vector space, then by a holomorphic representation of G in V we mean a complex analytic homomorphism of G into GL(F).The object of this paper is to present, in as simple and explicit a form as possible, a description of all irreducible holomorphic representations (up to equivalence) of the complex linear symplectic group Sp(,C).This realization is obtained by analyzing certain natural actions of Sp(:,C) on a symmetric algebra of polynomial functions of matrix argument.We rely on some general results of S. Helgason [3] and B. Kostant [4] to study this symmetric algebra.The techniques they used to decompose a symmetric algebra of polynomial functions were based on the existence of a symmetric bilinear form defined on this algebra.The situation is somewhat different in the context of our investigation, because

Let H⊂GL(V) be a connected complex reductive group where V is a finite-dimensional complex vector space. Let v∈V and let G={g∈ GL (V)∣gHv=Hv}. Following Raïs we say that the orbit Hv is characteristic for H if the identity component of G is H. If H is semisimple, we say that Hv is semi-characteristic for H if the identity component of G is an extension of H by a torus. We classify the H-orbits which are not (semi)-characteristic in many cases.