Products of Reflections and Triangularization of Bilinear Forms

Decompositions in categories of highest weight modules

This chapter studies, in the setting of vector spaces over a field, the basics concerning multilinear functions, tensor products, spaces of linear functions, and algebras related to tensor products. Sections 1–5 concern special properties of bilinear forms, all vector spaces being assumed to be finite-dimensional. Section 1 associates a matrix to each bilinear form in the presence of an ordered basis, and the section shows the effect on the matrix of changing the ordered basis. It then addresses the extent to which the notion of “orthogonal complement” in the theory of inner-product spaces applies to nondegenerate bilinear forms. Sections 2–3 treat symmetric and alternating bilinear forms, producing bases for which the matrix of such a form is particularly simple. Section 4 treats a related subject, Hermitian forms when the field is the complex numbers. Section 5 discusses the groups that leave some particular bilinear and Hermitian forms invariant. Section 6 introduces the tensor product of two vector spaces, working with it in a way that does not depend on a choice of basis. The tensor product has a universal mapping property—that bilinear functions on the product of the two vector spaces extend uniquely to linear functions on the tensor product. The tensor product turns out to be a vector space whose dual is the vector space of all bilinear forms. One particular application is that tensor products provide a basis-independent way of extending scalars for a vector space from a field to a larger field. The section includes a number of results about the vector space of linear mappings from one vector space to another that go hand in hand with results about tensor products. These have convenient formulations in the language of category theory as “natural isomorphisms.” Section 7 begins with the tensor product of three and then $n$ vector spaces, carefully considering the universal mapping property and the question of associativity. The section defines an algebra over a field as a vector space with a bilinear multiplication, not necessarily associative. If $E$ is a vector space, the tensor algebra $T(E)$ of $E$ is the direct sum over $n\geq0$ of the $n$-fold tensor product of $E$ with itself. This is an associative algebra with a universal mapping property relative to any linear mapping of $E$ into an associative algebra $A$ with identity: the linear map extends to an algebra homomorphism of $T(E)$ into $A$ carrying 1 into 1. Sections 8–9 define the symmetric and exterior algebras of a vector space $E$. The symmetric algebra $S(E)$ is a quotient of $T(E)$ with the following universal mapping property: any linear mapping of $E$ into a commutative associative algebra $A$ with identity extends to an algebra homomorphism of $S(E)$ into $A$ carrying 1 into 1. The symmetric algebra is commutative. Similarly the exterior algebra $\bigwedge(E)$ is a quotient of $T(E)$ with this universal mapping property: any linear mapping $l$ of $E$ into an associative algebra $A$ with identity such that $l(v)^2=0$ for all $v\in E$ extends to an algebra homomorphism of $\bigwedge(E)$ into $A$ carrying 1 into 1. The problems at the end of the chapter introduce some other algebras that are of importance in applications, and the problems relate some of these algebras to tensor, symmetric, and exterior algebras. Among the objects studied are Lie algebras, universal enveloping algebras, Clifford algebras, Weyl algebras, Jordan algebras, and the division algebra of octonions.

Given a bilinear form on , represented by a matrix , the problem of finding the largest dimension of a subspace of such that the restriction of A to this subspace is a non-degenerate skew-symmetric bilinear form is equivalent to finding the size of the largest invertible skew-symmetric matrix B such that the equation is consistent (here denotes the transpose of the matrix X). In this paper, we provide a characterization, by means of a necessary and sufficient condition, for the matrix equation to be consistent when B is a skew-symmetric matrix. This condition is valid for most matrices . To be precise, the condition depends on the canonical form for congruence (CFC) of the matrix A, which is a direct sum of blocks of three types. The condition is valid for all matrices A except those whose CFC contains blocks, of one of the types, with size smaller than 3. However, we show that the condition is necessary for all matrices A.

In this paper we describe an algorithm that quickly computes a maximal a \mathfrak {a} -valued lattice in an F F -vector space equipped with a non-degenerate bilinear form, where a \mathfrak {a} is a fractional ideal in a number field F F . We then apply this construction to give an algorithm to compute an a \mathfrak {a} -maximal lattice in a quadratic space over any number field F F where the prime p = 2 p=2 is unramified. We also develop the theory of p \mathfrak {p} -neighbors for a \mathfrak {a} -valued quadratic lattices at an arbitrary prime p \mathfrak {p} of O F \mathcal {O}_F (including when p ∣ 2 \mathfrak {p}\mid 2 ) and prove its close connection to the residual geometry of certain quadrics mod p \mathfrak {p} . Finally we give a well-known application of p \mathfrak {p} -neighboring lattices and exact mass formulas to compute a complete set of representatives for the classes in a given genus of (totally definite) quadratic O F \mathcal {O}_F -lattices.

The paper discusses a financial market model that generates stochastic volatility using a minimal number of factors. These factors model the dynamics of different denominations of a benchmark portfolio. Asset prices are specified as transformations of square root processes. Numerical results for the pricing and hedging of standard derivatives on indices for this class of models are documented. This includes cases where the standard risk neutral pricing methodology fails but a form of arbitrage still exists. However, payoffs can be perfectly hedged. In addition, the term structure of implied volatilities is documented.

Dual codes are defined with respect to non-degenerate sesquilinear or bilinear forms over a finite Frobenius ring. These dual codes have the properties one expects from a dual code: they satisfy a double-dual property, they have cardinality complementary to that of the primal code, and they satisfy the MacWilliams identities for the Hamming weight.

The set of Fortran subroutines discussed an implementation of the algorithm for finding the eigenvectors, x, and eigenvalues, lambda, such that Ax = lambdax, where A is a real skew-symmetric matrix or a real tridiagonal symmetric matrix with a constant diagonal. The algorithm uses only orthogonal similarity transformations and is believed to be the most efficient procedure available for computing all the eigenvalues or the complete eigensystem for the indicated classes of matrices. The three subroutines of the algorithm and their functions are described as follows: TRIZD--a subroutine that transforms an arbitrary real skew-symmetric matrix to skew-symmetric tridiagonal form by using orthogonal similarity transformations; IMZD--a subroutine that computes the eigenvalues and, optionally, the eigenvectors of a symmetric tridiagonal matrix with zeros on the diagonal or of a skew-symmetric tridiagonal matrix; TBAKZD--a subroutine that computes the eigenvectors of an arbitrary real skew-symmetric matrix by back-transforming the eigenvectors of the corresponding skew-symmetric tridiagonal matrix determined by TRIZD. The subroutines TRIZD, IMZD, and TBAKZD have been tested extensively on an IBM 360/91 computer using double precision arithmetic. Complete subroutine listings are available. (RWR)

Full field algebras, operads and tensor categories

We describe a new class of holonomy groups on pseudo-Riemannian manifolds. Namely, let $g$ be a nondegenerate bilinear form on a vector space $V$, and $L : V \to V$ a $g$-symmetric operator. Then the identity component of the centraliser of $L$ in $\mathrm{SO}(g)$ is a holonomy group for a suitable Levi-Civita connection.

Residually finite quantum group algebras

A Lie algebra is said to be metric if it admits a symmetric, invariant, and non-degenerate bilinear form. The harmonic oscillator algebra, which arises in the quantum mechanical description of a harmonic oscillator, is the smallest solvable non-abelian metric example. This algebra is the first step of a countable series of solvable Lie algebras that support invariant Lorentzian forms. Generalizing this situation, in this paper, we arrive at the oscillator Lie K -algebras as double extensions of metric spaces. The aim of this paper is to present some structural features, invariant metrics, and derivations of this class of algebras and to explore their possibilities of being extended to mixed metric Lie algebras.

Quadratic Dynamical Systems and Algebras

We show how both the contact structure and the metric structure of the thermodynamic phase space arise in a natural way from a generalized canonical probability distribution \ensuremath{\rho}. In particular, the metric form and the contact form are found to be derived from the microscopic entropy s=-ln\ensuremath{\rho}. Thus the first law and the second law of thermodynamics can be given the geometric interpretation that a thermodynamic system must possess both a contact and a compatible metric structure. We proceed to construct explicitly a new nondegenerate bilinear form on the thermodynamic phase space, whose restriction to state space yields the Weinhold-Ruppeiner metric, and whose restriction to Gibbs space can serve as an alternative to the metric proposed by Gilmore.

This article aims to solve the problem of the hyperspectral imagery (HSI) demosaicing under a novel subsampling hyperspectral sensing strategy. The existing method utilizes the periodic structure of subsampling to estimate a fixed subspace in matrix form from the measurement result, which reduces the representation ability of the subspace in iterations and destroys the intrinsic structure of the tensor. To overcome these drawbacks, we propose a tensor-based HSI demosaicing (T-Hy-demosaicing) model with tensor subspace representation, which takes the low-tubal-rankness and the nonlocal self-similarity into account. In particular, we suggest a tensor singular value decomposition based on orthogonal transformation (Tran-based t-SVD) to learn the tensor subspace that possesses a more powerful representation ability. In addition, we develop an effective algorithm to solve the proposed nonconvex model under the framework of the proximal alternating minimization algorithm. Experiments conducted on simulated datasets illustrate that the proposed method outperforms other comparative methods in both visual and quantitative terms.

Sparse orthogonal matrices