Indefinite Scalar Product Research Articles

The Jacobi-orthogonality in indefinite scalar product spaces

We generalize the property of Jacobi-orthogonality to indefinite scalar product spaces. We compare various principles and investigate relations between Osserman, Jacobi-dual, and Jacobi-orthogonal algebraic curvature tensors. We show that every quasi-Clifford tensor is Jacobi-orthogonal. We prove that a Jacobi-diagonalizable Jacobi-orthogonal tensor is Jacobi-dual whenever JX has no null eigenvectors for all nonnull X. We show that any algebraic curvature tensor of dimension 3 is Jacobi-orthogonal if and only if it is of constant sectional curvature. We prove that every 4-dimensional Jacobidiagonalizable algebraic curvature tensor is Jacobi-orthogonal if and only if it is Osserman.

A divide-and-conquer method for constructing a pseudo-Jacobi matrix from mixed given data

For the given signature operator H=Ir⊕−In−r, a pseudo-Jacobi matrix is a self-adjoint matrix relatively to a symmetric bilinear form 〈⋅,⋅〉H, and it is the counterpart of a classical Jacobi matrix to the indefinite scalar product space setting. In this article, we consider an inverse eigenvalue problem for this class of matrices. The main concern is to construct an n×n pseudo-Jacobi matrix from a prescribed n-tuple of distinct real numbers and a Jacobi matrix of order not less than ⌊n2⌋, such that its spectrum is this tuple and the given Jacobi matrix is its trailing principal submatrix. A divide-and-conquer scheme is used to solve this problem, and a necessary and sufficient condition under which the problem is solvable is presented. A numerical algorithm is designed to solve this pseudo-Jacobi matrix inverse eigenvalue problem according to the obtained results. Some illustrative numerical examples are also given to test the reconstructive algorithm.

On the existence of a curvature tensor for given Jacobi operators

It is well known that the Jacobi operators completely determine the curvature tensor. The question of existence of a curvature tensor for given Jacobi operators naturally arises, which is considered and solved in the previous work. Unfortunately, although the published theorem is correct, its proof is incomplete because it contains some omissions, and the aim of this paper is to present a complete and accurate proof. We also generalize the main theorem to the case of indefinite scalar product space. Accordingly, we generalize the proportionality principle for Osserman algebraic curvature tensors.

On the density of semisimple matrices in indefinite scalar product spaces

For an indefinite scalar product $[x,y]_B = x^HBy$ for $B= \pm B^H \in \mathbf{Gl}_n(\mathbb{C})$ on $\mathbb{C}^n \times \mathbb{C}^n$, it is shown that the set of diagonalizable matrices is dense in the set of all $B$-normal matrices. The analogous statement is also proven for the sets of $B$-selfadjoint, $B$-skewadjoint and $B$-unitary matrices.

Classification of linear operators satisfying (Au,v)=(u,Arv) or (Au,Arv)=(u,v) on a vector space with indefinite scalar product

We classify all linear operators A:V→V satisfying (Au,v)=(u,Arv) and all linear operators satisfying (Au,Arv)=(u,v) with r=2,3,… on a complex, real, or quaternion vector space with scalar product given by a nonsingular symmetric, skew-symmetric, Hermitian, or skew-Hermitian form.

Indefinite Ruhe’s Variant of the Block Lanczos Method for Solving the Systems of Linear Equations

In this paper, we equip Cn with an indefinite scalar product with a specific Hermitian matrix, and our aim is to develop some block Krylov methods to indefinite mode. In fact, by considering the block Arnoldi, block FOM, and block Lanczos methods, we design the indefinite structures of these block Krylov methods; along with some obtained results, we offer the application of this methods in solving linear systems, and as the testifiers, we design numerical examples.

Some Hyperbolic Iterative Methods for Linear Systems

The indefinite inner product defined by J=diagj1,…,jn, jk∈−1,+1, arises frequently in some applications, such as the theory of relativity and the research of the polarized light. This indefinite scalar product is referred to as hyperbolic inner product. In this paper, we introduce three indefinite iterative methods: indefinite Arnoldi’s method, indefinite Lanczos method (ILM), and indefinite full orthogonalization method (IFOM). The indefinite Arnoldi’s method is introduced as a process that constructs a J-orthonormal basis for the nondegenerated Krylov subspace. The ILM method is introduced as a special case of the indefinite Arnoldi’s method for J-Hermitian matrices. IFOM is mentioned as a process for solving linear systems of equations with J-Hermitian coefficient matrices. Finally, by providing numerical examples, the FOM, IFOM, and ILM processes have been compared with each other in terms of the required time for solving linear systems and also from the point of the number of iterations.

Some Properties of the Exchange Operator with respect to Structured Matrices defined by Indefinite Scalar Product Spaces

The properties of the exchange operator on some types of matrices are explored in this paper. In particular, the properties of exc(A,p,q), where A is a given structured matrix of size (p+q)×(p+q) and exc : M ×N×N → M is the exchange operator are studied. This paper is a generalization of one of the results in [N.J. Higham. J-orthogonal matrices: Properties and generation. SIAM Review, 45:504–519, 2003.].

Signature and Spectral Flow of J-Unitary $${\mathbb{S}^1}$$ S 1 -Fredholm Operators

Bijective operators conserving the indefinite scalar product on a Krein space \({(\mathcal{K}, J)}\) are called J-unitary. Such an operator T is defined to be \({\mathbb{S}^1}\) -Fredholm if T−z1 is Fredholm for all z on the unit circle \({\mathbb{S}^1}\) , and essentially \({\mathbb{S}^1}\) -gapped if there is only discrete spectrum on \({\mathbb{S}^1}\) . For paths in the \({\mathbb{S}^1}\) -Fredholm operators an intersection index similar to the Conley–Zehnder index is introduced. The strict subclass of essentially \({\mathbb{S}^1}\) -gapped operators has a countable number of components which can be distinguished by a homotopy invariant given by the signature of J restricted to the eigenspace of all eigenvalues on \({\mathbb{S}^1}\) . These concepts are illustrated by several examples.

Procrustes problems in finite dimensional indefinite scalar product spaces

Some applications from multidimensional scaling in an environment of indefinite scalar products are investigated. In particular, the construction of points from given distances is considered, and two variants of Procrustes problems are discussed. It is found that both Procrustes problems can be solved with the help of H-polar decompositions or ( G, H)-polar decompositions of specified matrices. Furthermore, a method for the numerical computation of H- or ( G, H)-polar decompositions of nonsingular matrices is described.

Definite and indefinite inner products on superspace (Hilbert–Krein superspace)

We present natural (invariant) definite and indefinite scalar products on the N=1 superspace which turns out to carry an inherent Hilbert-Krein structure. We are motivated by supersymmetry in physics but prefer a general mathematical framework.

Slow damping of internal waves in a stably stratified fluid

We study the damping of internal gravity waves in a stably stratified fluid with constant viscosity in two- and three-dimensional bounded domains. For the linearized Navier-Stokes equations for incompressible flow with no-slip boundary conditions that model this fluid, we prove there are non-oscillatory normal modes with arbitrarily small exponential decay rates. The proof is very different from that for a horizontally periodic layer and depends on a structure theorem for compact operators which are self-adjoint with respect to an indefinite scalar product in a Hilbert space. We give a complete proof of this theorem, which is closely related to results of Pontrjagin.

Stability of self-adjoint square roots and polar decompositions in indefinite scalar product spaces

Continuity properties of factors in polar decompositions of matrices with respect to indefinite scalar products are studied. The matrix having the polar decomposition and the indefinite scalar product are allowed to vary. Closely related properties of a self-adjoint (with respect to an indefinite scalar product) perturbed matrix to have a self-adjoint square root, or to have a representation of the form X*X, are also studied.

Classes of plus matrices in finite dimensional indefinite scalar product spaces

We describe in terms of canonical forms various classes of plus matrices in real and complex finite dimensional spaces with indefinite scalar product and study their topological structures (such as closure and interior).

Linearizations, realization, and scalar products for regular matrix polynomials

The connections between linearizations of regular matrix polynomials L( λ) under shift and inversion of the parameter are developed and used to obtain a new formula for the realization of L( λ) −1. In the self-adjoint case nondegenerate indefinite scalar products are obtained in which the fundamental (companion) linearizations are self-adjoint.

Errata for: Polar decomposition in finite dimensional indefinite scalar product spaces: Special cases and applications

Errata for: Polar decomposition in finite dimensional indefinite scalar product spaces: Special cases and applications

Polar decompositions in finite dimensional indefinite scalar product spaces: General theory

Several classes of polar decompositions of real and complex matrices with respect to a given indefinite scalar product are studied. Matrices that admit such polar decompositions are described in various ways. In particular, a full description of all polar decompositions of a given matrix up to the natural similarity between polar decompositions is given.

Nonnegative solutions of algebraic Riccati equations

Nonnegative Hermitian solutions of various types of continuous and discrete algebraic Riccati equations are studied. The Hamiltonian is considered with respect to two different indefinite scalar products. For the set of nonnegative solutions the order structure and the topology of the set and the stability of solutions is treated. For general Hermitian solutions a method to compute the inertia is given. Although most attention is payed to the classical types arising from LQ optimal control theory, the case where the quadratic term has an indefinite coefficient is studied as well.

Polar decompositions in finite dimensional indefinite scalar product spaces: General theory

Several classes of polar decompositions of real and complex matrices with respect to a given indefinite scalar product are studied. Matrices that admit such polar decompositions are described in various ways. In particular, a full description of all polar decompositions of a given matrix up to the natural similarity between polar decompositions is given.

Nonnegative solutions of algebraic Riccati equations

Nonnegative Hermitian solutions of various types of continuous and discrete algebraic Riccati equations are studied. The Hamiltonian is considered with respect to two different indefinite scalar products. For the set of nonnegative solutions the order structure and the topology of the set and the stability of solutions is treated. For general Hermitian solutions a method to compute the inertia is given. Although most attention is payed to the classical types arising from LQ optimal control theory, the case where the quadratic term has an indefinite coefficient is studied as well.