Spectral variation bounds for diagonalisable matrices

Summary This note is related to an earlier paper by Bhatia, Davis, and Kittaneh [4]. For matrices similar to Hermitian, we prove an inequality complementary to the one proved in [4, Theorem 3]. We also disprove a conjecture made in [4] about the norm of a commutator. This work was done when the first author visited the SFB 343 at University of Bielefeld in May and June 1994.     

Trace class multipliers and spectral variation of normal matrices

In this article we show how to estimate the trace multiplier norm of a rank 2 matrix. As an application, an alternative proof of a theorem of Holbrook et al. (Maximal spectral distance, Linear Algebra Appl., 249 (1996) 197–205) on the maximal spectral distance between two normal matrices with prescribed eigenvalues is given.     

Spectral geometry,link complements and surgery diagrams

We provide an upper bound on the Cheeger constant and first eigenvalue of the Laplacian of a finite-volume hyperbolic 3-manifold M, in terms of data from any surgery diagram for M. This has several consequences. We prove that a family of hyperbolic alternating link complements is expanding if and only if they have bounded volume. We also provide examples of hyperbolic 3-manifolds which require ‘complicated’ surgery diagrams, thereby proving that a recent theorem of Constantino and Thurston is sharp. Along the way, we find a new upper bound on the bridge number of a knot that is not tangle composite, in terms of the twist number of any diagram of the knot. The proofs rely on a theorem of Lipton and Tarjan on planar graphs, and also the relationship between many different notions of width for knots and 3-manifolds.     

Spectral operators of matrices

Mathematical Programming - The class of matrix optimization problems (MOPs) has been recognized in recent years to be a powerful tool to model many important applications involving structured low...     

The geometry of hypercomplex matrices

Hypercomplex matrices realize a matrix representation for quaternions and octonions. We establish a number of interesting properties of hypercomplex matrices.     

Rank reduction,normal forms and stokes matrices

We introduce the reduction of the rank of a linear differential system as the unique solution of a natural problem and we relate it to the formal meromorphic classification (normal forms) and to the convergent meromorphic classification (Stokes matrices). The reduction of the rank makes sense for connections as well.     

Spectral norm of random matrices

In this paper, we present a new upper bound for the spectral norm of symmetric random matrices with independent (but not necessarily identical) entries. Our results improve an earlier result of Füredi and Komlós. Research supported by an NSF CAREER award and by an Alfred P. Sloan fellowship.     

Spectral geometry for Riemannian foliations

Let F be a Riemannian foliation on a Riemannian manifold (M, g), with bundle-like metric g. Aside from the Laplacian △_g associated to the metric g, there is another differential operator, the Jacobi operator J▽, which is a second order elliptic operator acting on sections of the normal bundle. Its spectrum is discrete as a consequence of the compactness of M. Hence one has two spectra, spec (M, g) = spectrum of △_g (acting on functions), and spec (F, J▽) = spectrum of J▽. We discuss the following problem: Which geometric properties of a Riemannian foliation F on a Riemannian manifold (M, g) are determined by the two types of spectral invariants?     

Spectral methods with sparse matrices

Summary Spectral methods employ global polynomials for approximation. Hence they give very accurate approximations for smooth solutions. Unfortunately, for Dirichlet problems the matrices involved are dense and have condition numbers growing asO(N ⁴) for polynomials of degree N in each variable. We propose a new spectral method for the Helmholtz equation with a symmetric and sparse matrix whose condition number grows only asO(N ²). Certain algebraic spectral multigrid methods can be efficiently used for solving the resulting system. Numerical results are presented which show that we have probably found the most effective solver for spectral systems.     

Spectral symmetry in conference matrices

Designs, Codes and Cryptography - A conference matrix of order n is an $$n\times n$$ matrix C with diagonal entries 0 and off-diagonal entries $$\pm 1$$ satisfying $$CC^\top =(n-1)I$$ . If C is...     

Completely normal matrices

Normal matrices in which all submatrices are normal are said to be completely normal. We characterize this class of matrices, determine the possible inertias of a particular completely normal matrix, and show that real matrices in this class are closed under (general) Schur complementation. We provide explicit formulas for the Moore–Penrose inverse of a completely normal matrix of size at least four. A result on irreducible principally normal matrices is derived as well.     

Generalized normal matrices

A matrix A ∈ Mn（C） is called generalized normal provided that there is a positive definite Hermite matrix H such that HAH is normal. In this paper, these matrices are investigated and their canonical form, invariants and relative properties in the sense of congruence are obtained.     

Principally normal matrices

Normal matrices in which all principal submatrices are normal are said to be principally normal. Various characterizations of irreducible matrices in this class of are given. Notably, it is shown that an irreducible matrix is principally normal if and only if it is normal and all of its eigenvalues lie on a line in the complex plane. Such matrices provide a generalization of the Cauchy interlacing theorem.     

Spectral radius and infinity norm of matrices

Let Mn(R) be the linear space of all n×n matrices over the real field R. For any A∈Mn(R), let ρ(A) and ‖A_‖∞ denote the spectral radius and the infinity norm of A, respectively. By introducing a class of transformations φa on Mn(R), we show that, for any A∈Mn(R), ρ(A)<‖A_‖∞ if . If A∈Mn(R) is nonnegative, we prove that ρ(A)<‖A_‖∞ if and only if , and ρ(A)=‖A_‖∞ if and only if the transformation φ‖A_‖∞ preserves the spectral radius and the infinity norm of A. As an application, we investigate a class of linear discrete dynamic systems in the form of X(k+1)=AX(k). The asymptotical stability of the zero solution of the system is established by a simple algebraic method.     

Good distance graphs and the geometry of matrices

Denote by G=(V,∼) a graph which V is the vertex set and ∼ is an adjacency relation on a subset of V×V. In this paper, the good distance graph is defined. Let (V,∼) and (V^′,∼^′) be two good distance graphs, and φ:V→V^′ be a map. The following theorem is proved: φ is a graph isomorphism ⇔φ is a bounded distance preserving surjective map in both directions ⇔φ is a distance k preserving surjective map in both directions (where k<diam(G)/2 is a positive integer), etc. Let D be a division ring with an involution such that both |F∩Z_D|?3 and D is not a field of characteristic 2 with D=F, where and Z_D is the center of D. Let H_n(n?2) be the set of n×n Hermitian matrices over D. It is proved that (H_n,∼) is a good distance graph, where A∼B⇔rank(A-B)=1 for all A,B∈H_n.     

Quadratically normal and congruence-normal matrices

A matrix A ∈ C ^n×n is unitarily quasidiagonalizable if A can be brought by a unitary similarity transformation to a block diagonal form with 1 × 1 and 2 × 2 diagonal blocks. In particular, the square roots of normal matrices, i.e., the so-called quadratically normal matrices are unitarily quasidiagonalizable. A matrix A ∈ C ^n×n is congruence-normal if B = A[`(A)] B = A\overline A is a conventional normal matrix. We show that every congruence-normal matrix A can be brought by a unitary congruence transformation to a block diagonal form with 1 × 1 and 2 × 2 diagonal blocks. Our proof emphasizes andexploitsalikenessbetween theequations X <sup>2</sup> = B and X[`(X)] = B X\overline X = B for a normal matrix B. Bibliography: 13 titles.     

Spectral factorization of non-symmetric polynomial matrices

The topic of the paper is spectral factorization of rectangular and possibly non-full-rank polynomial matrices. To each polynomial matrix we associate a matrix pencil by direct assignment of the coefficients. The associated matrix pencil has its finite generalized eigenvalues equal to the zeros of the polynomial matrix. The matrix dimensions of the pencil we obtain by solving an integer linear programming (ILP) minimization problem. Then by extracting a deflating subspace of the pencil we come to the required spectral factorization. We apply the algorithm to most general-case of inner–outer factorization, regardless continuous or discrete time case, and to finding the greatest common divisor of polynomial matrices.