Invertibility Of Matrices Research Articles

The Invertibility of U-Fusion Cross Gram Matrices of Operators

Finding matrix representations is an important part of operator theory. Calculating such a discretization scheme is equally important for the numerical solution of operator equations. Traditionally in both fields, this was done using bases. Recently, frames have been used here. In this paper, we apply fusion frames for this task, a generalization motivated by a block representation, respectively, a domain decomposition. We interpret the operator representation using fusion frames as a generalization of fusion Gram matrices. We present the basic definition of U-fusion cross Gram matrices of operators for a bounded operator U. We give necessary and sufficient conditions for their (pseudo-)invertibility and present explicit formulas for the (pseudo-)inverse. More precisely, our attention is on how to represent the inverse and pseudo-inverse of such matrices as U-fusion cross Gram matrices. In particular, we characterize fusion Riesz bases and fusion orthonormal bases by such matrices. Finally, we look at which perturbations of fusion Bessel sequences preserve the invertibility of the fusion Gram matrix of operators.

Spectra of Deza graphs

A Deza graph with parameters is a k-regular graph with n vertices such that any two of its vertices have b or a common neighbours, where . In this paper we investigate spectra of Deza graphs. In particular, using the eigenvalues of a Deza graph we determine the eigenvalues of its children. Divisible design graphs are significant cases of Deza graphs. Sufficient conditions for Deza graphs to be divisible design graphs are given, a few families of divisible design graphs are presented and their properties are studied. Our special attention goes to the invertibility of the adjacency matrices of Deza graphs.

Положительная обратимость матриц и экспоненциальная устойчивость импульсных систем линейных дифференциальных уравнений Ито с ограниченными запаздываниями

Basing on the theory of positively invertible matrices, we study certain questions of the exponential 2p-stability $(1 \le p < \infty )$ of systems of Ito linear differential equations with bounded delays and impulse actions on certain solution components. We apply the ideas and methods developed by N.V. Azbelev and his followers for studying the stability of deterministic functional differential equations. For the systems of equations mentioned above, we establish sufficient conditions for the exponential 2p-stability ( $1 \le p < \infty$ ) stated in terms of the positive invertibility of matrices constructed from parameters of these systems. We verify the feasibility of these conditions for certain specific systems of equations.

On then-strong Drazin invertibility in rings

Let R be a ring and n be a positive integer. In this paper, further results on the n-strong Drazin inverse are obtained in a ring. We prove that a ∈ R is n-strongly Drazin invertible if and only if a-an+1 is nilpotent. In terms of this characterization, the extensions of Cline's formula and Jacobson's lemma for this inverse are proved. Moreover, the n-strong Drazin invertibility for the sums of two elements is considered. We prove that a,b ∈ R are n-strongly Drazin invertible if and only if a + b is n-strongly Drazin invertible, under the condition ab = 0. As applications for the additive results, we obtain some equivalent conditions of the n-strong Drazin invertibility of matrices over a ring.

Core invertibility of triangular matrices over a ring

We obtained several equivalent conditions for the existence of core inverses and dual core inverses of triangular matrices over a ring with involution. As applications, some necessary and sufficient conditions for the (2,2,0) core inverse problem are given.

Invertibility of 2 × 2 operator matrices

AbstractProperties of right invertible row operators, i.e., of 1 × 2 surjective operator matrices are studied. This investigation is based on a specific space decomposition. Using this decomposition, we characterize the invertibility of a 2 × 2 operator matrix. As an application, the invertibility of Hamiltonian operator matrices is investigated.

U-cross Gram matrices and their invertibility

The Gram matrix is defined for Bessel sequences by combining synthesis with subsequent analysis operators. If different sequences are used and an operator U is inserted we reach so called U-cross Gram matrices. This can be seen as reinterpretation of the matrix representation of operators using frames. In this paper we investigate some necessary or sufficient conditions for Schatten p-class properties and the invertibility of U-cross Gram matrices. In particular, we show that under mild conditions the pseudo-inverse of a U-cross Gram matrix can always be represented as a U-cross Gram matrix with dual frames of the given ones. We link some properties of U-cross Gram matrices to approximate duals. Finally, we state several stability results. More precisely, it is shown that the invertibility of U-cross Gram matrices is preserved under small perturbations.

Explicit inverse of nonsingular Jacobi matrices

We present here the necessary and sufficient conditions for the invertibility of tridiagonal matrices, commonly named Jacobi matrices, and explicitly compute their inverse. The techniques we use are related with the solution of Sturm–Liouville boundary value problems associated to second order linear difference equations. These boundary value problems can be expressed throughout a discrete Schrödinger operator and their solutions can be computed using recent advances in the study of linear difference equations. The conditions that ensure the uniqueness solution of the boundary value problem lead us to the invertibility conditions for the matrix, whereas the solutions of the boundary value problems provide the entries of the inverse matrix.

Some comments on tridiagonal (p,r)-Toeplitz matrices

The aim of this note is to extend the notion of a tridiagonal (p,r)-Toeplitz matrix recently introduced and to provide procedures for short proofs to results on the invertibility of such matrices.

Enhanced feature fusion through irrelevant redundancy elimination in intra-class and extra-class discriminative correlation analysis

Feature fusion aims to provide enhancements of data authenticity in both traditional and deep learning pattern analysis. Canonical correlation analysis (CCA) based feature fusion is a main technique for exploring the mutual relationships of multiple feature sets. In traditional CCA-based feature fusion, the dimensionality of each feature set is usually first reduced using principal component analysis (PCA), linear discriminant analysis (LDA) etc. to ensure non-singularity and invertibility of covariance matrices. One issue with the above standard CCA-based feature fusion is that the reduced feature sets generated by PCA or LDA may neglect certain correlation information among different feature sets which is useful for CCA, and this in turn may degrade the following classification performance. Another issue is that most CCA fused features may still contain redundancies due to the correlation criterion. These redundancies may be relevant or irrelevant to class labels. The irrelevant redundancies may degrade the pattern recognition performance, while the relevant redundancies can make the pattern recognition system more robust. In this paper, we propose an enhanced feature fusion scheme through irrelevant redundancy elimination in intra-class and extra-class discriminative correlation analysis (IEDCA-IRE) addressing the above two issues. IEDCA-IRE explores the intra-class correlation including both the pairs-wise correlation like CCA-based feature fusion approaches and the correlation across different features within the same class. By incorporating kernelized IEDCA into minimum redundancy maximum relevance (mRMR) criterion, only the relevant redundancy is retained in the fused feature. Our proposed IEDCA-IRE can be used in unimodal feature fusion, multimodal feature fusion, fusion of deep features extracted from different deep neural network models as well as fusion of deep features and handcrafted features. Extensive experiments have proved its effectiveness.

Left and Right Generalized Drazin Invertibility of an Upper Triangular Operator Matrices with Application to Boundary Value Problems

When A ∈ B(H) and B ∈ B(K) are given, we denote by MC the operator on the Hilbert space H ⊕ K of the form MC = (A C 0 B). In this paper we investigate the quasi-nilpotent part and the analytical core for the upper triangular operator matrix MC in terms of those of A and B. We give some necessary and sufficient conditions for MC to be left or right generalized Drazin invertible operator for some C ∈B(K,H). As an application, we study the existence and uniqueness of the solution for abstract boundary value problems described by upper triangular operator matrices with right generalized Drazin invertible component.

Consistency in invertibility and Fredholmness of operator matrices

In this paper we completely answer the following question: for given A \in \mathcal B(\mathcal H,\mathcal K) , C \in \mathcal B(\mathcal H,\mathcal K) does there exist operators T \in \mathcal B(\mathcal H,\mathcal L) and S \in \mathcal B(\mathcal K) such that the operator \left[ {\begin{array}{cc} A &amp; C \\ T &amp; S \\ \end{array} } \right] is consistent in invertibility and Fredholmness?

Coverings of random ellipsoids, and invertibility of matrices with i.i.d. heavy-tailed entries

Let A = (aij) be an n × n random matrix with i.i.d. entries such that Ea11 = 0 and Ea 11 2 = 1. We prove that for any δ > 0 there is L > 0 depending only on δ, and a subset N of B 2 of cardinality at most exp(δn) such that with probability very close to one we have $$A\left( {B_2^n} \right)\subset\mathop \cup \limits_{y \in A\left( \mathcal{N} \right)} \left( {y + L\sqrt n B_2^n} \right)$$ . In fact, a stronger statement holds true. As an application, we show that for some L' > 0 and u ∈ [0, 1) depending only on the distribution law of a11, the smallest singular value sn of the matrix A satisfies $$\mathbb{P}\left\{ {{s_n}\left( A \right) \leq \varepsilon {n^{ - 1/2}}} \right\} \leq L'\varepsilon + {u^n}$$ for all e > 0. The latter result generalizes a theorem of Rudelson and Vershynin which was proved for random matrices with subgaussian entries.

A lattice-theoretic approach to the Bourque–Ligh conjecture

ABSTRACTThe Bourque–Ligh conjecture states that if is a gcd-closed set of positive integers with distinct elements, then the LCM matrix is invertible. It is known that this conjecture holds for but does not generally hold for . In this paper, we study the invertibility of matrices in a more general matrix class, join matrices. At the same time, we provide a lattice-theoretic explanation for this solution of the Bourque–Ligh conjecture. In fact, let be a lattice, let be a subset of P and let be a function. We study under which conditions the join matrix on S with respect to f is invertible on a meet closed set S (i.e. .

Complex random matrices have no real eigenvalues

Let [Formula: see text] where [Formula: see text] are iid copies of a mean zero, variance one, subgaussian random variable. Let [Formula: see text] be an [Formula: see text] random matrix with entries that are iid copies of [Formula: see text]. We prove that there exists a [Formula: see text] such that the probability that [Formula: see text] has any real eigenvalues is less than [Formula: see text] where [Formula: see text] only depends on the subgaussian moment of [Formula: see text]. The bound is optimal up to the value of the constant [Formula: see text]. The principal component of the proof is an optimal tail bound on the least singular value of matrices of the form [Formula: see text] where [Formula: see text] is a deterministic complex matrix with the condition that [Formula: see text] for some constant [Formula: see text] depending on the subgaussian moment of [Formula: see text]. For this class of random variables, this result improves on the results of Pan–Zhou [Circular law, extreme singular values and potential theory, J. Multivariate Anal. 101(3) (2010) 645–656] and Rudelson–Vershynin [The Littlewood–Offord problem and invertibility of random matrices, Adv. Math. 218(2) (2008) 600–633]. In the proof of the tail bound, we develop an optimal small-ball probability bound for complex random variables that generalizes the Littlewood–Offord theory developed by Tao–Vu [From the Littlewood–Offord problem to the circular law: Universality of the spectral distribution of random matrices, Bull. Amer. Math. Soc.[Formula: see text]N.S.[Formula: see text] 46(3) (2009) 377–396; Inverse Littlewood–Offord theorems and the condition number of random discrete matrices, Ann. of Math.[Formula: see text] 169(2) (2009) 595–632] and Rudelson–Vershynin [The Littlewood–Offord problem and invertibility of random matrices, Adv. Math. 218(2) (2008) 600–633; Smallest singular value of a random rectangular matrix, Comm. Pure Appl. Math. 62(12) (2009) 1707–1739].

Invertibility of some circulant matrices

We consider several cases of circulant matrices and determine the invertibility of these matrices. We give a criterion of the invertibility of matrices of the form circ(a, b, c, …, c) and circ(a, b, c, …, c, a) where a; b and c are real numbers. By using a different approach, we are able to generalize the result of Carmona.

A proof of the Farkas–Minkowski theorem by a tandem method

AbstractThis note presents a proof of the Farkas–Minkowski theorem. Our proof does not presuppose the closedness of a finitely generated cone, nor employs separation theorems either. Even the concept of linear independence or invertibility of matrices is not necessary. Our new device consists in proving the Farkas–Minkowski theorem and the closedness of a finitely generated cone at the same time based upon mathematical induction. We make use of a minimization problem with an equality constraint, a method familiar to economics students.

Generalized Drazin invertibility of operator matrices

ABSTRACTLet and , where and are Banach or separable Hilbert spaces, and let be an operator acting on given by . We investigate the set , where is the generalized Drazin spectrum of .

Positive invertibility of matrices and stability of Itô delay differential equations

We study the global exponential p-stability (1 ≤ p < ∞) of systems of Ito nonlinear delay differential equations of a special form using the theory of positively invertible matrices. To this end, we apply a method developed by N.V. Azbelev and his students for the stability analysis of deterministic functional-differential equations. We obtain sufficient conditions for the global exponential 2p-stability (1 ≤ p < ∞) of systems of Ito nonlinear delay differential equations in terms of the positive invertibility of a matrix constructed from the original system. We verify these conditions for specific equations.

Stability and convex hulls of matrix powers

ABSTRACTInvertibility of all convex combinations of A and I is equivalent to the real eigenvalues of A, if any, being positive. Invertibility of all matrices whose rows are convex combinations of the respective rows of A and I is equivalent to A having positive principal minors (i.e. being a P-matrix). These results are extended by considering convex combinations of higher powers of A and of their rows. The invertibility of matrices in these convex hulls is associated with the eigenvalues of A lying in open sectors of the right-half plane and provides a general context for the theory of matrices with P-matrix powers.