Diagonal Entries Research Articles

Invertibility properties of operator matrices on Hilbert spaces

Denote by $$T_n^d(A)$$ an upper triangular operator matrix of dimension n whose diagonal entries are given and the others are unknown. In this article, we provide necessary and sufficient conditions for various types of Fredholm and Weyl completions of $$T_n^d(A)$$ . As consequences, we recover many known results of Zhang and Wu (J Math Anal Appl 392(2):103–110, 2012) for the case $$n=2$$ already existing in the literature, as well as some perturbation results of Wu and Huang (Acta Math Sin (Engl Ser) 36(7):783–796, 2020; Ann Funct Anal 11(3):780–798, 2020) given for the case of arbitrary $$n\ge 2$$ .

Matrix-valued nonstationary frames associated with the Weyl–Heisenberg group and the extended affine group

We study nonstationary frames of matrix-valued Gabor systems and wavelet systems in the matrix-valued function space [Formula: see text]. First, we show that a diagonal matrix-valued window function constitutes a frame for [Formula: see text] whenever each diagonal entry constitutes a frame for the space [Formula: see text]. This is not true for arbitrary nonzero matrix-valued function. Using this, we prove the existence of nonstationary matrix-valued Gabor frames associated with the Weyl–Heisenberg group in terms of density of real numbers. We give a representation of the frame operator of matrix-valued nonstationary Gabor system. A necessary condition with explicit frame bounds for nonstationary matrix-valued Gabor frames associated with the Weyl–Heisenberg group is given. We discuss matrix-valued frame preserving maps in terms of adjointablity, with respect to the matrix-valued inner product, of bounded linear operators acting on [Formula: see text]. It is shown that the image of a matrix-valued Gabor frame under bounded, linear and invertible operator on [Formula: see text] may not be a frame for [Formula: see text]. In this direction, we give sufficient conditions on bounded linear operators which can preserve frame conditions. Finally, we give necessary and sufficient condition for the existence of nonstationary matrix-valued wavelet frames associated with the extended affine group.

Estimates for the Weyl coefficient of a two-dimensional canonical system

For a two-dimensional canonical system $y'(t)=zJH(t)y(t)$ on some interval $(a,b)$ whose Hamiltonian $H$ is a.e. positive semi-definite and which is regular at $a$ and in the limit point case at $b$, denote by $q_H$ its Weyl coefficient. De~Branges' inverse spectral theorem states that the assignment $H\mapsto q_H$ is a bijection between Hamiltonians (suitably normalised) and Nevanlinna functions. We give upper and lower bounds for $|q_H(z)|$ and $\Im q_H(z)$ when $z$ tends to $i\infty$ non-tangentially. These bounds depend on the Hamiltonian $H$ near the left endpoint $a$ and determine $|q_H(z)|$ up to universal multiplicative constants. We obtain that the growth of $|q_H(z)|$ is independent of the off-diagonal entries of $H$ and depends monotonically on the diagonal entries in a natural way. The imaginary part is, in general, not fully determined by our bounds (in forthcoming work we shall prove that for most Hamiltonians also $\Im q_H(z)$ is fully determined). We translate the asymptotic behaviour of $q_H$ to the behaviour of the spectral measure $\mu_H$ of $H$ by means of Abelian--Tauberian results and obtain conditions for membership of growth classes defined by weighted integrability condition (Kac classes) or by boundedness of tails at $\pm\infty$ w.r.t. a weight function. Moreover, we apply our results to Krein strings and Sturm--Liouville equations.

Neighbors Degree Sum Energy of Commuting and Non-Commuting Graphs for Dihedral Groups

The neighbors degree sum (NDS) energy of a graph is determined by the sum of its absolute eigenvalues from its corresponding neighbors degree sum matrix. The non-diagonal entries of NDS−matrix are the summation of the degree of two adjacent vertices, or it is zero for non-adjacent vertices, whereas for the diagonal entries are the negative of the square of vertex degree. This study presents the formulas of neighbors degree sum energies of commuting and non-commuting graphs for dihedral groups of order 2n, D2n, for two cases−odd and even n. The results in this paper comply with the well known fact that energy of a graph is neither an odd integer nor a square root of an odd integer.

Nearly Optimal Block-Jacobi Preconditioning

The goal of Jacobi preconditioning of a symmetric positive definite matrix by a diagonal matrix is to choose to minimize the condition number . In 1969, van der Sluis proved that choosing so that the diagonal entries of are all ones reduces to within a factor of the minimum possible, where the factor depends on both the dimension and the norms used to define the condition number. We extend this result in two ways to block-Jacobi preconditioning, where is a block-diagonal matrix with blocks of given sizes, and we consider instead of to maintain the symmetric positive definite (spd) property. First, we extend van der Sluis’s original bound to include block-Jacobi. Second, we define a new norm in which choosing so that the corresponding diagonal blocks of are identity matrices minimizes the condition number. We use this to show that the condition number in the 2-norm of this optimally scaled is at least as large as the condition number in the new norm, and at most a factor larger, where is the number of diagonal blocks. We give an example where the optimal 2-norm condition number nearly attains this new upper bound, which for this example is tighter than van der Sluis’s bound by a factor equal to the matrix dimension. Finally, all these results generalize to the case of one-sided scaling of a full row-rank matrix .

Fluctuations of the diagonal entries of a large sample precision matrix

For a p×n data matrix Xn with i.i.d. centered entries and a population covariance matrix Σ, the corresponding sample precision matrix Σˆ−1 is defined as the inverse of the sample covariance matrix Σˆ=(1/n)Σ1/2XnXn⊤Σ1/2. We determine the joint distribution of a vector of diagonal entries of the matrix Σˆ−1 in the situation, where pn=p<n, p/n→y∈[0,1) for n→∞ and Σ is a diagonal matrix.

Sums of squares II: Matrix functions

This paper is the second in a series of three papers devoted to sums of squares and hypoellipticity of infinitely degenerate operators. In the first paper we established a sharp ω-monotonicity criterion for writing a smooth nonnegative function f that is flat at, and positive away from, the origin, as a finite sum of squares of C2,δ functions for some δ>0, namely that f is ω-monotone for some Hölder modulus of continuity ω. Counterexamples were provided for any larger modulus of continuity.In this paper we consider the analogous sum of squares problem for smooth nonnegative matrix functions M that are flat at, and positive away from, the origin. We show that such a matrix function M=[akj]k,j=1n can be written as a finite sum of squares of C2,δ vector fields if the diagonal entries akk are ω-monotone for some Hölder modulus of continuity ω, and if the off diagonal entries satisfy certain differential bounds in terms of powers of the diagonal entries. Examples are given to show that in some cases at least, these differential inequalities cannot be relaxed.Various refinements of these results are also given in which one or more of the diagonal entries need not be assumed to have any monotonicity properties at all. These sum of squares decompositions will be applied to hypoellipticity in the infinitely degenerate regime in the third paper in this series.

Quantitative inductive estimates for Green’s functions of non-self-adjoint matrices

We provide quantitative inductive estimates for Green's functions of matrices with (sub)expoentially decaying off diagonal entries in higher dimensions. Together with Cartan's estimates and discrepancy estimates, we establish explicit bounds for the large deviation theorem for non-self-adjoint Toeplitz operators. As applications, we obtain the modulus of continuity of the integrated density of states with explicit bounds and the pure point spectrum property for analytic quasi-periodic operators. Moreover, our inductions are self-improved and work for perturbations with low complexity interactions.

Derivative principles for invariant ensembles

In the present work we show that the joint probability distribution of the eigenvalues can be expressed in terms of a differential operator acting on the distribution of some other matrix quantities. Those quantities might be the diagonal or pseudo-diagonal entries as it is the case for Hermitian matrices. These representations are called derivative principles. We show them for the additive spaces of the Hermitian, Hermitian antisymmetric, Hermitian anti-self-dual, and complex rectangular matrices as well as for the two multiplicative matrix spaces of the positive definite Hermitian matrices and of the unitary matrices. In all six cases we prove the uniqueness of the derivative principles.

Large deviation principles via spherical integrals

In this article, we develop a framework to study the large deviation principle for matrix models and their quantized versions, by tilting the measures using the limits of spherical integrals obtained in [46,47]. As examples, we obtain 1. a large deviation principle for the empirical distribution of the diagonal entries of $UB_NU^*$, for a sequence of $N\times N$ diagonal matrices $B_N$ and unitary Haar distributed matrices $U$; 2. a large deviation upper bound for the empirical eigenvalue distribution of $A_N+UB_NU^*$, for two sequences of $N\times N$ diagonal matrices $A_N, B_N$, and their complementary lower bounds at measures which are described by the free product with amalgamation; 3. a large deviation principle for the Kostka number $K_{\boldsymbol\lambda_N \boldsymbol\eta_N}$, for two sequences of partitions $\boldsymbol\lambda_N, \boldsymbol\eta_N$ with at most $N$ rows; 4. a large deviation upper bound for the Littlewood-Richardson coefficients $c_{\boldsymbol\lambda_N \boldsymbol \eta_N}^{\boldsymbol \kappa_N}$, for three sequences of partitions $\boldsymbol\lambda_N, \boldsymbol \eta_N, \boldsymbol \kappa_N$ with at most $N$ rows, and their complementary lower bounds at nice measures.

Locally R-optimal designs for a class of nonlinear multiple regression models

This paper concerns with optimal designs for a wide class of nonlinear models with information driven by the linear predictor. The aim of this study is to generate an R-optimal design which minimizes the product of the main diagonal entries of the inverse of the Fisher information matrix at certain values of the parameters. An equivalence theorem for the locally R-optimal designs is provided in terms of the intensity function. Analytic solutions for the locally saturated R-optimal designs are derived for the models having linear predictors with and without intercept, respectively. The particle swarm optimization method has been employed to generate locally non-saturated R-optimal designs. Numerical examples are presented for illustration of the locally R-optimal designs for Poisson regression models and proportional hazards regression models.

Minimum upper bound of the Cayley transform of an orthogonal matrix multiplied by signed permutation matrices

O'Dorney has proved that for any orthogonal matrix U, there exists a signature matrix D (a diagonal matrix with diagonal entries ±1) by which |sij|≤1, where S=(sij)=Q(UD) is the Cayley transform of UD. In this paper, we prove that the upper bound can be reduced to 2−1 by multiplying by a certain signed permutation matrix, instead of a signature matrix.

Further inequalities for the 𝔸-numerical radius of certain 2 × 2 operator matrices

Abstract Let 𝔸 = ( A O O A ) \mathbb{A}={\bigl{(}\begin{smallmatrix}A&amp;O\\ O&amp;A\\ \end{smallmatrix}\bigr{)}} be a 2 × 2 2\times 2 diagonal operator matrix whose each diagonal entry is a bounded positive (semi-definite) linear operator A acting on a complex Hilbert space ℋ \mathcal{H} . In this paper, we derive several 𝔸 \mathbb{A} -numerical radius inequalities for 2 × 2 2\times 2 operator matrices whose entries are bounded with respect to the seminorm induced by the positive operator A on ℋ \mathcal{H} . Some applications of our inequalities are also given.

An ensemble of high rank matrices arising from tournaments

Suppose F is a field and let a≔(a1,a2,…) be a sequence of non-zero elements in F. For an≔(a1,…,an), we consider the family Mn(a) of n×n symmetric matrices M over F with all diagonal entries zero and the (i,j)th element of M either ai or aj for i<j. In this short paper, we show that all matrices in a certain subclass of Mn(a)—which can be naturally associated with transitive tournaments—have rank at least ⌊2n/3⌋−1. We also show that if char(F)≠2 and M is a matrix chosen uniformly at random from Mn(a), then with high probability rank(M)≥(12−o(1))n.

Left-Invertibility of Rank-One Perturbations

For each isometry V acting on some Hilbert space and a pair of vectors f and g in the same Hilbert space, we associate a nonnegative number c(V; f, g) defined by $$\begin{aligned} c(V; f,g) = (\Vert f\Vert ^2 - \Vert V^*f\Vert ^2) \Vert g\Vert ^2 + |1 + \langle V^*f , g\rangle |^2. \end{aligned}$$We prove that the rank-one perturbation \(V + f \otimes g\) is left-invertible if and only if $$\begin{aligned} c(V;f,g) \ne 0. \end{aligned}$$We also consider examples of rank-one perturbations of isometries that are shift on some Hilbert space of analytic functions. Here, shift refers to the operator of multiplication by the coordinate function z. Finally, we examine \(D + f \otimes g\), where D is a diagonal operator with nonzero diagonal entries and f and g are vectors with nonzero Fourier coefficients. We prove that \(D + f\otimes g\) is left-invertible if and only if \(D+f\otimes g\) is invertible.

Recovering the characteristic polynomial of a graph from entries of the adjugate matrix

The adjugate matrix of $G$, denoted by $\operatorname{adj}(G)$, is the adjugate of the matrix $x\mathbf{I}-\mathbf{A}$, where $\mathbf{A}$ is the adjacency matrix of $G$. The polynomial reconstruction problem (PRP) asks if the characteristic polynomial of a graph $G$ can always be recovered from the multiset $\operatorname{\mathcal{PD}}(G)$ containing the $n$ characteristic polynomials of the vertex-deleted subgraphs of $G$. Noting that the $n$ diagonal entries of $\operatorname{adj}(G)$ are precisely the elements of $\operatorname{\mathcal{PD}}(G)$, we investigate variants of the PRP in which multisets containing entries from $\operatorname{adj}(G)$ successfully reconstruct the characteristic polynomial of $G$. Furthermore, we interpret the entries off the diagonal of $\operatorname{adj}(G)$ in terms of characteristic polynomials of graphs, allowing us to solve versions of the PRP that utilize alternative multisets to $\operatorname{\mathcal{PD}}(G)$ containing polynomials related to characteristic polynomials of graphs, rather than entries from $\operatorname{adj}(G)$.

Inverse eigenvalue and related problems for hollow matrices described by graphs

A hollow matrix described by a graph $G$ is a real symmetric matrix having all diagonal entries equal to zero and with the off-diagonal entries governed by the adjacencies in $G$. For a given graph $G$, the determination of all possible spectra of matrices associated with $G$ is the hollow inverse eigenvalue problem for $G$. Solutions to the hollow inverse eigenvalue problems for paths and complete bipartite graphs are presented. Results for related subproblems such as possible ordered multiplicity lists, maximum multiplicity of an eigenvalue, and minimum number of distinct eigenvalues are presented for additional families of graphs.

Perturbing the spectrum of operator $$T_n^d(A)$$

Let $$T_n^d(A)$$ denote a partial upper triangular operator matrix whose diagonal entries are given and the others unknown. In this article we have aim to find characterizations of (left, right) invertibility of $$T_n^d(A)$$ in terms of diagonal entries solely, and hence we provide statements which generalize and correct results of Zhang and Wu (J. Math. Anal. Appl. 392:103–110, 2012). We pose our results without invoking separability condition, thus improving results of Zhang and Wu (J. Math. Anal. Appl. 392:103–110, 2012), and we give appropriate n-dimensional analogues, without assuming separability as well. We recover many perturbation results of Djordjević (J. Oper. Theory 48:467–486, 2002), and obtain some results of Du and Pan (Proc. Amer. Math. Soc. 121: 761–766, 1994) and Han et al. (Proc. Amer. Math. Soc. 128:119–23, 2000) in the case of the Hilbert space setting.

Generalized Drazin Invertibility of Operator Matrices

In this paper, we investigate the generalized Drazin invertibility of upper triangular operator matrices acting on Banach spaces. Among other things, we explicit the defect set with respect to the local spectral theory. Moreover, we exhibit some sufficient conditions which assure that the generalized Drazin spectrum of a 3 × 3 upper triangular block operator matrix is the union of its diagonal entries spectra.

Cluster-robust estimators for multivariate mixed-effects meta-regression

Meta-analyses frequently include trials that report multiple outcomes based on a common set of study participants. These outcomes will generally be correlated. Cluster-robust variance-covariance estimators are a fruitful approach for synthesizing dependent outcomes. However, when the number of studies is small, state-of-the-art robust estimators can yield inflated Type 1 errors. Therefore, two new cluster-robust estimators are presented, in order to improve small sample performance. For both new estimators the idea is to transform the estimated variances of the residuals using only the diagonal entries of the hat matrix. The proposals are asymptotically equivalent to previously suggested cluster-robust estimators such as the bias reduced linearization approach. The methods are applied to a dataset of 81 trials examining overall and disease-free survival in neuroblastoma patients with amplified versus normal MYC-N genes. Furthermore, their performance is compared and contrasted in an extensive simulation study. The focus is on bivariate meta-regression, although the approaches can be applied more generally.