Normal Matrices Research Articles

Operator ℓp → ℓq norms of random matrices with iid entries

We prove that for every p,q∈[1,∞] and every random matrix X=(Xi,j)i≤m,j≤n with iid centered entries satisfying the α-regularity assumption ‖Xi,j‖2ρ≤α‖Xi,j‖ρ for every ρ≥1, the expectation of the operator norm of X from ℓpn to ℓqm is comparable, up to a constant depending only on α, tom1/qsupt∈Bpn⁡‖∑j=1ntjX1,j‖q∧Logm+n1/p⁎sups∈Bq⁎m⁡‖∑i=1msiXi,1‖p⁎∧Logn. We give more explicit formulas, expressed as exact functions of p, q, m, and n, for the two-sided bounds of the operator norms in the case when the entries Xi,j are: Gaussian, Weibullian, log-concave tailed, and log-convex tailed. In the range 1≤q≤2≤p we provide two-sided bounds under the weaker regularity assumption (EX1,14)1/4≤α(EX1,12)1/2.

GENERALIZATION OF PELL SEQUENCE AND PELL-LUCAS SEQUENCE, NEW RESULTS AND CIRCULANT MATRICES ASPECTS

Some generalizations of Pell sequence and Pell-Lucas sequence, namely, (k,h)-Pell sequence and (k,h)-Pell-Lucas sequence are considered in this paper. We obtain generating functions, some identities and formulas for the sums of a finite number of terms and consecutive terms, sums of squares of consecutive terms and alternating sums of consecutive terms of these sequences. Then, we obtain the eigenvalues and determinants of particular circulant matrices involving (k,h)-Pell sequence and (k,h)-Pell-Lucas sequence. Finally, we obtain some upper and lower bounds for the spectral norms of these circulant matrices.

Interpolating inequalities for unitarily invariant norms of matrices

Interpolating inequalities for unitarily invariant norms of matrices

Polynomial Optimization Over Unions of Sets

AbstractThis paper studies the polynomial optimization problem whose feasible set is a union of several basic closed semialgebraic sets. We propose a unified hierarchy of Moment-SOS relaxations to solve it globally. Under some assumptions, we prove the asymptotic or finite convergence of the unified hierarchy. Special properties for the univariate case are discussed. The application for computing (p, q)-norms of matrices is also presented.

Generalized Quantum Signal Processing

Quantum signal processing (QSP) and quantum singular value transformation (QSVT) currently stand as the most efficient techniques for implementing functions of block-encoded matrices, a central task that lies at the heart of most prominent quantum algorithms. However, current QSP approaches face several challenges, such as the restrictions imposed on the family of achievable polynomials and the difficulty of calculating the required phase angles for specific transformations. In this paper, we present a generalized quantum signal processing (GQSP) approach, employing general SU(2) rotations as our signal-processing operators, rather than relying solely on rotations in a single basis. Our approach lifts all practical restrictions on the family of achievable transformations, with the sole remaining condition being that |P|≤1, a restriction necessary due to the unitary nature of quantum computation. Furthermore, GQSP provides a straightforward recursive formula for determining the rotation angles needed to construct the polynomials in cases where P and Q are known. In cases where only P is known, we provide an efficient optimization algorithm capable of identifying in under a minute of GPU time, a corresponding Q for polynomials of degree on the order of 107. We further illustrate GQSP simplifies QSP-based strategies for Hamiltonian simulation, offer an optimal solution to the ϵ-approximate fractional query problem that requires O((1/δ)+log(1/ϵ)) queries to perform where O(1/δ) is a proved lower bound, and introduces novel approaches for implementing bosonic operators. Moreover, we propose a novel framework for the implementation of normal matrices, demonstrating its applicability through synthesis of diagonal matrices, as well as the development of a new algorithm for convolution through synthesis of circulant matrices using only O(dlogN+log2N) 1 and 2-qubit gates for a filter of lengths d. Published by the American Physical Society 2024

Finding eigenvectors with a quantum variational algorithm

This paper presents a hybrid variational quantum algorithm that finds a random eigenvector of a unitary matrix with a known quantum circuit. The algorithm is based on the SWAP test on trial states generated by a parametrized quantum circuit. The eigenvector is described by a compact set of classical parameters that can be used to reproduce the found approximation to the eigenstate on demand. This variational eigenvector finder can be adapted to solve the generalized eigenvalue problem, to find the eigenvectors of normal matrices and to perform quantum principal component analysis on unknown input mixed states. These algorithms can all be run with low-depth quantum circuits, suitable for an efficient implementation on noisy intermediate-scale quantum computers and, with some restrictions, on linear optical systems. In full-scale quantum computers, where there might be optimization problems due to barren plateaus in larger systems, the proposed algorithms can be used as a primitive to boost known quantum algorithms. Limitations and potential applications are discussed.

On linear algebra of r-Hankel and r-Toeplitz matrices with geometric sequence

AbstractMatrix theory plays a crucial role in solving practical problems and performing computational operations. In particular, specific types of matrices and their linear algebraic properties are of paramount significance for these processes. In this paper, we study the properties of r-Hankel and r-Toeplitz matrices whose entries are geometric sequences, and then the determinants, inverse matrix, generalized inverse matrix (the Moore-Penrose inverse), and spectral norms of such matrices are obtained.

A parameter inversion method for the probability integral method based on robust ridge estimation

The probability integral method is one of the most widely used methods for predicting surface subsidence induced by underground mining in China. In its parameter calculation, the least square algorithm is commonly employed for fitting parameters. However, in the process of fitting parameters, the results are easily affected by ill-conditioned normal matrices and the interference of outliers, resulting in divergent problems. To solve these problems, the principle of robust ridge estimation was introduced in this paper, and a parameter calculation model for the probability integral method based on this principle (hereafter referred to as the established model) was established. Besides, a parameter calculation experiment with manual intervention was conducted in combination with engineering examples. The results demonstrate that the parameter calculation method based on robust ridge estimation can suppress the interference of outliers, overcome the problem of ill-conditioned matrix, and ensure the effectiveness and reliability of parameter estimation results. Compared with the conventional least squares method, the robust ridge estimation method demonstrates greater accuracy in predicting surface subsidence parameters, which validates its rationality and accuracy in underground mining engineering. The research findings provide technical support for obtaining similar parameters for surface subsidence in mining areas and hold significant engineering application value.

Procrustes problem for the inverse eigenvalue problem of normal (skew) J-Hamiltonian matrices and normal J-symplectic matrices

ABSTRACT A square complex matrix A is called (skew) J-Hamiltonian if AJ is (skew) Hermitian where J is a real normal matrix such that J 2 = − I , where I is the identity matrix. In this paper, we solve the Procrustes problem to find normal (skew) J-Hamiltonian solutions for the inverse eigenvalue problem. In addition, a similar problem is investigated for normal J-symplectic matrices.

Stability of solutions of systems of Volterra integral equations

In this paper, we propose new sufficient conditions for stability of solutions of systems of Volterra linear integral equations and systems of linear integro-differential Volterra equations. Solution stability conditions for systems of Volterra linear integral equations are studied with perturbed right hand sides of the equations. These new sufficient conditions are expressed in terms of logarithmic norms of matrices consisting of coefficients of equations and derivatives of their kernels. For Volterra integro-differential equations we also provide sufficient conditions for stability of solutions when the initial conditions are perturbed.

A revisit of the normalized eight-point algorithm and a self-supervised deep solution

The normalized eight-point algorithm has been widely viewed as the cornerstone in two-view geometry computation, where the seminal Hartley’s normalization has greatly improved the performance of the direct linear transformation algorithm. A natural question is, whether there are and how to find other normalization methods that can further improve the performance for each input sample. In this paper, we provide a novel perspective and propose two contributions to this fundamental problem: (1) we revisit the normalized eight-point algorithm and make a theoretical contribution by presenting the existence of different and better normalization algorithms; (2) we introduce a deep convolutional neural network with a self-supervised learning strategy for normalization. Given eight pairs of correspondences, our network directly predicts the normalization matrices, thus learning to normalize each input sample. Our learning-based normalization module can be integrated with both traditional (e.g., RANSAC) and deep learning frameworks (affording good interpretability) with minimal effort. Extensive experiments on both synthetic and real images demonstrate the effectiveness of our proposed approach.

Some new characterizations of normal matrices

This paper mainly studies some properties of normal matrix and gives the relation between the general solution of related matrix equations and normal matrices.

Further properties of PPT and (;1,;2)-normal matrices

Further properties of PPT and (;1,;2)-normal matrices

Chevet-type inequalities for subexponential Weibull variables and estimates for norms of random matrices

Chevet-type inequalities for subexponential Weibull variables and estimates for norms of random matrices

Spectral norm and von Neumann type trace inequality for dual quaternion matrices

Spectral norm and von Neumann type trace inequality for dual quaternion matrices

Some matrix inequalities related to norm and singular values

<abstract><p>In this short note, we presented a new proof of a weak log-majorization inequality for normal matrices and obtained a singular value inequality related to positive semi-definite matrices. What's more, we also gave an example to show that some conditions in an existing norm inequality are necessary.</p></abstract>

Berezin number and Berezin norm inequalities for operator matrices

We establish new upper bounds for the Berezin number and Berezin norm of operator matrices, which are refinements of existing bounds. Among other bounds, we prove that if is an operator matrix with for , then and where if i<j and if i>j. We also provide examples which illustrate these bounds for some concrete operators acting on the Hardy-Hilbert space.

Norm and numerical radius inequalities for operator matrices

Operator matrices have played a significant role in studying Hilbert space operators. In this paper, we discuss further properties of operator matrices and present new estimates for the operator norms and numerical radii of such operators. Moreover, operator matrices with positive real and imaginary parts will be discussed, and sharper bounds will be shown for such classes.

Tail bounds on the spectral norm of sub-exponential random matrices

Let [Formula: see text] be an [Formula: see text] symmetric random matrix with independent but non-identically distributed entries. The deviation inequalities of the spectral norm of [Formula: see text] with Gaussian entries have been obtained by using the standard concentration of Gaussian measure results. This paper establishes an upper tail bound of the spectral norm of [Formula: see text] with sub-exponential entries. Our method relies upon a crucial ingredient of a novel chaining argument that essentially involves both the particular structure of the sets used for the chaining and the distribution of coordinates of a point on the unit sphere.

Exponential moments for disk counting statistics at the hard edge of random normal matrices

We consider the multivariate moment generating function of the disk counting statistics of a model Mittag-Leffler ensemble in the presence of a hard wall. Let n be the number of points. We focus on two regimes: (a) the “hard edge regime” where all disk boundaries are at a distance of order \frac{1}{n} from the hard wall, and (b) the “semi-hard edge regime” where all disk boundaries are at a distance of order \frac{1}{\sqrt{n}} from the hard wall. As n \to + \infty , we prove that the moment generating function enjoys asymptotics of the form \exp (C_{1}n + C_{2}\ln n + C_{3} + \frac{C_{4}}{\sqrt{n}} + \mathcal{O}(n^{-\frac{3}{5}})) for the hard edge, \exp (C_{1}n + C_{2}\sqrt{n} + C_{3} + \frac{C_{4}}{\sqrt{n}} + \mathcal{O}(\frac{(\ln n)^{4}}{n})) for the semi-hard edge. In both cases, we determine the constants C_{1},\dots,C_{4} explicitly. We also derive precise asymptotic formulas for all joint cumulants of the disk counting function, and establish several central limit theorems. Surprisingly, and in contrast to the “bulk”, “soft edge”, and “semi-hard edge” regimes, the second and higher order cumulants of the disk counting function in the “hard edge” regime are proportional to n and not to \sqrt{n} .