相关论文: On the decay of the off-diagonal singular values i…
Cyclic block coordinate methods are a fundamental class of optimization methods widely used in practice and implemented as part of standard software packages for statistical learning. Nevertheless, their convergence is generally not well…
We study exponential decay of eigenfunctions of self-adjoint higher order elliptic operators on $\R^d$. We show that the possible critical decay rates are determined algebraically. In addition we show absence of super-exponentially decaying…
We present the explicit inverse of a class of symmetric tridiagonal matrices which is almost Toeplitz, except that the first and last diagonal elements are different from the rest. This class of tridiagonal matrices are of special interest…
In this paper we study the eigenvalues of Hermitian Toeplitz matrices with the entries $2,-1,0,\ldots,0,-\alpha$ in the first column. Notice that the generating symbol depends on the order $n$ of the matrix. If $|\alpha|\le 1$, then the…
The main focus of this work is the study of several cones relating the eigenvalues or singular values of a matrix to those of its off-diagonal blocks.
The problem of diagonalizing a class of complicated matrices, to be called ultrametric matrices, is investigated. These matrices appear at various stages in the description of disordered systems with many equilibrium phases by the technique…
In the time evolution of isolated quantum systems out of equilibrium, local observables generally relax to a long-time asymptotic value, governed by the expectation values (diagonal matrix elements) of the corresponding operator in the…
$k$-diagonal circulant matrices and cyclic banded matrices are widely used in numerical simulations and signal processing of circular linear systems. Algorithms that directly involve or specify linear or quadratic complexity for the…
A numerical technique used to solve boundary value problems is modified to find periodic steady-state solutions of nonautonomous dynamical systems. The technique uses a matrix representation of the time derivative obtained through…
This paper is concerned with decay estimate of solutions to the semilinear wave equation with strong damping in a bounded domain. Introducing an appropriate Lyaponuv function, we prove that when the damping is linear, we can find initial…
We derive sharp approximation error bounds for inverse block Toeplitz matrices associated with multivariate long-memory stationary processes. The error bounds are evaluated for both column and row sums. These results are used to prove the…
Multiple solutions are common in various non-convex problems arising from industrial and scientific computing. Nonetheless, understanding the nontrivial solutions' qualitative properties seems limited, partially due to the lack of efficient…
Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. This paper proposes adaptations of these tools focusing on solutions of the real analogue:…
Conventional ways to solve optimization problems on low-rank matrix sets which appear in great number of applications ignore its underlying structure of an algebraic variety and existence of singular points. This leads to appearance of…
Given $n$ pairwise openly disjoint triangles in 3-space, their vertical depth relation may contain cycles. We show that, for any $\varepsilon>0$, the triangles can be cut into $O(n^{3/2+\varepsilon})$ connected semi-algebraic pieces, whose…
The Eberlein diagonalization method is an iterative Jacobi-type method for solving the eigenvalue problem of a general complex matrix. In this paper we develop the block version of the Eberlein method. We prove the global convergence of our…
Almost block diagonal linear systems of equations can be exemplified by two modules. This makes it possible to construct all sequential forms of band and/or block elimination methods, six old and fourteen new. It allows easy assessment of…
Diagonalization, or eigenvalue decomposition, is very useful in many areas of applied mathematics, including signal processing and quantum physics. Matrix decomposition is also a useful tool for approximating matrices as the product of a…
Singular value decompositions of matrices are widely used in numerical linear algebra with many applications. In this paper, we extend the notion of singular value decompositions to finite complexes of real vector spaces. We provide two…
We show that nonlocal seminorms are strictly decreasing under the continuous Steiner rearrangement. This implies that all solutions to nonlocal equations which arise as critical points of nonlocal energies are radially symmetric and…