The classical Cayley-Hamilton theorem (Gantmacher, 1974; Kaczorek, 1988; Lancaster, 1969) says that every square matrix satisfies its own characteristic equation. Let A ∈ Cn×n (the set of n × n complex matrices) and p(s) = det[Ins − A] = ∑n i=0 a si, (an = 1) be the characteristic polynomial of A. Then p(A) = ∑n i=0 aiA i = 0n (the n × n zero matrix). The Cayley Hamilton theorem was extended to...

As an emerging approach of signal processing, not only has compressed sensing (CS) successfully compressed and sampled signals with few measurements, but also has owned the capabilities of ensuring the exact recovery of signals. However, the above-mentioned properties are based on the (compressed) sensing matrices. Hence the construction of sensing matrices is the key problem. Compared with the...

In this paper, we first review the PU and Uzawa-SAOR relaxation methods with singular or nonsingular preconditioning matrices for solving singular saddle point problems, and then we provide numerical experiments to compare performance results of the relaxation iterative methods using nonsingular preconditioners with those using singular preconditioners. Mathematics Subject Classification: 65F10...

1.1 Singular values and matrix inversion For non-symmetric matrices, the eigenvalues and singular values are not equivalent. However, they share one important property: Fact 1 A matrix A, NxN, is invertible iff all of its singular values are non-zero.

In this paper, we consider the singular values and singular vectors of finite, low rank perturbations of large rectangular random matrices. Specifically, we prove almost sure convergence of the extreme singular values and appropriate projections of the corresponding singular vectors of the perturbed matrix. As in the prequel, where we considered the eigenvalues of Hermitian matrices, the non-ra...

Let $(A)$ be a complex $(ntimes n)$ matrix and assume that the numerical range of $(A)$ lies in the set of a sector of half angle $(alpha)$ denoted by $(S_{alpha})$. We prove the numerical ranges of the conjugate, inverse and Schur complement of any order of $(A)$ are in the same $(S_{alpha})$.The eigenvalues of some kinds of matrix product and numerical ranges of hadmard product, star-congruen...

2 Preliminaries 7 2.1 Matrices and their singular values . . . . . . . . . . . . . . . . . . 7 2.2 Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Sub-gaussian random variables . . . . . . . . . . . . . . . . . . . 9 2.4 Sub-exponential random variables . . . . . . . . . . . . . . . . . . 14 2.5 Isotropic random vectors . . . . . . . . . . . . . . . . . . . . . . ...

In this work, the convection-diffusion integro-differential equation with a weakly singular kernel is discussed. The  Legendre spectral tau method is introduced for finding the unknown function. The proposed method is based on expanding the approximate solution as the elements of a shifted Legendre polynomials. We reduce the problem to a set of algebraic equations by using operational matrices....

This paper proposes a harmonic Lanczos bidiagonalization method for computing some interior singular triplets of large matrices. It is shown that the approximate singular triplets are convergent if a certain Rayleigh quotient matrix is uniformly bounded and the approximate singular values are well separated. Combining with the implicit restarting technique, we develop an implicitly restarted ha...

In this work, we deal with normalizable-balanced singular systems, that is, singular systems where the obtained closed-loop system is balanced by an appropriate feedback. In the invariant case, we use a proportional and derivative feedback to obtain systems without infinite poles. In addition, the existence and construction of state-feedbacks to obtain normalizable-balanced N-periodic singular ...