Displacement structure approach to q-adic polynomial-Vandermonde and related matrices

In the present paper a new class of the so-called q-adic polynomial-Vandermonde-like matrices over an arbitrary non-algebraically closed field is introduced. This class generalizes both the simple and the confluent polynomial-Vandermonde-like matrices over the complex field, and the q-adic Vandermonde and the q-adic Chebyshev-Vandermonde-like matrices studied earlier by different authors. Three kinds of displacement structures and two kinds of fast inversion formulas are obtained for this class of matrices by using displacement structure matrix method, which generalize the corresponding results of the polynomial-Vandermonde-like and the q-adic Vandermonde-like matrices.     

The norm of polynomials in large random and deterministic matrices

Let ${{\bf X}_N =(X_1^{(N)}, \ldots, X_p^{(N)})}$ be a family of N × N independent, normalized random matrices from the Gaussian Unitary Ensemble. We state sufficient conditions on matrices ${{\bf Y}_N =(Y_1^{(N)}, \ldots, Y_q^{(N)})}$ , possibly random but independent of X _N , for which the operator norm of ${P({\bf X}_N, {\bf Y}_N, {\bf Y}_N^*)}$ converges almost surely for all polynomials P. Limits are described by operator norms of objects from free probability theory. Taking advantage of the choice of the matrices Y _N and of the polynomials P, we get for a large class of matrices the ??no eigenvalues outside a neighborhood of the limiting spectrum?? phenomena. We give examples of diagonal matrices Y _N for which the convergence holds. Convergence of the operator norm is shown to hold for block matrices, even with rectangular Gaussian blocks, a situation including non-white Wishart matrices and some matrices encountered in MIMO systems.     

Surjectivity and invertibility properties of totally positive matrices

A new “finite section” type theorem is used to show that the members of an interesting class of bounded totally positive matrices map l_∞ onto l_∞ if and only if their range contains a vector which alternates in sign and has coordinates bounded away from zero. The class of matrices studied contains all banded totally positive matrices, and thus all infinite spline collocation matrices. Connections to related work and extension to matrices which are not sign regular are indicated.     

Monotone positive stable matrices

The class of real matrices which are both monotone (inverse positive) and positive stable is investigated. Such matrices, called N-matrices, have the well-known class of nonsingular M-matrices as a proper subset. Relationships between the classes of N-matrices, M-matrices, nonsingular totally nonnegative matrices, and oscillatory matrices are developed. Conditions are given for some classes of matrices, including tridiagonal and some Toeplitz matrices, to be N-matrices.     

On matrices having equal spectral radius and spectral norm

In this paper we characterize all nxn matrices whose spectral radius equals their spectral norm. We show that for n?3 the class of these matrices contains the normal matrices as a subclass.     

Sparse recovery in probability via <Emphasis Type="Italic">l</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>-minimization with Weibull random matrices for 0 &lt; <Emphasis Type="Italic">q</Emphasis> ≤ 1

Although Gaussian random matrices play an important role of measurement matrices in compressed sensing, one hopes that there exist other random matrices which can also be used to serve as the measurement matrices. Hence, Weibull random matrices induce extensive interest. In this paper, we first propose the l_2,q robust null space property that can weaken the D-RIP, and show that Weibull random matrices satisfy the l_2,q robust null space property with high probability. Besides, we prove that Weibull random matrices also possess the l _q quotient property with high probability. Finally, with the combination of the above mentioned properties, we give two important approximation characteristics of the solutions to the l _q -minimization with Weibull random matrices, one is on the stability estimate when the measurement noise e ∈ ? ⁿ needs a priori ||e||₂ ≤ ?, the other is on the robustness estimate without needing to estimate the bound of ||e||₂. The results indicate that the performance of Weibull random matrices is similar to that of Gaussian random matrices in sparse recovery.     

Regular sets of matrices and applications

SupposeA ₁,...,A _s are (1, - 1) matrices of order m satisfying 1 $$A_i A_j = J, i,j \in \left\{ {1,...s} \right\}$$ 2 $$A_i^T A_j = A_j^T A_i = J, i \ne j, i,j \in \left\{ {1,...,s} \right\}$$ 3 $$\sum\limits_{i = 1}^s {(A_i A_i^T = A_i^T A_i ) = 2smI_m } $$ 4 $$JA_i = A_i J = aJ, i \in \left\{ {1,...,s} \right\}, a constant$$ Call A₁,…,A _s ,a regular s- set of matrices of order m if Eq. 1-3 are satisfied and a regular s-set of regular matrices if Eq. 4 is also satisfied, these matrices were first discovered by J. Seberry and A.L. Whiteman in “New Hadamard matrices and conference matrices obtained via Mathon’s construction”, Graphs and Combinatorics, 4(1988), 355-377. In this paper, we prove that

1. if there exist a regular s-set of order m and a regulart-set of order n there exists a regulars-set of ordermn whent =sm
2. if there exist a regular s-set of order m and a regulart-set of order n there exists a regulars-set of ordermn when 2t = sm (m is odd)
3. if there exist a regulars-set of order m and a regulart-set of ordern there exists a regular 2s-set of ordermn whent = 2sm As applications, we prove that if there exist a regulars-set of order m there exists
4. an Hadamard matrices of order4hm whenever there exists an Hadamard matrix of order4h ands =2h
5. Williamson type matrices of ordernm whenever there exists Williamson type matrices of ordern and s = 2n
6. anOD(4mp;ms₁,…,ms_u whenever anOD (4p;s₁,…,s_u)exists and s = 2p
7. a complex Hadamard matrix of order 2cm whenever there exists a complex Hadamard matrix of order 2c ands = 2c

This paper extends and improves results of Seberry and Whiteman giving new classes of Hadamard matrices, Williamson type matrices, orthogonal designs and complex Hadamard matrices.     

Covering properties and square principles

Covering matrices were used by Viale in his proof that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom and later by Sharon and Viale to investigate the impact of stationary reflection on the approachability ideal. In the course of this work, they isolated two reflection principles, CP and S, which may hold of covering matrices. In this paper, we continue previous work of the author investigating connections between failures of CP and S and variations on Jensen’s square principle. We prove that, for a regular cardinal λ > ω ₁, assuming large cardinals, □(λ, 2) is consistent with CP(λ, θ) for all θ with θ ⁺ < λ. We demonstrate how to force nice θ-covering matrices for λ which fail to satisfy CP and S. We investigate normal covering matrices, showing that, for a regular uncountable κ, □ _κ implies the existence of a normal ω-covering matrix for κ ⁺ but that cardinal arithmetic imposes limits on the existence of a normal θ-covering matrix for κ ⁺ when θ is uncountable. We introduce the notion of a good point for a covering matrix, in analogy with good points in PCF-theoretic scales. We develop the basic theory of these good points and use this to prove some non-existence results about covering matrices. Finally, we investigate certain increasing sequences of functions which arise from covering matrices and from PCF-theoretic considerations and show that a stationary reflection hypothesis places limits on the behavior of these sequences.     

<Emphasis Type="Italic">e</Emphasis>-Invertible Matrices Over Commutative Semirings

In this article, the e-invertible matrices over commutative semirings are studied. Some properties and equivalent characterizations of the e-invertible matrices are given. Also, the interrelationships between invertible matrices and e-invertible matrices over commutative semirings are discussed. The main results obtained in this article generalize and enrich the corresponding results about invertible matrices over commutative semirings.     

Generalized Hankel matrices of Markov parameters and their applications to control problems

The concept of Hankel matrices of Markov parameters associated with two polynomials is generalized for matrices. The generalized Hankel matrices of Markov parameters are then used to develop methods for testing the relative primeness of two matrices A and B, for determining stability and inertia of a matrix, and for constructing a class of matrices C such that A + C has a desired spectrum. Neither the method of construction of the generalized Hankel matrices nor the methods developed using these matrices require explicit computation of the characteristic polynomial of A (or of B).     

Fast computation of eigenvalues of companion,comrade, and related matrices

The class of eigenvalue problems for upper Hessenberg matrices of banded-plus-spike form includes companion and comrade matrices as special cases. For this class of matrices a factored form is developed in which the matrix is represented as a product of essentially 2×2 matrices and a banded upper-triangular matrix. A non-unitary analogue of Francis’s implicitly-shifted QR algorithm that preserves the factored form and consequently computes the eigenvalues in O(n ²) time and O(n) space is developed. Inexpensive a posteriori tests for stability and accuracy are performed as part of the algorithm. The results of numerical experiments are mixed but promising in certain areas. The single-shift version of the code applied to companion matrices is much faster than the nearest competitor.     

Displacement structure approach to Chebyshev-Vandermonde and related matrices

In this paper we use the displacement structure concept to introduce a new class of matrices, designated asChebyshev-Vandermonde-like matrices, generalizing ordinary Chebyshev-Vandermonde matrices, studied earlier by different authors. Among other results the displacement structure approach allows us to give a nice explanation for the form of the Gohberg-Olshevsky formulas for the inverses of ordinary Chebyshev-Vandermonde matrices. Furthermore, the fact that the displacement structure is inherited by Schur complements leads to a fastO(n ²) implementation of Gaussian elimination withpartial pivoting for Chebyshev-Vandermonde-like matrices.     

G-matrices,J-orthogonal matrices,and their sign patterns

A real matrix A is a G-matrix if A is nonsingular and there exist nonsingular diagonal matrices D₁ and D₂ such that A^?T = D₁AD₂, where A^?T denotes the transpose of the inverse of A. Denote by J = diag(±1) a diagonal (signature) matrix, each of whose diagonal entries is +1 or ?1. A nonsingular real matrix Q is called J-orthogonal if Q^TJQ = J. Many connections are established between these matrices. In particular, a matrix A is a G-matrix if and only if A is diagonally (with positive diagonals) equivalent to a column permutation of a J-orthogonal matrix. An investigation into the sign patterns of the J-orthogonal matrices is initiated. It is observed that the sign patterns of the G-matrices are exactly the column permutations of the sign patterns of the J-orthogonal matrices. Some interesting constructions of certain J-orthogonal matrices are exhibited. It is shown that every symmetric staircase sign pattern matrix allows a J-orthogonal matrix. Sign potentially J-orthogonal conditions are also considered. Some examples and open questions are provided.     

Tridiagonal and upper triangular inverse M-matrices

In this paper we characterize the nonnegative nonsingular tridiagonal matrices belonging to the class of inverse M-matrices. We give a geometric equivalence for a nonnegative nonsingular upper triangular matrix to be in this class. This equivalence is extended to include some reducible matrices.     

Finite Diagonal Random Matrices

The goal of this article is to extend some results of Popescu (Probab. Theory Relat. Fields 144:179, 2009) in several directions. We establish the limiting spectral distribution (LSD) for r-diagonal matrices under reduced moment conditions compared to those required by Popescu. We also deal with the joint convergence of several sequences of such matrices. In particular, we show that there is a large class of such matrices where the joint limit is not free while the marginals are semicircular. We also consider matrices of the form $X_{n}X_{n}^{T}$ where X _n is a sequence of nonsymmetric r-diagonal random matrices and establish their limiting spectral distribution.     

Geometry and inequalities of geometric mean

We study some geometric properties associated with the t-geometric means A ?_tB:= A^1/2(A^?1/2BA^?1/2)^tA^1/2 of two n × n positive definite matrices A and B. Some geodesical convexity results with respect to the Riemannian structure of the n × n positive definite matrices are obtained. Several norm inequalities with geometric mean are obtained. In particular, we generalize a recent result of Audenaert (2015). Numerical counterexamples are given for some inequality questions. A conjecture on the geometric mean inequality regarding m pairs of positive definite matrices is posted.     

A kind of nonnegative matrices and its application on the stability of discrete dynamical systems

In this paper we introduce a new kind of nonnegative matrices which is called (sp) matrices. We show that the zero solutions of a class of linear discrete dynamical systems are asymptotically stable if and only if the coefficient matrices are (sp) matrices. To determine that a matrix is (sp) matrix or not is very simple, we need only to verify that some elements of the coefficient matrices are zero or not. According to the result above, we obtain the conditions for the stability of several classes of discrete dynamical systems.     

Optimally scaled and optimally conditioned Vandermonde and Vandermonde-like matrices

Vandermonde matrices with real nodes are known to be severely ill-conditioned. We investigate numerically the extent to which the condition number of such matrices can be reduced, either by row-scaling or by optimal configurations of nodes. In the latter case we find empirically the condition of the optimally conditioned n×n Vandermonde matrix to grow exponentially at a rate slightly less than \((1+\sqrt{2})^{n}\). Much slower growth—essentially linear—is observed for optimally conditioned Vandermonde-Jacobi matrices. We also comment on the computational challenges involved in determining condition numbers of highly ill-conditioned matrices.