An annotated bibliography on convergence of matrix products and the theory of joint/generalized spectral radius (original) (raw)
Related papers
On the joint spectral radius of nonnegative matrices
Linear Algebra and its Applications
A 1. .. A n) i,j , where D ×D is the dimension of the matrices, U, V are respectively the largest entry and the smallest entry over all the positive entries of the matrices in A, and C is taken over all components in the dependency graph. The dependency graph is a directed graph where the vertices are the dimensions and there is an edge from i to j if and only if A i,j = 0 for some matrix A ∈ A. Furthermore, a bound on the norm is also given: There exists a nonnegative integer r so that for every n, const n r ρ(A) n ≤ max A1,...,An∈A A 1. .. A n ≤ const n r ρ(A) n. Corollaries of the approach include a simple proof for the joint spectral theorem for finite sets of nonnegative matrices and the convergence rate of some sequences. The method in use is mostly based on Fekete's lemma.
Spectral convergence for a general class of random matrices
Statistics & Probability Letters, 2011
Let X be an M N complex random matrix with i.i.d. entries having mean zero and variance 1=N and consider the class of matrices of the type B = A + R 1=2 XTX H R 1=2 , where A, R and T are Hermitian nonnegative de…nite matrices, such that R and T have bounded spectral norm with T being diagonal, and R 1=2 is the nonnegative de…nite square-root of R. Under some assumptions on the moments of the entries of X, it is proved in this paper that, for any matrix with bounded trace norm and for each complex z outside the positive real line, Tr h (B zI M) 1 i M (z) ! 0 almost surely as M; N ! 1 at the same rate, where M (z) is deterministic and solely depends on ; A; R and T. The previous result can be particularized to the study of the limiting behavior of the Stieltjes transform as well as the eigenvectors of the random matrix model B. The study is motivated by applications in the …eld of statistical signal processing.
$\Ell_{\Infty}$-Norm of Iterates and the Spectral Radius of Matrices
Časopis pro pěstování matematiky
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz časopis pro pěstování matematiky, roč. 105 (1980), Praha Z^-NORM OF ITERATES AND THE SPECTRAL RÁDIUS OF MATRICES ZDENĚK DOSTÁL, Ostrava
On explicit a priori estimates of the joint spectral radius by the generalized Gelfand formula
Differential Equations and Dynamical Systems, 2010
In various problems of control theory, non-autonomous and multivalued dynamical systems, wavelet theory and other fields of mathematics information about the rate of growth of matrix products with factors taken from some matrix set plays a key role. One of the most prominent quantities characterizing the exponential rate of growth of matrix products is the so-called joint or generalized spectral radius. In the work some explicit a priori estimates for the joint spectral radius with the help of the generalized Gelfand formula are obtained. These estimates are based on the notion of the measure of irreducibility (quasi-controllability) of matrix sets proposed previously by A. Pokrovskii and the author.
Journal of the London Mathematical Society, 1986
of identically distributed independent random matrices with a common distribution fi, the limit A(M)= lim /rMog || * " . . . * , || n-» oo exists with probability 1. We show that if fi has compact support in GL(/w, R) and if for k ^ 1 {^} fc> }( are 'locally perturbed' i.i.d. sequences which have laws /i k , and which satisfy || X\ k) -X t || ^ \/k almost surely and a further technical condition, then A(jx k ) -> A(ji) as k -* oo. Extensions are proved for Markovian matrix sequences and for some nonlinearly perturbed matrix sequences. JGL(m,R) J (0) Furstenberg and Kesten [7] showed the existence of the almost-sure constant (the 'maximal Lyapunov exponent' of the sequence Here and in what follows we write \\g\\ = sup 2eS || gz || for geGL(m, U), where m z = (z 1 ,...,z m )eU m , ||2 i -l If the support of fi (supp^) is irreducible in the sense that the minimal closed subgroup Gp of GL (m, U) containing the support of ft leaves no proper subspace of U m invariant, then Furstenberg [6] has shown that A(M)=\ f \og\\gz\\v(dz)Kdg) (1.1) J G L ( T O , R) J S for any Borel probability measure v on S (and there is at least one) satisfying, for all Borel subsets B cz S, v(B)= M *v(B) = f J\geGL(m, U):-^-eB\)v(dz). (1.2) Kifer [10] has applied this formula of Furstenberg to show that if {fi k }f^i is a sequence
A limit formula for joint spectral radius with p-radius of probability distributions
Linear Algebra and its Applications
In this paper we show a characterization of the joint spectral radius of a set of matrices as the limit of the p-radius of an associated probability distribution when p tends to ∞. Allowing the set to have infinitely many matrices, the obtained formula extends the results in the literature. Based on the formula, we then present a novel characterization of the stability of switched linear systems for an arbitrary switching signal via the existence of stochastic Lyapunov functions of any higher degrees. Numerical examples are presented to illustrate the results.
Rank-one Approximation of Joint Spectral Radius of Finite Matrix Family
2011
In this paper we study the joint/generalized spectral radius of a finite set of matrices in terms of its rank-one approximation by singular value decomposition. In the first part of the paper, we show that any finite set of matrices with at most one element's rank being greater than one satisfies the finiteness property under the framework of (invariant) extremal norm. Formula for the computation of joint/generalized spectral radius for this class of matrix family is derived. Based on that, in the second part, we further study the joint/generalized spectral radius of finite sets of general matrices through constructing rank-one approximations in terms of singular value decomposition, and some new characterizations of joint/generalized spectral radius are obtained. Several benchmark examples from applications as well as corresponding numerical computations are provided to illustrate the approach.
Some Properties of the Spectral Radius of a Set of Matrices
International Journal of Applied Mathematics and Computer Science, 2006
In this paper we show new formulas for the spectral radius and the spectral subradius of a set of matrices. The advantage of our results is that we express the spectral radius of any set of matrices by the spectral radius of a set of symmetric positive definite matrices. In particular, in one of our formulas the spectral radius is expressed by singular eigenvalues of matrices, whereas in the existing results it is expressed by eigenvalues.
A characterization of the generalized spectral radius with Kronecker powers
Automatica, 2011
Based on Turán's power sum theory, we extend a recent result obtained by . Computationally efficient approximations of the joint spectral radius, SIAM Journal on Matrix Analysis and Applications, 27,[256][257][258][259][260][261][262][263][264][265][266][267][268][269][270][271][272], by deriving a new characterization of the generalized spectral radius in terms of Kronecker powers.