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A note on the Joint Spectral Radius
[article]

2015
*
arXiv
*
pre-print

A brief summary on the properties of the so called Joint

arXiv:1502.01506v1
fatcat:xmneo25uwnbqzlm2a4su5wpk5i
*Spectral**Radius*... eigenvalue, then m = 1*and*the*Jordan*canonical*form*of A is [λ ] ⊕ J. ... The finiteness property holds for pairs of*matrices*with entries in {−1, 0, +1} (the so-called sign-*matrices*).*Jordan*canonical*form*of A. ...##
###
Matrix Powers in Finite Precision Arithmetic

1995
*
SIAM Journal on Matrix Analysis and Applications
*

We derive a sufficient condition for fl(Ak) 0 as k x)

doi:10.1137/s0895479893256347
fatcat:qvjcsjrokrd45bawa3zucyurb4
*and*a bound on [[fl(Ak)[[, both expressed in terms of the*Jordan*canonical*form*of A. Examples show that the results can be sharp. ... If A is a square matrix with*spectral**radius*less than 1 then A k 0 as k c, but the powers computed in finite precision arithmetic may or may not converge. ... We thank Des Higham*and*Nick Trefethen for their comments on the manuscript. ...##
###
Page 7691 of Mathematical Reviews Vol. , Issue 2000k
[page]

2000
*
Mathematical Reviews
*

Finally, the largest

*spectral**radius**and*the two small- est*spectral*radii of reducible football tournament*matrices*are determined.” ... Ipsen (1-NCS; Raleigh, NC) 2000k:15639 15A42 05C20 05C35 Li, Jiong Sheng (PRC-HEF; Hefei); He, Li Feng (PRC-HEF; Hefei)*Spectral**radius*of football tournament*matrices*. (Chinese. ...##
###
Page 1007 of Mathematical Reviews Vol. , Issue 2000b
[page]

2000
*
Mathematical Reviews
*

*Jordan*canonical

*forms*of some lower triangular

*matrices*with repeated eigenvalues. (English summary) Int. J. Math. Game

*Theory*Algebra 8 (1999), no.4, 175-185. ... In this paper, for any n x k irreducible matrix, a

*normal*

*form*(ar-equivalent) is defined

*and*then all (k +1) xk ar-polynomial

*matrices*are characterized. ...

##
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Page 5567 of Mathematical Reviews Vol. , Issue 2002H
[page]

2002
*
Mathematical Reviews
*

The joint

*spectral**radius*max{||4o(1),---»4etm lf": o:{1,..., m}— {I,..., k}} of a finite set of complex*matrices*Aj,...,. ...*Normal*(n x n)-*matrices*A*and*B are (Hermite-) congruent if*and*only if rank(A) = rank(B) =r,*and*the nonzero eigenvalues A), A2,..., 4,*and*y), “2,..., u, of both*matrices*can be For the web version of ...##
###
A Perron–Frobenius Theorem for Deciding Matrix Growth

2021
*
Journal of Logical and Algebraic Methods in Programming
*

So far it only has been proven for small

doi:10.1016/j.jlamp.2021.100699
fatcat:tfarrkr4xzb6jozn74yisrpmbi
*matrices*,*and*here we present a proof for the general case. ... We further verify both the algorithm*and*the new Perron-Frobenius theorem in the proof assistant Isabelle/HOL,*and*integrate it into Ce T A, a verified certifier for various properties, including complexity ... Shortly after finding the proof in Section 3*and*finishing its formalization, Hans Zwart sent us an alternative proof of Theorem 4. ...##
###
Page 7311 of Mathematical Reviews Vol. , Issue 2001J
[page]

2001
*
Mathematical Reviews
*

The proofs are based on the use of the properties of centralizers of

*matrices*reduced to*Jordan**normal**form*. ... The search for these functions*forms*an important part of the problem of reducing a family of*matrices*to a*normal**form**and*is necessary for the effective use of*normal**forms*. ...##
###
The combinatorial structure of eventually nonnegative matrices

2002
*
The Electronic Journal of Linear Algebra
*

Graph-theoretic

doi:10.13001/1081-3810.1088
fatcat:rdsxljw7ubezdkkvvxut7scloe
*spectral**theory*of*matrices*continues to develop,*and*in this paper we are interested in extending many of these ideas to the class of eventually nonnegative*matrices*. ... For example, there are irreducible eventually nonnegative*matrices*for which the*spectral**radius*is a multiple eigenvalue. ... Case II: Every initial class has*spectral**radius*ρ, but there exists a noninitial class with*spectral**radius*ρ. By that y T (ρI − A) = 0. If (ρI − A)z > 0, then y T [(ρI − A)z] > 0. ...##
###
A solution of a nonlinear system arising in spectral perturbation theory of nonnegative matrices

1997
*
Linear Algebra and its Applications
*

It is known that the

doi:10.1016/s0024-3795(96)00154-1
fatcat:ngxivkxjczhydjgik2iduvfoom
*spectral**radius*of P + EE*and*corresponding (*normalized*) eigenvector have fractional power series expansions. ...*radius**and*that the (unique up to scalar multiples) left*and*right eigenvectors of P corresponding to its*spectral*... The*spectral**radius**and*corresponding*normalized*eigenveetors of*matrices*govern the evolution of dynamic systems,*and*hence they are important characteristics of such systems; see numerous examples in ...##
###
Matrices Leaving a Cone Invariant
[chapter]

2013
*
Discrete Mathematics and Its Applications
*

Generalizations of the Perron-Frobenius

doi:10.1201/b16113-40
fatcat:xh4akfie4jh6vbrvijdm3rfaqi
*theory*of nonnegative*matrices*to linear operators leaving a cone invariant were first developed for operators on a Banach space by Krein*and*Rutman [KR48], Karlin ... [Kar59]*and*Schaefer [Sfr66] ... the Frobenius*normal**form*. ...##
###
Matrices Leaving a Cone Invariant
[chapter]

2006
*
Handbook of Linear Algebra
*

Generalizations of the Perron-Frobenius

doi:10.1201/9781420010572.ch26
fatcat:rv2m7hgi7ncu7hofymcou3b2ue
*theory*of nonnegative*matrices*to linear operators leaving a cone invariant were first developed for operators on a Banach space by Krein*and*Rutman [KR48], Karlin ... [Kar59]*and*Schaefer [Sfr66] ... the Frobenius*normal**form*. ...##
###
Page 142 of Mathematical Reviews Vol. , Issue 2001A
[page]

2001
*
Mathematical Reviews
*

Juan Miguel Gracia (E-PAIV; Vitoria)
15 LINEAR

*AND*MULTILINEAR ALGEBRA; MATRIX*THEORY*2001a:15013 15A21 Mutzbauer, Otto (D-WRZB-IM; Wiirzburg)*Normal**forms*of*matrices*with applications to almost completely ... Cain (1-L[ASU; Ames, IA) 2001a:15011 15A21 15A03 15A30 Gekhtman, Michael I. (1-NDM; Notre Dame, IN); Rodman, Leiba (1-CWM; Williamsburg, VA)*Normal**forms*of generic triangular band*matrices**and**Jordan*...##
###
Page 5258 of Mathematical Reviews Vol. , Issue 99h
[page]

1999
*
Mathematical Reviews
*

*normal*

*form*. ... (E-UPMTC; Madrid)

*Jordan*

*normal*

*form*via elementary transformations. (English summary) SIAM Rev. 40 (1998), no. 4, 947-956 (electronic). ...

##
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A parallelizable SOR-like method: Systems with plus-shaped and linear spectra

1991
*
Linear Algebra and its Applications
*

is circular,

doi:10.1016/0024-3795(91)90394-c
fatcat:2d7qqgng5vhnvg4q2vhzw6phzu
*and*2' = col [O, x]. As in the SOR*theory*, the*spectral**radius*of g is less than that of B. ... Among other things, we recapture the results of the SOR*theory*, improve on some known results,*and*extend our*theory*directly to*matrices*I, -B which need not be symmetric yet have real (plus-*and*minus-valued ... The author wishes to thank the referee for the many suggestions*and*helpful comments which influenced this paper. ...##
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Page 3507 of Mathematical Reviews Vol. , Issue 98F
[page]

1998
*
Mathematical Reviews
*

Summary: “An algorithm for determining the

*Jordan**normal**form*of a matrix is presented. The algorithm implements the geomet- ric approach given by Gantmakher. ... Summary: “In this paper we compare the*spectral**radius*of a weighted additive mean L of order ¢ involving Hadamard powers of nonnegative*matrices*with the corresponding mean R of the respective*spectral*...
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