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A note on the Joint Spectral Radius [article]

Antonio Cicone
2015 arXiv   pre-print
A brief summary on the properties of the so called Joint 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.  ... 
arXiv:1502.01506v1 fatcat:xmneo25uwnbqzlm2a4su5wpk5i

Matrix Powers in Finite Precision Arithmetic

Nicholas J. Higham, Philip A. Knight
1995 SIAM Journal on Matrix Analysis and Applications  
We derive a sufficient condition for fl(Ak) 0 as k x) 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.  ... 
doi:10.1137/s0895479893256347 fatcat:qvjcsjrokrd45bawa3zucyurb4

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.  ... 

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

René Thiemann
2021 Journal of Logical and Algebraic Methods in Programming  
So far it only has been proven for small 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.  ... 
doi:10.1016/j.jlamp.2021.100699 fatcat:tfarrkr4xzb6jozn74yisrpmbi

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

Sarah Carnochan Naqvi, Judith J. McDonald
2002 The Electronic Journal of Linear Algebra  
Graph-theoretic 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.  ... 
doi:10.13001/1081-3810.1088 fatcat:rdsxljw7ubezdkkvvxut7scloe

A solution of a nonlinear system arising in spectral perturbation theory of nonnegative matrices

Uriel G. Rothblum, Hans Schneider
1997 Linear Algebra and its Applications  
It is known that the 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  ... 
doi:10.1016/s0024-3795(96)00154-1 fatcat:ngxivkxjczhydjgik2iduvfoom

Matrices Leaving a Cone Invariant [chapter]

Bit-Shun Tam, Hans Schneider
2013 Discrete Mathematics and Its Applications  
Generalizations of the Perron-Frobenius 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.  ... 
doi:10.1201/b16113-40 fatcat:xh4akfie4jh6vbrvijdm3rfaqi

Matrices Leaving a Cone Invariant [chapter]

Bit-Shun Tam, Hans Schneider
2006 Handbook of Linear Algebra  
Generalizations of the Perron-Frobenius 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.  ... 
doi:10.1201/9781420010572.ch26 fatcat:rv2m7hgi7ncu7hofymcou3b2ue

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).  ... 

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

J. de Pillis
1991 Linear Algebra and its Applications  
is circular, 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.  ... 
doi:10.1016/0024-3795(91)90394-c fatcat:2d7qqgng5vhnvg4q2vhzw6phzu

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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