On sum-symmetric matrices

Afriat, S. N.

doi:10.1016/0024-3795(74)90051-2

Cited by 44 publications

(4 citation statements)

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“…Note that if A is definite, then every column of A * can be used to obtain a visualization of A, which may not be strict. This result was known to Afriat [2,3] and Fiedler and Pták [21], and it has been a source of inspiration for many works on scaling problems, see [18,19,27,35,37,38].…”

Section: Subeigenvectors Visualization and Convexitymentioning

confidence: 81%

On visualization scaling, subeigenvectors and Kleene stars in max algebra

Sergeev

Schneider

Butkovič

2009

Linear Algebra and its Applications

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The purpose of this paper is to investigate the interplay arising between max algebra, convexity and scaling problems. The latter, which have been studied in nonnegative matrix theory, are strongly related to max algebra. One problem is that of strict visualization scaling, defined as, for a given nonnegative matrix A, a diagonal matrix X such that all elements of X −1 AX are less than or equal to the maximum cycle geometric mean of A, with strict inequality for the entries which do not lie on critical cycles. In this paper such scalings are described by means of the max algebraic subeigenvectors and Kleene stars of nonnegative matrices as well as by some concepts of convex geometry.

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Section: Subeigenvectors Visualization and Convexitymentioning

confidence: 81%

On visualization scaling, subeigenvectors and Kleene stars in max algebra

Sergeev

Schneider

Butkovič

2009

Linear Algebra and its Applications

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“…In reviewing the literature on Levinger's Theorem, a number of overlaps are noted. The elementary proof of Levinger's Theorem in [16] defines 'balanced' matrices, which is the same as 'sum-symmetric' introduced in [1] and 'line-sum-symmetric' from [13]. Lemmas 2.1 and 2.2 in [16] correspond to Corollaries 3 and 5 in [13], but the proofs are quite distinct.…”

Section: The Assumption Of Simultaneous Symmetrizability Meansmentioning

confidence: 99%

On the Ordering of Spectral Radius Product $r({\bf A})\,r({\bf AD})$ versus $r({\bf A}^2{\bf D})$ and Related Applications

Altenberg¹

2013

SIAM J. Matrix Anal. & Appl.

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For a nonnegative matrix A and real diagonal matrix D, two known inequalities on the spectral radius, r(A 2 D 2 ) ≥ r(AD) 2 and r(A) r(AD 2 ) ≥ r(AD) 2 , leave open the question of what determines the order of r(A 2 D 2 ) with respect to r(A) r(AD 2 ). This is a special case of a broad class of problems that arise repeatedly in ecological and evolutionary dynamics. Here, sufficient conditions are found on A that determine orders in either direction. For a diagonally symmetrizable nonnegative matrix A with all positive eigenvalues and nonnegative D, r(A 2 D) ≤ r(A) r(AD). The reverse holds if all of the eigenvalues of A are negative besides the Perron root. This is a particular case of the more general result that r(A[(1 − m) r(B) I + mB]D) is monotonic in m when all non-Perron eigenvalues of A have the same sign -decreasing for positive signs and increasing for negative signs, for symmetrizable nonnegative A and B that commute. Commuting matrices include the Kronecker products A, B ∈ {⊗ L i=1 M ti i }, t i ∈ {0, 1, 2, . . .}, which comprise a class of application for these results. This machinery allows analysis of the sign of ∂/∂m j r(. The eigenvalue sign conditions also provide lower or upper bounds to the harmonic mean of the expected sojourn times of Markov chains. These inequalities appear in the asymptotic growth rates of viral quasispecies, models for the evolution of dispersal in random environments, and the evolution of site-specific mutation rates over the entire genome. 1

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“…The equivalence of conditions (i) and (ii) of Theorem (5.2) for the case that Go(P) is the full (directed) graph with vertex set (n ) , is found in Mriat [1,2,3] at level (A). Fiedler-Ptak [12,13] and Engel-Schneider [10,11] also proved this equivalence.…”

mentioning

confidence: 92%

“…For diagonal similarity, Afriat [1], [2], [3] has given solutions to the special scaling problem in a somewhat restricted form at level (A). Independently, Fiedler-Ptak [12], [13] proved some results at level (M) which contain the special case, see also Engel-Schneider [10], [11].…”

Section: Introductionmentioning

confidence: 99%

Cones, graphs and optimal scalings of matrices

Saunders

Schneider

1979

Linear and Multilinear Algebra

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Characterizations are given of the optimal scaiings of a complex square matrix within its diagonal similarity class and its restricted diagonal equivalence class with respect to the maximum element norm. The characterizations are in terms of a finite number of products, principally circuit and diagonal products. The proofs proceed by reducing the optimal scaling problems from the multiplicative matrix level in succession to an additive matrix level, a graph theoretic level, and a geometric level involving duality theorems for cones. At the geometric level, the diagonal similarity and the restricted diagonal equivalence problems are unified.

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On sum-symmetric matrices

Cited by 44 publications

References 3 publications

On visualization scaling, subeigenvectors and Kleene stars in max algebra

On visualization scaling, subeigenvectors and Kleene stars in max algebra

On the Ordering of Spectral Radius Product $r({\bf A})\,r({\bf AD})$ versus $r({\bf A}^2{\bf D})$ and Related Applications

Cones, graphs and optimal scalings of matrices

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