scholarly journals Optimal Gersgorin-style estimation of the largest singular value. II

2016 ◽  
Vol 31 ◽  
pp. 679-685
Author(s):  
Charles Johnson ◽  
J. Pena ◽  
Tomasz Szulc
Keyword(s):  
Spectral Radius ◽  
Linear Algebra ◽  
Nonnegative Matrix ◽  
Singular Value ◽  
Nonnegative Matrices ◽  
Electronic Journal ◽  
Complex Matrix ◽  
Class Of Matrices

In estimating the largest singular value in the class of matrices equiradial with a given $n$-by-$n$ complex matrix $A$, it was proved that it is attained at one of $n(n-1)$ sparse nonnegative matrices (see C.R.~Johnson, J.M.~Pe{\~n}a and T.~Szulc, Optimal Gersgorin-style estimation of the largest singular value; {\em Electronic Journal of Linear Algebra Algebra Appl.}, 25:48--59, 2011). Next, some circumstances were identified under which the set of possible optimizers of the largest singular value can be further narrowed (see C.R.~Johnson, T.~Szulc and D.~Wojtera-Tyrakowska, Optimal Gersgorin-style estimation of the largest singular value, {\it Electronic Journal of Linear Algebra Algebra Appl.}, 25:48--59, 2011). Here the cardinality of the mentioned set for $n$-by-$n$ matrices is further reduced. It is shown that the largest singular value, in the class of matrices equiradial with a given $n$-by-$n$ complex matrix, is attained at one of $n(n-1)/2$ sparse nonnegative matrices. Finally, an inequality between the spectral radius of a $3$-by-$3$ nonnegative matrix $X$ and the spectral radius of a modification of $X$ is also proposed.

scholarly journals ON NONNEGATIVE MATRICES WITH A FULLY CYCLIC PERIPHERAL SPECTRUM

1990 ◽  
Vol 21 (1) ◽  
pp. 65-70
Author(s):  
Bit-Shun Tam
Keyword(s):  
Spectral Radius ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Nonnegative Matrix ◽  
Nonnegative Matrices ◽  
Complex Matrix ◽  
Peripheral Spectrum ◽  
Rho A ◽  
Square Complex

Let $A$ be a square complex matrix. We denote by $\rho(A)$ the spectral radius of $A$. The set of eigenvalues of $A$ with modulus $\rho(A)$ is called the peripheral spectrum of $A$. The latter set is said to to be fully cyclic if whenever $\rho(A)\alpha x =Ax$, $x\neq 0$, $|a|= 1$, then $|x|(sgn \ x)^k$ is an eigenvector of $A$ corresponding to $\rho(A)\alpha^k$ for all integers $k$. In this paper we give some necessary conditions and a set of sufficient conditions for a nonnegative matrix to have a fully cyclic peripheral spectrum. Our conditions are given in terms of the reduced graph of a nonnegative matrix.

scholarly journals Sharp Bounds on the Spectral Radius of Nonnegative Matrices and Comparison to the Frobenius’ Bounds

2020 ◽  
Vol 14 ◽  
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Discrete Time ◽  
Upper Bound ◽  
Nonnegative Matrix ◽  
Nonnegative Matrices ◽  
Sharp Bounds ◽  
Similarity Transformations ◽  
Linear Invariant ◽  
The Stability

In this paper, a new upper bound and a new lower bound for the spectral radius of a nοnnegative matrix are proved by using similarity transformations. These bounds depend only on the elements of the nonnegative matrix and its row sums and are compared to the well-established upper and lower Frobenius’ bounds. The proposed bounds are always sharper or equal to the Frobenius’ bounds. The conditions under which the new bounds are sharper than the Frobenius' ones are determined. Illustrative examples are also provided in order to highlight the sharpness of the proposed bounds in comparison with the Frobenius’ bounds. An application to linear invariant discrete-time nonnegative systems is given and the stability of the systems is investigated. The proposed bounds are computed with complexity O(n2).

A Lagrange series approach to the spectrum of the Kite

2015 ◽  
Vol 30 ◽  
pp. 934-943
Author(s):  
Piet Van Mieghem
Keyword(s):  
Spectral Radius ◽  
Linear Algebra ◽  
Characteristic Polynomial ◽  
Clique Number ◽  
Electronic Journal ◽  
Largest Eigenvalue ◽  
Eigenvalues Of Graphs ◽  
Lagrange Series

A Lagrange series around adjustable expansion points to compute the eigenvalues of graphs, whose characteristic polynomial is analytically known, is presented. The computations for the kite graph P_nK_m, whose largest eigenvalue was studied by Stevanovic and Hansen [D. Stevanovic and P. Hansen. The minimum spectral radius of graphs with a given clique number. Electronic Journal of Linear Algebra, 17:110–117, 2008.], are illustrated. It is found that the first term in the Lagrange series already leads to a better approximation than previously published bounds.

Parallel Nonnegative Matrix Factorization via Newton Iteration

2016 ◽  
Vol 26 (03) ◽  
pp. 1650014 ◽  
Author(s):  
Markus Flatz ◽  
Marián Vajteršic
Keyword(s):  
Shared Memory ◽  
Matrix Factorization ◽  
Message Passing ◽  
Nonnegative Matrix Factorization ◽  
Nonnegative Matrix ◽  
Newton Iteration ◽  
Parallel Execution ◽  
Kkt Conditions ◽  
Nonnegative Matrices ◽  
First Order

The goal of Nonnegative Matrix Factorization (NMF) is to represent a large nonnegative matrix in an approximate way as a product of two significantly smaller nonnegative matrices. This paper shows in detail how an NMF algorithm based on Newton iteration can be derived using the general Karush-Kuhn-Tucker (KKT) conditions for first-order optimality. This algorithm is suited for parallel execution on systems with shared memory and also with message passing. Both versions were implemented and tested, delivering satisfactory speedup results.

scholarly journals Maximizing the spectral radius of fixed trace diagonal perturbations of nonnegative matrices

1996 ◽  
Vol 241-243 ◽  
pp. 635-654 ◽  
Author(s):  
Charles R. Johnson ◽  
Raphael Loewy ◽  
D.D. Olesky ◽  
P. van den Driessche
Keyword(s):  
Spectral Radius ◽  
Nonnegative Matrices

Stability indicatrices of nonnegative matrices and some of their applications in problems of biology and epidemiology

2019 ◽  
Vol 34 (5) ◽  
pp. 269-275
Author(s):  
Valery N. Razzhevaikin
Keyword(s):  
Nonnegative Matrix ◽  
Host Population ◽  
Nonnegative Matrices ◽  
Biological Communities ◽  
Evolutionary Selection ◽  
Evolutionary Optimality ◽  
Selection For ◽  
The Stability ◽  
Series Of Functions

Abstract The method of constructing a stability indicatrix of a nonnegative matrix having the form of a polynomial of its coefficients is presented. The algorithm of construction and conditions of its applicability are specified. The applicability of the algorithm is illustrated on examples of constructing the stability indicatrix for a series of functions widely used in simulation of the dynamics of discrete biological communities, for solving evolutionary optimality problems arising in biological problems of evolutionary selection, for identification of the conditions of the pandemic in a distributed host population.

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