Note on the computation of the maximal eigenvalue of a non-negative irreducible matrix

1971 ◽  
Vol 17 (5) ◽  
pp. 382-386 ◽  
Author(s):  
Takashi Noda
Keyword(s):  
Irreducible Matrix ◽  
Maximal Eigenvalue

scholarly journals Non-negative integral matrices with given spectral radius and controlled dimension

2021 ◽  
pp. 1-24
Author(s):  
MEHDI YAZDI
Keyword(s):  
Spectral Radius ◽  
Positive Real Number ◽  
Algebraic Integer ◽  
Log P ◽  
Irreducible Matrix ◽  
Positive Real ◽  
Shift Of Finite Type ◽  
Primitive Matrix ◽  
Integral Matrices ◽  
Perron Number

Abstract A celebrated theorem of Douglas Lind states that a positive real number is equal to the spectral radius of some integral primitive matrix, if and only if, it is a Perron algebraic integer. Given a Perron number p, we prove that there is an integral irreducible matrix with spectral radius p, and with dimension bounded above in terms of the algebraic degree, the ratio of the first two largest Galois conjugates, and arithmetic information about the ring of integers of its number field. This arithmetic information can be taken to be either the discriminant or the minimal Hermite-like thickness. Equivalently, given a Perron number p, there is an irreducible shift of finite type with entropy $\log (p)$ defined as an edge shift on a graph whose number of vertices is bounded above in terms of the aforementioned data.

scholarly journals On the construction of matrix representations and their extensions II

1991 ◽  
Vol 33 (3) ◽  
pp. 323-341
Author(s):  
Eric B. Kuisch
Keyword(s):  
Semidirect Product ◽  
Irreducible Matrix ◽  
Matrix Representations

In [5] we exhibited the construction of faithful irreducible matrix representations of p-groups E and constructed their extensions to a semidirect product E. H, in case E and H satisfied suitable conditions. One of the major conditions was that the prime p had to be odd.In this paper we assume the same conditions as in [5], but now with p = 2, in order to see if similar results can be obtained. Henceforth we will work with the following hypothesis.

scholarly journals Two Proofs and One Algorithm Related to the Analytic Hierarchy Process

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Miron Pavluš ◽  
Rostislav Tomeš ◽  
Lukáš Malec
Keyword(s):  
Analytic Hierarchy Process ◽  
Pairwise Comparisons ◽  
Problem Solution ◽  
Maximal Eigenvalue ◽  
Analytic Hierarchy ◽  
Solution Vector ◽  
Decision Making Processes ◽  
Hierarchy Process ◽  
Activity Areas

36 years ago, Thomas Saaty introduced a new mathematical methodology, called Analytic Hierarchy Process (AHP), regarding the decision-making processes. The methodology was widely applied by Saaty and by other authors in the different human activity areas, like planning, business, education, healthcare, etc. but, in general, in the area of management. In this paper, we provide two new proofs for well-known statement that the maximal eigenvalue λmax is equal to n for the eigenvector problem Aw=λw, where A is, so-called, the consistent matrix of pairwise comparisons of type n×n (n≥ 2) with the solution vector w that represents the probability components of disjoint events. Moreover, we suggest an algorithm for the determination of the eigenvalue problem solution Aw=nw as well as the corresponding flowchart. The algorithm for arbitrary consistent matrix A can be simply programmed and used.

scholarly journals The Aα-Spectral Radii of Graphs with Given Connectivity

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 44 ◽  
Author(s):  
Chunxiang Wang ◽  
Shaohui Wang
Keyword(s):  
Spectral Radius ◽  
Diagonal Matrix ◽  
Extremal Graphs ◽  
Maximal Eigenvalue ◽  
Edge Connectivity

The A α -matrix is A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) with α ∈ [ 0 , 1 ] , given by Nikiforov in 2017, where A ( G ) is adjacent matrix, and D ( G ) is its diagonal matrix of the degrees of a graph G. The maximal eigenvalue of A α ( G ) is said to be the A α -spectral radius of G. In this work, we determine the graphs with largest A α ( G ) -spectral radius with fixed vertex or edge connectivity. In addition, related extremal graphs are characterized and equations satisfying A α ( G ) -spectral radius are proposed.

Sign in / Sign up

Export Citation Format

Share Document