scholarly journals Calculating the minimal/maximal eigenvalue of symmetric parameterized matrices using projection

2019 ◽  
Vol 26 (5) ◽  
Author(s):  
Koen Ruymbeek ◽  
Karl Meerbergen ◽  
Wim Michiels
Keyword(s):  
Maximal Eigenvalue

Computing the Maximal Eigenvalue and Eigenvector of a Positive Matrix

1968 ◽  
Vol 5 (2) ◽  
pp. 269-274 ◽  
Author(s):  
C. A. Hall ◽  
T. A. Porsching
Keyword(s):  
Maximal Eigenvalue ◽  
Positive Matrix ◽  
Eigenvalue And Eigenvector

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.

Counts of long aligned word matches among random letter sequences

1987 ◽  
Vol 19 (2) ◽  
pp. 293-351 ◽  
Author(s):  
Samuel Karlin ◽  
Friedemann Ost
Keyword(s):  
Poisson Distribution ◽  
Maximal Length ◽  
Finite Alphabet ◽  
Maximal Eigenvalue ◽  
Schur Product ◽  
Distributional Properties ◽  
Random Letter

Asymptotic distributional properties of the maximal length aligned word (a contiguous set of letters) among multiple random Markov dependent sequences composed of letters from a finite alphabet are given. For sequences of length N, Cr,s(N) defined as the longest common aligned word found in r or more of s sequences has order growth log N/(–logλ) where λis the maximal eigenvalue of r-Schur product matrices from among the collections of Markov matrices that generate the sequences. The count Z∗r,s(N, k) of positions that initiate an aligned match of length exceeding k = log N/(–logλ) + x but fail to match at the immediately preceding position has a limiting Poisson distribution. Distributional properties of other long aligned word relationships and patterns are also discussed.

On the maximal numerical range of some matrices

2018 ◽  
Vol 34 ◽  
pp. 288-303 ◽  
Author(s):  
Ali Hamed ◽  
Ilya Spitkovsky
Keyword(s):  
Numerical Range ◽  
Maximal Eigenvalue ◽  
Direct Sums ◽  
Codimension One ◽  
Empty Intersection ◽  
Insight Into

The maximal numerical range $W_0(A)$ of a matrix $A$ is the (regular) numerical range $W(B)$ of its compression $B$ onto the eigenspace $\mathcal L$ of $A^*A$ corresponding to its maximal eigenvalue. So, always $W_0(A)\subseteq W(A)$. Conditions under which $W_0(A)$ has a non-empty intersection with the boundary of $W(A)$ are established, in particular, when $W_0(A)=W(A)$. The set $W_0(A)$ is also described explicitly for matrices unitarily similar to direct sums of $2$-by-$2$ blocks, and some insight into the behavior of $W_0(A)$ is provided when $\mathcal L$ has codimension one.

scholarly journals A family of anisotropic integral operators and behavior of its maximal eigenvalue

2011 ◽  
pp. 443-460 ◽  
Author(s):  
Boris Mityagin ◽  
Alexander Sobolev
Keyword(s):  
Integral Operators ◽  
Maximal Eigenvalue ◽  
And Behavior

scholarly journals The maximal eigenvalue of 0–1 matrices with prescribed number of ones

1985 ◽  
Vol 69 ◽  
pp. 33-69 ◽  
Author(s):  
Shmuel Friedland
Keyword(s):  
Maximal Eigenvalue

scholarly journals Optimal Multi-port-based Teleportation Schemes

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 477
Author(s):  
Marek Mozrzymas ◽  
Michał Studziński ◽  
Piotr Kopszak
Keyword(s):  
High Efficiency ◽  
Deterministic Case ◽  
Local Dimension ◽  
Maximal Eigenvalue ◽  
Probability Of Success ◽  
Average Probability ◽  
Primal And Dual ◽  
Optimal Set ◽  
Global Parameters ◽  
Final Expression

In this paper, we introduce optimal versions of a multi-port based teleportation scheme allowing to send a large amount of quantum information. We fully characterise probabilistic and deterministic case by presenting expressions for the average probability of success and entanglement fidelity. In the probabilistic case, the final expression depends only on global parameters describing the problem, such as the number of ports N, the number of teleported systems k, and local dimension d. It allows us to show square improvement in the number of ports with respect to the non-optimal case. We also show that the number of teleported systems can grow when the number N of ports increases as o(N) still giving high efficiency. In the deterministic case, we connect entanglement fidelity with the maximal eigenvalue of a generalised teleportation matrix. In both cases the optimal set of measurements and the optimal state shared between sender and receiver is presented. All the results are obtained by formulating and solving primal and dual SDP problems, which due to existing symmetries can be solved analytically. We use extensively tools from representation theory and formulate new results that could be of the separate interest for the potential readers.

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