November 2024

Applied Mathematics and Computation

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November 2024

Applied Mathematics and Computation

October 2024

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2 Reads

Advances in Operator Theory

In decision making a weight vector is often obtained from a reciprocal matrix A that gives pairwise comparisons among n alternatives. The weight vector should be chosen from among efficient vectors for A. Since the reciprocal matrix is usually not consistent, there is no unique way of obtaining such a vector. It is known that all weighted geometric means of the columns of A are efficient for A. In particular, any column and the standard geometric mean of the columns are efficient, the latter being an often used weight vector. Here we focus on the study of the efficiency of the vectors in the (algebraic) convex hull of the columns of A. This set contains the (right) Perron eigenvector of A, a classical proposal for the weight vector, and the Perron eigenvector of $AA^{T}$ (the right singular vector of A), recently proposed as an alternative. We consider reciprocal matrices A obtained from a consistent matrix C by modifying at most three pairs of reciprocal entries contained in a 4-by-4 principal submatrix of C. For such matrices, we give necessary and sufficient conditions for all vectors in the convex hull of the columns to be efficient. In particular, this generalizes the known sufficient conditions for the efficiency of the Perron vector. Numerical examples comparing the performance of efficient convex combinations of the columns and weighted geometric means of the columns are provided.

September 2024

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2 Reads

The longstanding \emph{nonnegative inverse eigenvalue problem} (NIEP) is to determine which multisets of complex numbers occur as the spectrum of an entry-wise nonnegative matrix. Although there are some well-known necessary conditions, a solution to the NIEP is far from known. An invertible matrix is called a \emph{Perron similarity} if it diagonalizes an irreducible, nonnegative matrix. Johnson and Paparella developed the theory of real Perron similarities. Here, we fully develop the theory of complex Perron similarities. Each Perron similarity gives a nontrivial polyhedral cone and polytope of realizable spectra (thought of as vectors in complex Euclidean space). The extremals of these convex sets are finite in number, and their determination for each Perron similarity would solve the diagonalizable NIEP, a major portion of the entire problem. By considering Perron similarities of certain realizing matrices of Type I Karpelevich arcs, large portions of realizable spectra are generated for a given positive integer. This is demonstrated by producing a nearly complete geometrical representation of the spectra of $4 \times 4$ stochastic matrices. Similar to the Karpelevich region, it is shown that the subset of complex Euclidean space comprising the spectra of stochastic matrices is compact and star-shaped. \emph{Extremal} elements of the set are defined and shown to be on the boundary. It is shown that the polyhedral cone and convex polytope of the \emph{discrete Fourier transform (DFT) matrix} corresponds to the conical hull and convex hull of its rows, respectively. Similar results are established for multifold Kronecker products of DFT matrices and multifold Kronecker products of DFT matrices and Walsh matrices. These polytopes are of great significance with respect to the NIEP because they are extremal in the region comprising the spectra of stochastic matrices.

August 2024

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3 Reads

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1 Citation

International Journal of Approximate Reasoning

July 2024

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3 Reads

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1 Citation

Electronic Journal of Linear Algebra

We focus on the relationship between Hamiltonian cycle products and efficient vectors for a reciprocal matrix $A$, to more deeply understand the latter. This facilitates a new description of the set of efficient vectors (as a union of convex subsets), greater understanding of convexity within this set and of order reversals in efficient vectors. A straightforward description of all efficient vectors for an $n$-by-$n$, column perturbed consistent matrix is given; it is the union of at most $(n-1)(n-2)/2$ convex sets.

July 2024

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4 Reads

In decision making a weight vector is often obtained from a reciprocal matrix A that gives pairwise comparisons among n alternatives. The weight vector should be chosen from among efficient vectors for A. Since the reciprocal matrix is usually not consistent, there is no unique way of obtaining such a vector. It is known that all weighted geometric means of the columns of A are efficient for A. In particular, any column and the standard geometric mean of the columns are efficient, the latter being an often used weight vector. Here we focus on the study of the efficiency of the vectors in the (algebraic) convex hull of the columns of A. This set contains the (right) Perron eigenvector of A, a classical proposal for the weight vector, and the Perron eigenvector of AA^{T} (the right singular vector of A), recently proposed as an alternative. We consider reciprocal matrices A obtained from a consistent matrix C by modifying at most three pairs of reciprocal entries contained in a 4-by-4 principal submatrix of C. For such matrices, we give necessary and sufficient conditions for all vectors in the convex hull of the columns to be efficient. In particular, this generalizes the known sufficient conditions for the efficiency of the Perron vector. Numerical examples comparing the performance of efficient convex combinations of the columns and weighted geometric means of the columns are provided.

May 2024

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7 Reads

Involve a Journal of Mathematics

February 2024

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5 Reads

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5 Citations

SIAM Journal on Matrix Analysis and Applications

January 2024

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27 Reads

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9 Citations

Annals of Operations Research

The Analytic Hierarchy Process (AHP) is a much discussed method in ranking business alternatives based on empirical and judgemental information. We focus here upon the key component of deducing efficient vectors for a reciprocal matrix of pair-wise comparisons. It has been shown that the entry-wise geometric mean of all columns is efficient for any reciprocal matrix. Here, by combining some new basic observations with some known theory, we (1) give a method for inductively generating large collections of efficient vectors, and (2) show that the entry-wise geometric mean of any collection of distinct columns of a reciprocal matrix is efficient. We study numerically, using different measures, the performance of these geometric means in approximating the reciprocal matrix by a consistent matrix. We conclude that, as a general method to be chosen, independent of the data, the geometric mean of all columns performs well when compared with the geometric mean of proper subsets of columns.

October 2023

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21 Reads

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9 Citations

Linear Algebra and its Applications

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... If a graph is a tree T , a Parter vertex v for λ relative to A T ( ) ∈ is characterized by the existence of a downer branch at v [5]. However, when G is a general graph, a necessary and sufficient condition for a vertex to be a Parter vertex is given in [12]. We give the proof of it to be self-contained here. ...

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February 2023

Linear and Multilinear Algebra

... We also present an example of a matrix A that is not a simple perturbed consistent matrix and for which E(A) is convex. In the more recent paper [19], we have described the efficient vectors for reciprocal matrices obtained from a consistent matrix by modifying a 3-by-3 principal submatrix and have provided a class of efficient vectors if the modified block is of size greater than 3. ...

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February 2024

SIAM Journal on Matrix Analysis and Applications

... However, when A is inconsistent, there will be infinitely many, projectively distinct efficient vectors. Denote by E(A) the collection of all vectors efficient for A. It is now known that every column of A is efficient [14], the entry-wise geometric convex hull of the columns [15] and, in particular, the simple entry-wise geometric mean of all the columns [5], are efficient. However, the Perron vector is not always efficient [6,16]. ...

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- Full-text available
January 2024

Annals of Operations Research

- Citing Article
October 2023

Linear Algebra and its Applications

... Fathi et al. (2020) investigated two distinct inverse eigenvalue problems related to a nonsymmetric tridiagonal matrix and explored the applications of these problems in graph theory and perturbation theory. Furtado et al. (2023) examined square matrices exhibiting the inverse diagonal property. Qi et al. (2019) provided two formulas for Chebyshev polynomials of the second kind. ...

- Citing Article
September 2023

Kuwait Journal of Science

... We cite three works concerning these issues: [1,3,4]. The two rst papers are recent, from 2023, and the third is from 2003. ...

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January 2023

Special Matrices

... There is remarkable information about the catalog of linear trees based upon the so-called linear superposition principle (LSP) [8,13]. In view of recent work on U(T) for linear trees [2,3], our interest here is in better understanding U(T) for NL trees. ...

- Citing Article
March 2022

Linear and Multilinear Algebra

... Despite huge interest and extensive literature on the problem, the IEP-G has been solved only for a few selected families of graphs that include paths [12][13][14], cycles [11], generalized stars [17], complete graphs [6], lollipop and barbell graphs [20], linear trees [16], and graphs with at most five vertices [4,7]. For the background to the problem we refer the reader to [15]. ...

- Citing Article
April 2022

Discrete Mathematics

... That is a contradiction to (8). So, k cannot be Parter for λ in G e ij ( ). ...

- Citing Article
March 2021

Linear Algebra and its Applications

... In this article, we are particular about in a 2-downer edge cycle and focus on the effect of removing a 2downer edge in a cycle or a 2-downer edge triangle. In [11], the possible classification for an edge e ij for λ relative to A G ( ) ∈ is given as in Table 1, when the classifications of adjacent vertices i and j are known. Here, we refer to the two theorems we require later. ...

- Citing Article
September 2020

Linear and Multilinear Algebra