Pál Rózsa

ON BAND MATRICES AND THEIR INVERSES

Abstract Structural properties of the inverses of band matrices are discussed. The definition of semiseparable matrices is given, and the theorem is proved that the inverse of a strict band matrix is a semiseparable matrix and vice versa. Finally, a recurrence algorithm is recommended for computing the blocks of the inverses of strict band matrices.

Consistency adjustments for pairwise comparison matrices

This paper is concerned with the development of a ‘best’ rank one transitive approximation to a general paired comparison matrix in a least-squares sense. A direct attack on the non-linear problem is frequently replaced by a sub-optimal linear problem and, here, the best procedure of this kind is obtained. The Newton–Kantorovich method for the solution of the non-linear problem is also studied, including the role of the best linear approximation as a starting point for this method. Numerical examples are included. Copyright

Transitive matrices and their applications

Abstract Transitive matrices and symmetrically reciprocal (SR) matrices are defined and spectral properties of certain SR perturbations of transitive matrices are studied. The results are applied to a multicriteria decision making method, the analytic hierarchy process (AHP), which uses an SR matrix with positive entries. A proof is given that rank reversal is inherent in this method, if its input matrix is perturbed.

On the Inverse of Block Tridiagonal Matrices with Applications to the Inverses of Band Matrices and Block Band Matrices

In the present paper the authors make an attempt to give a uniform description of the main properties of tridiagonal, band, block tridiagonal and block band matrices and their inverses. Some basic concepts are recalled and also some new results are presented.

Data Perturbations of Matrices of Pairwise Comparisons

This paper deals with data perturbations of pairwise comparison matrices (PCM). Transitive and symmetrically reciprocal (SR) matrices are defined. Characteristic polynomials and spectral properties of certain SR perturbations of transitive matrices are presented. The principal eigenvector components of some of these PCMs are given in explicit form. Results are applied to PCMs occurring in various fields of interest, such as in the analytic hierarchy process (AHP) to the paired comparison matrix entries of which are positive numbers, in the dynamic input–output analysis to the matrix of economic growth elements of which might become both positive and negative and in vehicle system dynamics to the input spectral density matrix whose entries are complex numbers.

On periodic block-tridiagonal matrices☆

Abstract Periodic block-tridiagonal matrices are defined, and conditions are given for factorizing their characteristic polynomial by means of the zeros of Chebyshev polynomials of the second kind. These conditions are expressed by the block centrosymmetry of certain submatrices along the main diagonal. The main theorems can be considered as generalizations of an earlier result published by Elsner and Redheffer and by Rozsa.

The spectrum and stability of a vibrating rail supported by sleepers

Abstract Models for the vibrations of a thin (axially loaded, damped) rail resting on sleepers are discussed with special reference to the distributions of eigenvalues arising as boundary conditions are changed, and also to the parameter sets which produce oscillatory and nonoscillatory solutions.

A recursive least-squares algorithm for pairwise comparison matrices

Pairwise comparison matrices are commonly used for setting priorities among competing objects. In a leading decision making method called the analytic hierarchy process the principal right eigenvector components represent the weights of the alternatives. The direct least-squares method extracts the weight vector by first finding a rank-one matrix which minimizes the Euclidean distance from the original ratio matrix. We develop a recursive least-squares algorithm and reveal a striking correspondence between these two approaches for these matrices. The recursion applies for merely positive matrices also. We prove that a convergent iteration leads to matrices by which the Perron-eigenvectors and the Perron approximation of the original matrix may be produced. We show that certain useful properties of the recursion advance the development of reliable measures of perturbations of transitive matrices. Numerical analysis is included for a macroeconomic problem taken from the literature.