The algorithm for finding the optimal consistent approximation of an inconsistent pairwise comaprisons matrix is based on a logarithmic transformation of a pairwise comparisons matrix into a vector space with the Euclidean metric. Orthogonal basis is introduced in the vector space. The orthogonal projection of the transformed matrix onto the space formed by the images of consistent matrices is ...

We characterize the orthogonal frames and orthogonal multiwavelet frames in L2 R with matrix dilations of the form Df x √ |detA|f Ax , where A is an arbitrary expanding d × d matrix with integer coefficients. Firstly, through two arbitrarily multiwavelet frames, we give a simple construction of a pair of orthogonal multiwavelet frames. Then, by using the unitary extension principle, we present ...

Abstract Multiple orthogonal polynomials with respect to two weights on the step-line are considered. A connection between different dual spectral matrices, one banded (recursion matrix) and Hessenberg, respectively, Gauss–Borel factorization of moment matrix is given. It shown a hidden freedom exhibited by system related multiple polynomials. Pearson equations discussed, Laguerre–Freud conside...

We develop a method that allows us to construct families of orthogonal matrix polynomials of size N ×N satisfying second order difference equations with polynomial coefficients. The existence (and properties) of these orthogonal families strongly depends on the non commutativity of the matrix product, the existence of singular matrices and the matrix size N .

In this paper, the relationship between the (P,Q)-orthogonal symmetric and symmetric matrices is derived. By applying the generalized singular value decomposition, the general expression of the least square (P,Q)-orthogonal symmetric solutions for the matrix equation AXB = C is provided. Based on the projection theorem in inner space, and by using the canonical correlation decomposition, an ana...

We extend results concerning orthogonal matrices to a more general class of matrices that will be called P -orthogonal. This is a large class of matrices that includes, for instance, orthogonal and symplectic matrices as particular cases. We study the elementary properties of P -orthogonal matrices and give some exponential representations. The role of these matrices in matrix decompositions, w...