TY - JOUR

T1 - Reciprocal matrices: properties and approximation by a transitive matrix

AU - Bebiano, Natália

AU - Fernandes, Rosário

AU - Furtado, Susana

N1 - N. Bebiano: partially supported by project UID/MAT/00324/2019.
R. Fernandes: partially supported by project UID/MAT/00297/2019.
S. Furtado: partially supported by project UID/MAT/04721/2019.

PY - 2020/5/1

Y1 - 2020/5/1

N2 - Reciprocal matrices and, in particular, transitive matrices, appear in several applied areas. Among other applications, they have an important role in decision theory in the context of the analytical hierarchical process, introduced by Saaty. In this paper, we study the possible ranks of a reciprocal matrix and give a procedure to construct a reciprocal matrix with the rank and the off-diagonal entries of an arbitrary row (column) prescribed. We apply some techniques from graph theory to the study of transitive matrices, namely to determine the maximum number of equal entries, and distinct from ± 1 , in a transitive matrix. We then focus on the n-by-n reciprocal matrix, denoted by C(n, x), with all entries above the main diagonal equal to x> 0. We show that there is a Toeplitz transitive matrix and a transitive matrix preserving the maximum possible number of entries of C(n, x), whose distances to C(n, x), measured in the Frobenius norm, are smaller than the one of the transitive matrix proposed by Saaty, constructed from the right Perron eigenvector of C(n, x). We illustrate our results with some numerical examples.

AB - Reciprocal matrices and, in particular, transitive matrices, appear in several applied areas. Among other applications, they have an important role in decision theory in the context of the analytical hierarchical process, introduced by Saaty. In this paper, we study the possible ranks of a reciprocal matrix and give a procedure to construct a reciprocal matrix with the rank and the off-diagonal entries of an arbitrary row (column) prescribed. We apply some techniques from graph theory to the study of transitive matrices, namely to determine the maximum number of equal entries, and distinct from ± 1 , in a transitive matrix. We then focus on the n-by-n reciprocal matrix, denoted by C(n, x), with all entries above the main diagonal equal to x> 0. We show that there is a Toeplitz transitive matrix and a transitive matrix preserving the maximum possible number of entries of C(n, x), whose distances to C(n, x), measured in the Frobenius norm, are smaller than the one of the transitive matrix proposed by Saaty, constructed from the right Perron eigenvector of C(n, x). We illustrate our results with some numerical examples.

KW - Analytical hierarchical process

KW - Frobenius norm

KW - Perron eigenvalue

KW - Rank

KW - Reciprocal matrix

KW - Toeplitz matrix

KW - Transitive matrix

UR - http://www.scopus.com/inward/record.url?scp=85079229743&partnerID=8YFLogxK

U2 - 10.1007/s40314-020-1075-2

DO - 10.1007/s40314-020-1075-2

M3 - Article

AN - SCOPUS:85079229743

SN - 2238-3603

VL - 39

JO - Computational and Applied Mathematics

JF - Computational and Applied Mathematics

IS - 2

M1 - 50

ER -