Isolated and structured families of models for stochastic symmetric matrices

Abstract

Stochastic symmetric matrices with a dominant eigenvalue, 𝛂,can be written as the sum of λ𝛂𝛂 t (where λ is the first eigenvalue), with a symmetric error matrix E. The information in the stochastic matrix will be condensed in its structured vectors, λ𝛂, and the sum of square of residues, V. When the matrices of a family correspond to the treatments of a base design, we say the family is structured. The action of the factors, which are considered in the base design, on the structure vectors of the family matrices will be analyzed. We use ANOVA (Analysis of Variance) and related techniques, to study the action under linear combinations of the components of structure vectors of the m matrices of the model. Orthogonal models with m treatments are associated to orthogonal partitions. The hypothesis to be tested, on the action of the factors in the base design, will be associated to the spaces in the orthogonal partitions. We will show how to carry out transversal and longitudinal analysis for families of stochastic symmetric matrices with dominant eigenvalue associated to orthogonal models.