A Matrix Theoretical Theorem and its Application in Medical Diagnosis

Abstract

Let A be a finite set and t a symmetric function on A×A with values in [0,1]. Let us denote the elements of A by x₁,…,x_n. Now form the symmetric matrix

$$ {\text{M}}_{\text{1}} \, = \,\left[ {{\text{t(x}}_{\text{k}} {\text{,x}}_{\text{s}} )} \right]_{k,s = 1}^n $$

Let us give an arbitrary number 0<p ≦ 1. We transform M₁ into another matrix N₁ as follows: If for an element t(x_k,x_s) in M₁ the relation t(x_k,x_s) ≦ p holds, then let us put in N₁ the number 0, and if t(x_k,x_s) ˃ p let us put 1. In this way we obtain an incidence matrix, the rows of which define the vectors y₁, ⁽¹⁾ y₂ ⁽¹⁾,…,y_n ⁽¹⁾. We denote the set of these vectors by B⁽¹⁾.