Let P be an n x n (0, I)-matrix with at least one 1 in each row. Lei S(P) be the set of all stochastic eigenvectors belonging to n X n stochastic matrices A so that A 6 I? It is shown that S(P) is a convex polytope. Further, some work on computing this polytope is given.
In recent years, much work has been done in qualitative matrix theory. The basic work of this field is to develop results about a matrix A when only some qualitative information about A, such as all entries of A are nonnegative, is given. A first such result [3] was due to Oskar Perron when he established that any positive matrix has a positive eigenvalue. Other such results [1] were developed more recently. Qualitative matrix results also show some relation to interval analysis problems involving matrices whose entries are known to lie only within specified intervals [2].
The work of this paper further develops the area of qualitative matrix theory. The problem of interest is as follows: Let P be an n x n (0, 1)-matrix with at ieast one 1 in each row. Define S(P) as the set of all left stochastic eigenvefi=tors belonging to n x n row stochastic matrices A so that A G P. Describe the set S(P).
In resolving this problem, it is first shown that S(P) is a compact convex set.
Lemma. The set S(P) is a compact convex set.
Proof. First suppose that A and B are n x n stochastic matrices, with A G P and B =S P, having stochastic eigenvectors x and y respectively, where x + y > 0. For any vector 2 = (z,, . . . . z,) in IR", let 2 = diag(z,, . . . , z,,). Pick any pair of positive numbers cu and @ so that cx + @ = 1. Set c = (cwX+ pY)-'(&A + PYB) 20. Since Ce=e, where e=(l,l,..., l)', it follows that C is stochastic. Further, it follows from the definition that C G P. Finally, (ax + Py)C = e'(acX + fJ WC = e'(aXA + pYB) = (YX + py. Hence, (11x + By E S(P). Now suppose x + y has at least one zero component. Without loss of generality,