Graph-theoretic methods for determining the distinguished eigenvalues to nonnegative reducible matrices with applications to certain linear systems

The proposed research consists of designing an algorithm, or computational procedure, for determining the distinguished eigenvalues of a nonnegative, nonnull reducible matrix. Nonnegative reducible matrices are those containing nonnegative entries, and are usually represented in upper triangular block form (to which they can be converted by a sequence of permutation transformations). The diagonal blocks in such a representation are either 1 x 1—null matrices or nonnegative irreducible matrices (which do not admit such a "triangular" deconposition). Distinguished eigenvalues are those nonnegative eigenvalues which have an associated nonnegative eigenvector. The algorithm will incorporate the graph-theoretic results of H. Dean Victory, Jr., and the numerical results of W. Bunse, B. Schulzendorff, L. Eisner, and Pham van At on the computation of the dominant eigenvalue, with normalized eigenvector, for nonnegative irreducible matrices. The results of this thesis can then be applied to con5)uting nonnegative solutions X to conditional equations of the form Xx = Ax + b where X is a positive parameter, b a nontrivial vector given with nonnegative components, A a nontrivial and nonnegative matrix.