The Frobenius normal form of a matrix is an important tool in analyzing its properties. When a matrix is powered up, the Frobenius normal form of the original matrix and that of its powers need not be the same. In this dissertation, conditions on a matrix A and the power q are provided so that for any invertible matrix S, if S−1AqS is block upper triangular, then so is S−1AS when partitioned conformably. The result is established for general matrices over any field. It is also observed that the contributions of the index of cyclicity to the spectral properties of a matrix hold over any field. We apply the block upper triangular powers result to the cone Frobenius normal form of powers of an eventually cone nonnegative matrix. Furthermore, we give a counterexample to the statement of Barker (1974) Theorem 7, and extend the Barker theorem to a larger class of cones called orthogonal face free cone using the necessary and sufficient conditions on the closedness of the linear image of a closed convex cone. In addition, we provide the generators for a proper polyhedral cone K, such that A is eventually K-irreducible with a small number of extreme vectors. We conclude this dissertation by a brief discussion of the general construction of Frobenius normal form with respect to cones which incorporate the idea of level sets.
|Commitee:||Tsatsomeros, Michael, Hudelson, Matthew G.|
|School:||Washington State University|
|School Location:||United States -- Washington|
|Source:||DAI-B 81/1(E), Dissertation Abstracts International|
|Keywords:||Block upper triangular matrices, Cones, Eventually nonnegative matrices, Fields, Frobenius normal form, Nonnegative matrices|
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