Year: 1997
Type: dissertation
Abstract: This thesis studies the problem of estimating the largest possible dimension of a linear space of real matrices under the assumption that every non-zero matrix in the space has (the same) fixed rank.The complex version of this problem has been studied by R. Westwick and J. Sylvester.Sylvester introduced a technique based on the theory of Chern classes for estimating the dimension from above.The question of determining the largest dimension of a linear space of maximal-rank real n x n matrices (or, equivalently, of determining the largest number of nonsingular nxn matrices all of whose nontrivial linear combinations are nonsingular) was solved by J.F. Adams, P. Lax and R. Phillips.Their proof uses Adams' solution of the vector fields on spheres problem to show that the linear spaces constructed by J. Radon and A. Hurwitz are of the largest possible dimension under this hypothesis.A number of general results on the dimensions of real linear spaces of fixed-rank singular matrices, are due to R. Meshulam, E. Rees, K.Y.Lam and P. Yiu.The method used to provide upper bounds for the dimension is analogous to the complex case; here Stiefel-Whitney classes and K-theory are used for the calculations.Clifford Algebras are then used to construct spaces and so provide lower bounds for the dimension.For spaces of large rank the maximum dimension is related to the Radon-Hurwitz function, p(ri).In particular, for spaces of n x n matrices of rank nk with n > 8 and k = 1,2, the maximum dimension is given by the largest of the Ik + 1 integers We show that for k = 3 or 4 this relation continues to hold for almost all values of n.Sufficient conditions for the remaining cases are formulated in terms of lower bounds for the geometric dimensions of certain sums of line bundles over real projective spaces.Also considered are similar questions for spaces of rectangular matrices.We show how calculations with Stiefel-Whitney classes together with information about the existence of certain bilinear maps enable us to determine the dimensions of spaces of real n x k matrices of fixed-rank k for all n and k with k < 9.The case of fixed-rank symmetric matrices is also investigated.The main result here is that every space of real symmetric nxn matrices of fixed rank Ik -f 1 must have dimension 1.
Author: P.G. Moseley
Language: English
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2269272697