Title:

On the dimensions of linear spaces of real matrices of fixed rank

This thesis studies the problem of estimating the largest possible dimension of a linear space of real matrices under the assumption that every nonzero 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 maximalrank real n x n matrices (or, equivalently, of determining the largest number of nonsingular n x n 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 linear spaces of fixedrank real matrices, as well as related questions concerning linear spaces whose nonzero matrices have rank bounded below, are due to E. Rees and K.Y. Lam. The method used to provide upper bounds for the dimension is analogous to the complex case; here StiefelWhitney classes and Ktheory are used for the calculations. Clifford Algebras are then used to construct spaces and so provide lower bounds for the dimension. We show how calculations with StiefelWhitney 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 fixedrank k for all n and k with k ≤ 9. The case of fixedrank symmetric matrices is also investigated. The main result here is that every space of real symmetric n x n matrices of fixed rank 2k + 1 must have dimension 1.
