Distribution of Kronecker products of matrices
Is Version Of
Consider the set W of all square matrices of order n. Each matrix A will represent a point in an n2-dimensional Euclidean space. Let the set of points represented by all matrices of order n be denoted by E w. There exists a subset of W all of whose elements can be written as a Kronecker product K = X x Y. The set of points represented by the matrices in this subset of W is denoted by Ep. The topological structure and the density of the set Ep in the set Ew is considered in this thesis. The function F = tr(A - K) (A - K)' is used as a norm for the distance between a point in Ew and a point in Ep;A study of the topological structure of the set Ep in the set in Ew is made in Chapter II. It is found that the points in Ep form an arcwise connected set. One obtains a subset Ep(s,r) of the set Ep when the orders, r and s, of the factors of K are fixed. It is shown that the set of points Ep(s,r) which lie on a given hypersphere form an arcwise connected set. The final result in this chapter is that the set of points Ep(s,r) which lie inside and on a unit hypersphere form a connected, closed, and bounded set;The density of the set Ep in the set Ew is considered in Chapters III and IV. It is shown that for the set of A's in the set W for which ‖A‖ = R, where R is a constant, the max∥A∥ =Rmin F⩽ r2-1 R2r2 where r is the order of the matrix Y in the Kronecker product K = X x Y and min F is the absolute minumum of F for a given matrix A. The max∥A∥= R (min F) is used as a measurement of the density of the set Ep in the set Ew. In Chapter IV the computation of min F is demonstrated for the case where Y is of order 2 and A is of even order. Results are obtained for the case where A is symmetric and for the case where A consists of two square blocks down the diagonal and zeros elsewhere;In Chapter V a given matrix A of order rs is interpreted to represent r2 points in an s2-dimensional space. It is pointed out that the problem of determining a K which produces a min F is equivalent to the problem of finding a ray through the origin such that the sum of the squares of the distance of the given r2 points to the line will be a minumum. Comments are made which show the relation between the given matrix A, the function F, the Ellipsoid of Inertia, and the positive definite matrix H which is associated with the positive definite matrix H which is associated with the Ellipsoid of Inertia. Bounds are obtained for the ‖H‖ in terms of the ‖A‖. Finally, a number of decompositions of A into a sum of Kronecker products are shown.