Distribution of Kronecker products of matrices

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)\u27 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\u27s 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

MODEL BUILDING TO MEASURE IMPACT OF WEATHER ON CROP YIELDS

The object of this research was to identify and evaluate alternatives when building mathematical models to measure the impact of weather on crop yields. Alternatives exist relative to selection of: (1) observational units with attention to size and coverage (areal and temporal), (2) observational periods for defining weather variables, and (3) mathematical forms and types of weather variables to measure impacts of moisture and temperature. The study involved an analysis of four weather-yield functions for winter wheat. The functions represented combinations of levels of two factors: (1) size and coverage of the observational units (plot yields from a multi-state area vs. average farm yields over Agricultural Statistics Districts in Kansas) and (2) weather variables used to represent moisture impacts (precipitation vs. evapotranspiration). From an eight-year test, using data from Kansas, we concluded that functions developed from a broad coverage (plot yields from a multi-state area) may have had a slight edge in precision

GENOTYPE X WEATHER INTERACTIONS IN GRAIN YIELDS OF WHEAT

The purpose of this paper is to demonstrate the advantage of using weather elements as covariates in studying yield differentials between varieties of wheat over different climatological regions. Using regression methods, the dependence of varietal yield differences on weather elements was demonstrated with a relatively small sample consisting of yield and weather data over a 3-year period from nine locations in Kansas. For each location, the sample-derived regression equation was used to calculate predicted yield differentials and 95% confidence intervals for the mean (CLM) for each year from 1950 through 1989. The proportion of CLMs that covered positive (or negative) values only was considered an important statistic. For each location, it estimated the proportion of years when the average yield of one variety was quite certain to exceed that of another . The procedure was applied to the problem of choosing new varieties for release to wheat growers. Results showed that a new variety, Karl, could be expected to outyield a popular variety, Newton, in more than 50% of the years in climates with mean annual precipitation exceeding 28 inches. Further, the mean yield of Karl could be expected to exceed that of another popular variety, Arkan, in over 50% of the years at almost all locations across the state