By A Seidenberg

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A. : I I r- xXi H 1 -39- where Ll indicates that the sum is taken over indices with eigenvalues Ai I O. b. Prove that if A is normal. (An)+ c. If \ Xl = 2i-2. 3. [ /2 -1 x2 = 1/3[~1· 2 x3 =_1 312 [-~l· 1 construct the Moore-Penrose inverse of the matrix. A. 3. Applications with Matrices of Special Structure For many applications of mathematics it is required to solve systems of equations Ax = b in which A or b or both A and b have some special structure resulting from the physical considerations of the particular problem.

X .. 24) . =L1C ij· + np n L i=l p Llc ij j= for i = 1, ... ,n and j=l, ... ,p. 19 (Continuation): The transportation problem has been generalized in a number of different ways, and one of these extensions follows directly using matrices of the form T = T{p,W). Suppose that we are given q transportation problems, each with n origins and p destinations, and let aik,bjk,Cijk and x ijk be the row sums, column sums, costs and variables, respectively, associated with the kth tableau, k=l, ... ,q. 27) for i=l, ...

19. 2). 6 in which at least one of the matrices has this form occur frequently in statistical design of experiments [1] [4]. For example, suppose it is requi red to examine the effect of p different fertil izers on soy bean yield. One approach to this problem is to divide a field into pq subsections (called plots), randomly assign each of the p type of fertil izers to q plots, and measure the yield from each. 16) m + t. + e .. I I J where y .. is the yield of the jth plot to which ferti lizer i has IJ been appl ied, m is an estimate of an overall "main" effect, ti is an estimate of the effect of the particular fertil izer treatment and e ij is the experimental error associated with the particular plot.