By D. K. Faddeev, V.N. Faddeeva, C. Robert
.Hardback,Ex-Library,with ordinary stamps markings, ,in reliable all-round condition,dust jacket in reasonable condition,621pages.
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Q. 27) for i=l, ... ,n and j=l, ... ,p, where dll, ... ,d np are given positive integers. ) a. Show that the conditions n L a· k i=l I p L b· k , j=l j k=l, ... ,q, p Jla ik q L b· k k=l j L d .. , j=l Ij n L d .. , i=l Ij i=l, ... ,n, j=l , ... ,p, are necessary in order for a three-dimensional transportation problems to have a solution. b. Show that the conditions which the x ijk must satisfy if there is a solution can be written as Tx = b where T = T{q,W) with -54- Figure 7. The parallelopiped requirements for the tableau of a three-dimensional transportation problem.
1J and Q = (qk~)' then P X Q is obtained by replacing each element qk~ by the matrix qk~P, whereas Q X P is obtained by replacing each element Pij by the matrix PijQ. Consequently P X Q and Q X P differ only in the order in which rows and columns appear, and there exist permutation matrices Rand S, say, such that QXP = R[PXQ]S. (We remark also that some authors, for example, Thrall and Tornheim [13], define the Kronecker product of P and Q alternately as P X Q = (p .. Q) , 1J that is, our Q X P.
I=O 2 then SSR I Given the data -I o -5 -4 2 -3 10 construct the best I inear, quadratic and cubic least squares approximations. For each case determine SSR and SSE. What conclusions can you draw from the data available? 19 Let A, ZI and Z2 be any matrices. a. Prove that a solution, X, to the equations XAX X, AX ZI and XA = Z2' if it exists, is unique. b. 20 For what choices of ZI and Z2 is X a general ized inverse of A? 2): -27- a. The equations of XAX = X and (AX)H = AX are equivalent to the single equations XXHAH X.