Elementary Matrices And Some Applications To Dynamics And by R. A. Frazer, W. J. Duncan, A. R. Collar

By R. A. Frazer, W. J. Duncan, A. R. Collar

This ebook develops the topic of matrices with precise connection with differential equations and classical mechanics. it's meant to convey to the coed of utilized arithmetic, with out earlier wisdom of matrices, an appreciation in their conciseness, energy and comfort in computation. labored numerical examples, a lot of that are taken from aerodynamics, are incorporated.

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Extra info for Elementary Matrices And Some Applications To Dynamics And Differential Equations

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Let u i = [cos a, sin a] [cost, suit] and u2 = fcostf, cosal. i&int, sinaj (i) (i) Then ux = f" 0, 0 1 and w2 = r - s i n ^ O l . [ — sin£, cosJj [ cos£, 0J Hence, ifw = u±u2, we have (i) (i) (i) w= uxu2-\-uxu2 — fcos a, sin al f" — sin t, 01 + f" 0, [cos t, sin 2J [ cos t, Oj [ ~ in(a-^), 01 + TO, 0, 0 sm 0 1 fcos t, cos al ^ cos d L s m ^> sm a\ 1 = /sin(a-£). Oj [o,sin(a-*)J This may be verified by differentiation of w = fcosa, sinal Fcos^, cosal = |~cos(a — t), 1 1. [ 1, cos (a — £)J [cos t, sin £j [sin t, sin a] (ii) Taylor's Theorem for Matrices.

Iii) Multiply Degenerate Matrices. 0 0 0 0" 0 0 0 0 1 0 0 0 lJ is of degeneracy 2 and rank 2. The equations ax = 0 can here be satisfied by x(l) = {1,0,0,0} and x(2) = {0,1,0,0}, and the most general solution is x = c1x(l) + c2x(2). As another illustration we may take ' 1 2 3-1" 2 4 6-2 -1 -2 -3 1 . 0 0 0 OJ 20 SINGULAR MATRICES AS PRODUCTS 1*9 This matrix is of degeneracy 3 and rank 1. The equations ax = 0 are satisfied by the three columns as(2) = {3, 0 , - 1 , 0 } , 05(3) = {1, 0, 0, 1}, and the most general solution is x = (iv) Singular Matrices Expressed as Products.

Then the submatrices occupying corresponding positions will be of the same order, and may be added. Since each element in the sum is the sum of the individual elements, it is clear that the sum of the partitioned matrices is equal to the sum of the original matrices. Multiplication of partitioned matrices requires more detailed consideration. Let BA = P, where B is of type (m,p) and A of type (p, n). Suppose now that B is partitioned by, say, two vertical lines into three submatrices Bl9 B2, J53; thus B = [Bl9 B2, BZ].

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