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

This booklet develops the topic of matrices with detailed connection with differential equations and classical mechanics. it truly is meant to carry to the coed of utilized arithmetic, without prior wisdom of matrices, an appreciation in their conciseness, strength and comfort in computation. labored numerical examples, a lot of that are taken from aerodynamics, are incorporated.

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O -%\ [ - 1 oj [» oj CHAPTER II POWERS OF MATRICES, SERIES, AND INFINITESIMAL CALCULUS 2-1. Introductory. In the present chapter we shall consider some of the properties of matrices which are expressible as functions of a given matrix or which have elements functionally dependent on real or complex parameters. In the latter connection the ideas of differentiation and integration of matrices will be developed. g. that defining the exponential function) which are of importance in the infinitesimal calculus.

For instance [2 3j[5 6J [23 28J * The term "conjugate" is used by some writers as meaning "transposed". f For a more complete account see, for example, Chap, rv of Ref. 2. 34 ORTHOGONAL MATRICES 1-17 If [ai:j] is a symmetrical matrix of order n, the adjoint [Ait] is also n n a irA*T = 0 and 2 a%rAir = \a\, we have by symmetrical. Since 2 r=l r=l direct multiplication 1 0 0 1 0 . 0 . 0 0 iJ = A21 A2 •^22 ""-3 0 \a a 32 0 Ail 1 0 MJ 1 Moreover, j An | = | aI *- . Hence %4 #'43 a44 a ^3 n4 — a m It follows from this equation that if An = 0 the two determinants on the left have opposite signs.

Y\ — XT T T rl 1-1 2 1 -1 3 0-3 2 0 9-6 1 - 3 - 6 19J x9 Here we may take A1 = 1 and1 -1 -1 3 = 2; 1-1 - 1 3 2 0 1 - 1 2 = 6; - 1 3 0 2 0 9 2 1 0-3 9-6 = 24. 1 - 3 - 6 19 These particular discriminants are all positive, and A(x,x) is accordingly a positive function. It can be verified that A (x, x) is expressible as A (x, x) = {x1 -x2 + 2xs + #4)2 + 2(x2 + xs- xtf (ii) Case of Proportional Rows and Columns. J±(X,X) — 1-1 2 1 -1 1 -2 -1 2-2 7-4 1 - 1 - 4 17J x3 Since the second row of a is proportional to the first, the variable x2 will be absent from B.