By R. A. Frazer, W. J. Duncan, A. R. Collar
This e-book develops the topic of matrices with specific connection with differential equations and classical mechanics. it's meant to carry 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, lots 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
That defining the exponential function) which are of importance in the infinitesimal calculus. 2*2. Powers of Matrices. utom factors is written um. In addition u° is interpreted to mean the unit matrix In. By the associative law „,,, of multiplication, 7 m m 7 U X VL == U m 1 XU l m (u ) = (u ) and == M *~ = /^m/. Hence in the multiplication of powers of matrices the usual index laws of scalar algebra hold for positive integral and zero indices. The reciprocal matrix u~x exists provided u is not singular, and the higher negative integral powers of u are then defined as the powers of the reciprocal.
For instance, 11 T4 11 = [21 151. 59j -3j 2 7 LI - 3 _ An obvious extension is that if a is a symmetrical matrix, then the product ufau is symmetrical. 1-14. Linear Substitutions. Suppose that Vi = + ^2 2/2 = Vn = U + • • • + ^2 nl Then the set of variables y is said to be derived from the set # by a linear transformation, and the equations are also said to define a linear 1-14 27 LINEAR SUBSTITUTIONS substitution. The whole set of equations can be represented by the single matrix equation __ ,,x where x and y are column matrices.
If t receives the increment 8t, there will be corresponding increments of the elements of u, and the matrix of the increments may be written 8u. The differential coefficient (or derivative) of u with respect to t is then defined by the equation du ,. 8u Hence the elements of dujdt are the differential coefficients of the corresponding elements of u. An immediate deduction is that d dux du2 Iiw = uxu2i then w + 8w = (u1 + Sux) (u2 + 8u2) and on neglect of the second order term, this gives More generally the derivative of a continued product is formed as for scalar expressions, except that the original order of the factors must be preserved throughout.
Elementary Matrices And Some Applications To Dynamics And Differential Equations by R. A. Frazer, W. J. Duncan, A. R. Collar