A University Algebra: An Introduction to Classic and Modern by D. E. Littlewood

By D. E. Littlewood

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Extra resources for A University Algebra: An Introduction to Classic and Modern Algebra

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An and that the corresponding poles are *^i> -^2» ••••» X n. Suppose further that each pole X i is multiplied by a scalar o f such a magnitude as will ensure that i A = i. Then it will automatically follow that the poles are orthogonal, for since _ _ A X i = A*Xt-, XjA = XjXj, therefore _ ^ ^ X sA Xt = XiXjXi = XjXjXj, and hence since X{ 7^ Xj, therefore X,X< = 0. 50 A UNIVERSITY ALGEBRA Hence the matrix T = [ Xl9 X 2, . . , X n] is an orthogonal matrix and T -'A T = diag. [Xl9 A2, ----- , AJ.

Then A = cos 9 — i sin 9 will also be a latent root and the roots will be coupled in pairs. I f the number o f rows is odd, however, there will be one latent root left over which is necessarily real. Hence it will be equal to ± 1. For an even number o f rows, if all the roots are coupled in pairs in the form (cos 0 db i sin 9), then the determinant, being the product o f the latent roots will be + 1. I f the determinant is negative then it is necessary that at least two latent roots should be real, and these must be + 1 and — 1 respectively.

The rank is 4 unless a = 1 or a = 5. If a = 1, the matrix is R o w (2) — row (3) — row (4) — "“1,1, 1,1" 1, 3, - 2, 1 2, 0, - 3, 2 • _3, 3, - 3, 3_ row (1) is [0, 2, — 3,0], 2 row (1) is [0,- 2 , - 5 , 0], 3 row (1) is [0, 0, — 6, 0]. These last three rows are clearly linearly dependent, but contain 2 inde­ pendent rows. The first row of the matrix is linearly independent of them all. Hence if a = 1, the rank is 3. For a = 5, the matrix is ri, l, 1, 1" 1, 3 , - 2 , 5 2, 8, - 7, 14 * _3, 7, - 3, 11_ Thence row (2) — row (3) — row (4) — row (1) = [0, 2, — 3, 4], 2 row (1) = [0, 6, — 9, 12], 3 row (1) = [0, 4, — 6, 8].

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