By D. E. Littlewood

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

**Example text**

Since the only elements in the last three rows are the three elements — 1 in the bottom left-hand corner, the determinant becomes equal to the three-rowed determinant in the top right-hand corner, and this proves thetheorem. /2 4- bty 4- b2, c0y2 + cxy + c2 , a0z2 4* axz 4- a 2,b0z2 -f bxz -f 62, c0z2 4- cxz + c2 MATRICES 31 this can be factorized in the form I &0* K Cp ®1> C1 ®a> c 2 The first factor is the product of the differences (x — y)(x — z)(y — z), while the second factor cannot be simplified.

V = A~lL. The matrix o f transformation o f the coefficients is A “ 1. The above results hold for any axes, rectangular or oblique. For rectangular Cartesian coordinates a special type o f matrix o f trans formation must be employed. Clearly the elements p ly p 2, p 8 are unrestricted since any translation is allowable. The restriction concerns the coefficients o f x, y, z, namely li9 mi9 n{. , transformations which leave the origin fixed. , y, z) from the origin is x2 + y2 + z2 = X X . The value o f X X is called the norm o f the vector X .

Consider the matrix f 5 , 2, 2-| 2, 2 , 1 L -2 , . 1, 2 _ The characteristic equation is A3 - 9À2 + 15A - 7 = 0, (A - l)2 (A — 7) = 0. Corresponding to A = 7 r - 2, 2, 2“ 2 ,-5 , 1 _ 2, 1, - 5 _ ~2~ 1 1 Corresponding to A = 1 f 4 , 2, 2“ r 2, 1, 1 _2 , 1, 1_ _ 2, o, - on 1 1_ Each of these poles corresponding to A = 1, is orthogonal to [2, 1, 1], but they are not orthogonal to each other. For the second column choose either of these or any linear combination, say 0“ 1 . For the third column a linear combination must be taken which is orthogonal to this.