By D. E. Littlewood

ISBN-10: 0486627152

ISBN-13: 9780486627151

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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. Again, if St. — xr + yr -f- zr, then X 2, X , y2, y, 1 sa, 2, 1 So, ^ * &2> 1 $2» 1, X2 1. y, y2 1. z. zs 1, 1, 1 ^2 — X, y, Z X 2, y 2, Z2 ^4 = (* - y )\ * ~ *)2(y - *)*• Exercise lie Find the products of the following matrices and verify that the deter­ minant o f the product is equal to the product o f the determinants.

Firstly if the transformation o f the coefficients is o f the form T [a JT = diag. [Alt A2, ----- - AJ, then the rank o f [a#] is equal to the rank o f the diagonal matrix. I f the rank is r then the number o f zero terms among the A/s is (n — r) and independent o f the mode o f reduction. Next suppose that in one reduction ZatFix¡ = Z\Vi2> where Ax, ___ _ are positive and the remaining coefficients negative or zero, while in another reduction = Z p fr1, where [il9 [iQ are positive and the remaining coefficients zero or negative with p < q.

Similarly for A = — 1, the right factor o f zero of “ 9, - 12, 5” 15, - 24, 11 24, - 42, 20_ - Lastly, for A = 2, Thence “ 6, - 12, 5 “ "1 ~ 3 15, - 27, 11 24, - 42, 17_ _ 6 8, - 12, 5 “ 1 . - 1 . 2“ | r i, H 1 ,-2 , 6 15, - 25, 11 1, 2, 3 = 24, - 42, 19_ _ 1 , 3, 6 _ _ 1 , - 3, 12_ “ 1, 1, 1“ “ 1 - 1 1. 2, 3 2__ _ 1, 3, 6_ so that ~ i , l, i~ 1, 2, 3 _ 1, 3, 6 _ -1 “ 8, 15, 24, - 12, 5“ 25, 11 42, 1 9 _ r i . 1, 1“ 1, 2, 3 1, 3, 6 ” 1 = - 1 _ 2_ The Spur o f a Matrix The sum o f the leading diagonal elements o f a square matrix is called the spur o f the matrix.

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


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