定理4.4设A∈Cm"是Hermite.正定矩阵，则存在下 三角矩阵L∈Cx",使得A=LL,称为A的Cholesky分解。 以n=3为例 [a1a2 937 d 2 a23 121 l22 as a32 13112 1 k 42 11 =42141h2+1zf 1☑21l1+12l2 g141g1+122h+h2+ha月 This documentis prduced byria version ofPrnhVisit www.prinashcfor more informion
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Ax=b→L(UX)=b三 Ly=b Ux=y 例.试用Doolittle分解求解方程组 2 -6 10 13 -19 19 -6 -3 -6Lx3] -30 2 5 -6 「1 0 04 42 413 3 -19 0 V22 2423 -6 -3 6 12 133」 This documentis produced bytril versinofPrVisit www.prinashmformore infomio
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7「2 -6 A=LU= 21 3-7 -341 4 (2)解Ux=y (1)解Ly=b 1 7y1「10 21 y2 19 1-341y」-30」 得y=10,y2=19-20=-1,y=34-30=4 解得：x3=1,x2=2,x=3 即y=10,-1,4) 所以方程组的解为x=(3,2,1)。 This documentis prduced byria version ofPrnhVisit www.prinashcfor more informion
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练习1：将A分解为A=LU,其中L为单位 下三角矩阵，U为上三角矩阵： 22 23 4= =LU 6 练习2：将A分解为A=L,其中L为正线 下三角矩阵。 This document is produced by trial version of Print2Flash Visit www.print2flash.com for more informatio
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4.2矩阵的QR分解 定义：设A∈Cx”,如果存在n阶酉矩阵Q 和n阶上三角矩阵R,使得A=OR 则称之为A的QR分解或酉-三角分解。当 A∈Rx”时，称为A的正交三角分解。 定理4.5任意A∈Cmx”都可以作QR分解。 定理4.6设A∈C”,则A可唯一分解为 A=OR 其中Q是n阶酉矩阵，R∈C"是具有正对角元的 上三角矩阵
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