第四章 矩阵的分解 本章我们主要讨论矩阵的四种分解：矩 阵的三角分解，QR分解，满秩分解，奇异 值分解。 4.1 矩阵的三角分解 4.1.1 三角分解及其存在唯一性问题 定义4.1设 A∈C"”,如果存在下三角矩阵 L∈C""和上三角矩阵 U∈C"" 使得A=LU, 则称A可以作三角分解。 This document is produced by trial version of Print2Flash. Visit www.print2flash.com for more information
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定理4.1设A∈Cm",则A有三角分解的充要条件 是A的各阶顺序主子式△，…△均不为零， 定理表明并不是每个可逆矩阵都可以作三角 分解。如 01 A= 10 不能作三角分解。 定理4.2设A∈C”,且A的前r个顺序主子式不为零， 即△k≠0(k=1,2,…,r),则A可以作三角分解， 证明：A= (A.A2A Az A,A2)（BABA2
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上述定理只是充分条件，如： 00)=0011)_0011 12气11八01(12八0 矩阵的三角分解不唯一，如A=LU=(LD)(DU), 其中D为可逆的对角矩阵 定义4.2将L是单位下三角矩阵的LU三角分解 称为矩阵的Doolittle分解。将U是单位上三角 矩阵的LU三角分解称为矩阵的Crout分解。若 A=LDR,其中L是单位下三角矩阵，D是对角 矩阵，U是单位上三角矩阵，则称为A的LDR 分解。 This document is produced bytriaversion of PrinFah Visit www.printfash for more infomation
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定理4.3设A为n阶非奇异矩阵，则A有唯一LDR分解 的充要条件是A的各阶顺序主子式△，…△均不为零. 此时，D=diag(d,d,,dn)的元素满足 4=△，4=Ak=2.…以 推论设A为n阶非奇异矩阵，则A有唯一的Doolittle分解或 Crout2分解的充要条件是A的各阶顺序主子式△，…△均不为零. 4.1.2三角分解的紧凑计算格式 计算Doolittle分解：以n=3为例 This documentis prduced byriaversion ofPrnh Visit www.prinashcfor more informion
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47 4243 422 21 1 u22 23 a33 2412 3 12141 121412+u2 1243+423 1312411131412+132u22 13i413+132423+33 由a,=4,→4，=4，(j=1,2,3) 由a=4h1→41=24：41=4k1→41=4 141 由a22=242+42→42=a2-l2142a23=l214g+423→423=a23-h214 由aa=%14a+l→a--42 1422 由a3=131413+l32423+43→43=43-(3143+l2423)
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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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