第7章矩陣的特征值和特征向量(參考版)
2024-09-05 09:06本頁面
【正文】 數(shù) 學(xué) 系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS Givens旋轉(zhuǎn)變換 對(duì)稱陣 ),( ?qpQ 為正交陣 ????????????????????????1c o ss i ns i nc o s1),(????????qpQp列 q列 數(shù) 學(xué) 系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS 記： )(),(),( , )( ijTij bqpAQqpQBaA ??? ??則： ???????????????????????????????????????2s i n22c o s2s i nc o ss i n2s i ns i nc o s,c o ss i n,s i nc o s2222qqpppqqppqpqqqpppppqqqppppqipiqiiqqipipiipaaabbaaabaaabqpiaabbqpiaabb變換的目的是為了減少非對(duì)角元的分量，則 02s i n22c os ????? ?? qqpppqqppq aaabb數(shù) 學(xué) 系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS ?t a n,2??? taaaspqppqq記 則 ?????????1 , 0012 , 0 2ststst的按模較小根 所以： ?????????????dttct221s i n11c o s??數(shù) 學(xué) 系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS ???????????????????????0,qppqpqqqpppqppppqipiqiiqqipipiipbbtaabtaabqpicadabbqpidacabb數(shù) 學(xué) 系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS Jacobi迭代 取 p,q使 ijjipq aa ?? m a x，則 ),(),( )()1( ?? qpQAqpQA kTk ??定理： 若 A對(duì)稱，則 },{ 1)1( nk d ia gA ?? ???數(shù) 學(xué) 系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS 解 記 A(0)=A,取 p=1,q=2,apq(0)=a12(0)=2,于是有 例 用 Jacobi 方法計(jì)算對(duì)稱矩陣 的全部特征值 . ???????????612152224A )0(12)0(22)0(11 ????aaa? )1|/ ( |)s g n (, 2 ????? ???t7 8 8 2 0 )1(c o s 212 ??? ?t? 61 541 o ssi n, ??? ?? t從而有 數(shù) 學(xué) 系 University of Science and Technology of China DEPARTMENT OF MATHEMATICS 所以 再取 p=2,q=3,apq(1)=a23(1)=,類似地可得 ?????????????????????? ???10007 8 8 2 0 1 5 4 1 06 1 5 4 1 8 8 2 0 1000c o ssin0sinc o s)(1 ?????pqRR( 1 ) ( 0 )11 0 0 6. 56 15 52 2. 02 01 900. 96 1 2. 02 01 90 6T??
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