new矩阵教案Ch4P1

返回1特征值的估计证：nnCAHUTUAniiiniit1212||||njiijniiitt212||||2||||FT定理1(Shur不等式)的特征值为设nnCA211212||||||||FninjijniiAa则,,,,21n.为正规矩阵且等号成立当且仅当A返回nii12||2||||FT令)(21),(21AACAABHH},,,,{,,21nCBA的特征值分别为且满足},,,,{},,,,{2121nniii2||||FA2||||HFUTU返回|,|||||21n12,n12.n定理2(Hirsch)的特征值为设nnCA则,,,2n,1|,|max||)1,ijjiian|,|max|Re|)2,ijjiibn|,|max|Im|)3,ijjiicn返回证：nnCA)2HHHHTUAUTAUU,)(21)(21TTUAAUBUUHHHH)(21)(21HHHHTTUAAUCUUniii122||||)1ninjija112||2,2||maxijjian||max||,ijjiian返回ninjijniininjijniicb1121211212|||Im||||Re|2,2||maxijjibn2,2||maxijjicn221112221112111111||||2|||2||2|||2nnniiijiijnnniiijiijjnijjijnijjibttc返回||max|Im|)3,ijjiicn||max|Re|)2,ijjiibn定理3(Bendixson)的任一特则设ARAnn,满足值i||max2)1(|Im|,ijjiicnn返回证：ninjijniic11212|||Im|jijiijc,2||2,||max)1(ijjicnnsiii122|Im|2|Im|2nii12|Im|2,||max)1(ijjicnn返回定理4则定义同上设,,,,,,iiinnCBCA11Im,Reinin证：)1||(||2xxAxi),(Axx),(xxi),(xxiiiHAxxiHHxAx为正规矩阵BDdiagBUUnH),,,(21返回xUDUxHHiRe)2,(xAAxH),(Bxx),(xUDUxHDyyHniiiy12||niiiniinyy12112||Re||1Rein返回定理4(Browne):的特征值为设nnCA则奇异值为,,,,212nn,1),,2,1(||1niin证：矩阵为HermiteAAHDdiagUUAAnHH),,,(22221HHHUUAUAUDHUAUBHBBDijintjtitbb21返回xBx),,2,1(1nixxbinttit),,2,1(1nixxbinttitniiintsnitsisitxxxxbb11,1)(niiiniiiixxxx1122122||in返回定理6(Hadamard不等式)则设,nnCA2/11121)]||([|det||)(|njniijniiaAA或的某一列全为且等号成立当且仅当,0A.的列向量彼此正交A证：),,,(21naaaA设线性相关naaa,,,)1210|det|A结论成立返回线性无关naaa,,,)221正交化11,11232131331212211nnnnnnbpbpbabpbpbabpbaba返回100101),,,(212121nnnpppbbbABbbbAndet),,,det(det21211,112||||||||iiiiiibpbpba211,2112||||||||||||||||iiiiibpbpb2||||ib返回BBBHdetdet|det|2BBHdetniib12||||niib12||)||(niia12||)||(2/11121)]||([|det||)(|njniijniiaAA