返回一、相似矩阵的基本概念二、 矩阵的相似对角化返回返回.**********AA求设矩阵???????????????一、相似矩阵的基本概念例返回?????????????????????????????????????????1210110211000100021010110211PPA110111110???????PPPPPPPPPPA??????????????????????????????????????????????121011021100010002**********?.1?与什么样的矩阵有这样的问题:P??.2???p解返回一、矩阵相似的定义与性质使矩阵阶矩阵,如果存在可逆都是与设,PnBABAPP??1.~BABA,记为相似与则称简单性质:??;~1AA反身性??;对称性ABBA~~2???.~~~3CACBBA?且传递性??113????QCQBPBPA:证??.111PQDDCDPPQCQA??????定义返回定理1相似矩阵有相同的特征值..,~APPBBA1??则设??PAIPAPPIBI??????????11PAIP????1AI???思考:相似矩阵有相同的行列式?证返回二、矩阵的相似对角化????????????????nA????21~,,则An????定理2返回nI??????????????21??????n???????????????,,,的全部特征值是:??,的特征值相同与?A?.21nA???,,,的全部特征值是:??证返回定理3n阶矩阵A与对角矩阵相似的充分必要条件是A有n个线性无关的特征向量.:个线性无关的特征向量有设充分性nAnPPP,,,21???nipAPiii,,2,1??????nPPPP?21?令???????????????n?????21?????APPPAP1则????????????????nA?????21~证返回?????????????????nAPP?????211设必要性?PAP?则??nPPPP?21?设????nnnPPPAPAPAP?????221121?则??niPAPiii,,2,1????.,,,21个线性无关的特征向量的是nAPPPn??返回定理4 矩阵A 不同特征值的特征向量线性无关.,,,,mmmAAA?????????????,,,且m???????????kkm时,设当??22112211????AkAkkkA???则??20222111???????kk????301212111??????kk:又由式??????0:322122??????k,0221?????且?,0,012???kk同理,.,21线性无关???.21线性无关,,,由归纳法可证:m????证
线性几何课件5.2矩阵的相似对角化 来自淘豆网m.daumloan.com转载请标明出处.
