矩陣教案Ch5P

1、2 矩陣函數(shù)矩陣函數(shù)定義定義設(shè)設(shè)冪冪級(jí)級(jí)數(shù)數(shù)且且當(dāng)當(dāng)收收斂斂半半徑徑為為 , r 0kkkzcrzzczfkkk |,)(0則稱(chēng)收斂的矩陣冪級(jí)則稱(chēng)收斂的矩陣冪級(jí)滿足滿足如果如果,)(rArCAnn 即即記記為為的的和和為為矩矩陣陣函函數(shù)數(shù)數(shù)數(shù)),(,0AfAakkk 即即冪冪級(jí)級(jí)數(shù)數(shù)收收斂斂于于時(shí)時(shí)),(,|zfrz 一、矩陣函數(shù)的定義一、矩陣函數(shù)的定義常用的矩陣函數(shù)：常用的矩陣函數(shù)：,)(0 kkkAcAfnnkkACAAke ,!1)1(0nnkkkCAAkA ,)!12()1(sin)2(012則得到則得到為參數(shù)為參數(shù)換為換為的方陣的方陣把把,)(tAtAAf.)()(0 kkkAtcA

2、tf1)(,)()4(01 ArAAEkk1)(,1)1()ln()5(01 ArAkAEkkk二、矩陣函數(shù)值的計(jì)算二、矩陣函數(shù)值的計(jì)算1、利用相似對(duì)角化、利用相似對(duì)角化:DdiagAPPn ),(211 設(shè)設(shè)nnkkkCAAkA ,)!2()1(cos)3(02 0)(kkkAcAf 01)(kkkPDPc10 PDcPkkk1001 PccPkknkkkk 11)()( PffPn 同理同理).(,),(),()(21tftftfPdiagAtfn 1例例.,163053064AteA求求設(shè)設(shè) :解解2)1)(2()det()1 AE1, 2321 對(duì)對(duì)應(yīng)應(yīng)的的特特征征向向量量：)2T)1

3、 , 1 , 1(:211 TT)1 , 0 , 0(,)0 , 1 , 2(:13232 101011021P ttttttttttttteeeeeeeeeeeee2202022222222212 PeeePetttAt2、Jordan 標(biāo)準(zhǔn)形法標(biāo)準(zhǔn)形法: 1)(kkikiJaJf),(211sJJJdiagJAPP 設(shè)設(shè) 111)1(111kkikikkimkimkkikkikCCCaii )()()!2(1)()()!1(1)(! 11)()2() 1(iimiiimiiiffmffmffii 01)(kkkPPJaAf10 PJaPkkk1001 PJaJaPkkskkkk11)()(

4、 PJfJfPs2例例.sin,00001000000000AA求求設(shè)設(shè) :解解標(biāo)標(biāo)準(zhǔn)準(zhǔn)形形化化為為Jordan)1A 0010,321JJJ iJsin)2 計(jì)計(jì)算算33210sin00cos! 110sinsin, 0sin, 0sinJJJJ 0000100000000000sin A3、數(shù)項(xiàng)級(jí)數(shù)求和法：、數(shù)項(xiàng)級(jí)數(shù)求和法：哈密爾頓哈密爾頓-凱萊定理：凱萊定理：nnPA 上上的的一一個(gè)個(gè)是是數(shù)數(shù)域域設(shè)設(shè)則則的的特特征征多多項(xiàng)項(xiàng)式式是是矩矩陣陣,|)(,AAEf 0)(0111 EbAbAbAAfnnnEbAbAbAnnn0111 )(0)(11)(1)1(0)1(11)1(11llnlnlnnnnbAbAbAEbAbAbA1100)()( nnnkkkcAcAcEcAcAf )(011EbAbnn11)(111)(111)(00)()()( nllnlnnlllnlllnAbccAbccEbcc三、矩陣函數(shù)的一些性質(zhì)三、矩陣函數(shù)的一些性質(zhì)性質(zhì)性質(zhì)1：.,BAABBAeeeeeBAAB 則則如如果