矩陣運(yùn)算ppt課件

1、數(shù)數(shù)=線線性性代代1.2 矩陣的運(yùn)算定義定義1.3 矩陣加法矩陣加法對(duì)于對(duì)于nmijaA)(nmijbB)(,規(guī)定規(guī)定nmijijbaBA)(如如8885202342016543211.2 矩陣的運(yùn)算矩陣的運(yùn)算留意：留意：.111112不能相加不能相加與與例如，例如， BA數(shù)數(shù)=線線性性代代1.2 矩陣的計(jì)算定義定義1.4 數(shù)乘矩陣數(shù)乘矩陣對(duì)于對(duì)于nmijaA)(,k為常數(shù),規(guī)定規(guī)定kA(或記為或記為Ak)nmijka)(kBkABAklAkAAlkAklkAlAA)()()()(1數(shù)乘矩陣的運(yùn)算滿足數(shù)乘矩陣的運(yùn)算滿足:123246如 245681012數(shù)數(shù)=線線性性代代1.2 矩陣的計(jì)算定義

2、定義1.5 矩陣乘法矩陣乘法nsijsmijbBaA)(,)(規(guī)定規(guī)定nmijcCAB)(,其中skkjiksjisjijiijbabababac12211闡明闡明(1)AB有意義有意義,必需是必需是A的列數(shù)等于的列數(shù)等于B的行數(shù)的行數(shù)(2)當(dāng)當(dāng)AB有意義時(shí)有意義時(shí),AB的行數(shù)等于的行數(shù)等于A的行數(shù)的行數(shù),AB的列數(shù)等于的列數(shù)等于B的列數(shù)的列數(shù)如如,例例1.1中的旋轉(zhuǎn)變換可寫成中的旋轉(zhuǎn)變換可寫成yxyxcossinsincos或yxyxcossinsincos數(shù)數(shù)=線線性性代代1.2 矩陣的計(jì)算例例:設(shè)設(shè)ABBA求,3211,21432173911553211214321:AB解但是但是 BA

3、無意義無意義111246121111,00340034003311110011111211341234,003400123412O 但 是例例 : :數(shù)數(shù)=線線性性代代1.2 矩陣的計(jì)算1111222233331 12233:abbaabbaabbaa ba ba b 例小結(jié)小結(jié): (1)矩陣乘法無交換律矩陣乘法無交換律,有有“左乘和左乘和“右乘之分右乘之分 (2)AB=0,未必有未必有A=0或或B=0 (3)AB=AC,且且A0時(shí)未必有時(shí)未必有B=C數(shù)數(shù)=線線性性代代1.2 矩陣的計(jì)算矩陣乘法滿足以下運(yùn)算規(guī)律矩陣乘法滿足以下運(yùn)算規(guī)律BCACCBAACABCBAABkkBABkABCACABO

4、BOOOAAIAAIpmpnnmpmpnnmnnmnmm)(6()()5()()()(4()()(3(,)2()1(.),(),6(,)(,)()(,)(,)(,)(: )6(元相等即可只要證兩端的明故要證矩陣亦為矩陣為則設(shè)下證jipmBCACpmCBAbaBAcCbBaAnmijijpnijnmijnmij數(shù)數(shù)=線線性性代代1111()( ,)()()nikikk jknikk jikk jknnikk jikk jkkABCijabcacbcacbc的元1.2矩陣的計(jì)算上式右端第上式右端第1項(xiàng)為項(xiàng)為AC的的(i,j)元元,上式右端第上式右端第2項(xiàng)為項(xiàng)為BC的的(i,j)元元,故上式右端為故

5、上式右端為AC+BC的的(i,j)元元(6)式兩端的式兩端的(i,j)元相等元相等pjmi, 2 , 1;, 2 , 1證畢數(shù)數(shù)=線線性性代代1.2 矩陣的計(jì)算方陣的冪方陣的冪AAAAAAAAAAIAAmnn,210規(guī)定對(duì)于m個(gè)個(gè)但對(duì)于同階方陣但對(duì)于同階方陣A、B來說來說,以下等式不一定成立：以下等式不一定成立：22222)(;2)( ;)(BABABABABABABAABmmm這些等式在這些等式在ABBA時(shí)才成立時(shí)才成立,()klk lklklA AAAA(k,l為非負(fù)整數(shù)為非負(fù)整數(shù))方陣的冪滿足：方陣的冪滿足：數(shù)數(shù)=線線性性代代1.2 矩陣的計(jì)算11,(2 , 3 ,)01nAAn 求23

6、2111112010101121113010101101nAAAAnA 猜 測(cè)112,0111111010101kkkknAkkAAA 時(shí) 成 立設(shè)成 立則故 結(jié) 論 成 立例例解：解：證之證之?dāng)?shù)數(shù)=線線性性代代1.2 矩陣的計(jì)算定義定義1.6 矩陣的轉(zhuǎn)置矩陣的轉(zhuǎn)置的轉(zhuǎn)置矩陣為矩陣稱對(duì)于AaaaaaaaaaAmnaAmnnnmmTnmij212221212111,)(TTTTTTTTTTABABkAkABABAAA)(4()(3()(2()(1(轉(zhuǎn)置的運(yùn)算規(guī)律數(shù)數(shù)=線線性性代代1.2 矩陣的計(jì)算可將可將(4)推行為：推行為：TTTmTmAAAAAA1221)()()()TTTTTTTABCAB

7、 CCABC B A例如 BCAAICAAIBATT求例2,2100219 . 1()(2)222()12()2TTTTTTTTTTTBCIAAIAAIAAAAAAAAIAAAAAAIAAAAI解：解：數(shù)數(shù)=線線性性代代1.2 矩陣的計(jì)算21410041210021210021TAA其中對(duì)稱矩陣對(duì)稱矩陣:njiaaAAaAjiijTnnij, 1,)(、其中或滿足同階對(duì)稱矩陣之和、差及數(shù)乘對(duì)稱矩陣，所得同階對(duì)稱矩陣之和、差及數(shù)乘對(duì)稱矩陣，所得矩陣仍為對(duì)稱矩陣，但對(duì)稱矩陣之積卻不一定矩陣仍為對(duì)稱矩陣，但對(duì)稱矩陣之積卻不一定是對(duì)稱矩陣。是對(duì)稱矩陣。數(shù)數(shù)=線線性性代代1.2 矩陣的計(jì)算反對(duì)稱矩陣：反

8、對(duì)稱矩陣：032301210), 1(0,),2, 1,(,)(BnibbbjinjibbBBbBiiiiiijiijTnnij如有故當(dāng)或滿足例：例： 試證：任一試證：任一n階方陣階方陣A可表示為一個(gè)對(duì)稱矩陣可表示為一個(gè)對(duì)稱矩陣 與一反對(duì)稱矩陣之和與一反對(duì)稱矩陣之和證：證：11(),()2211()() )2211()()22TTTTTTTTTTBAACAABAAAAAAAAB令 則 =即即B為對(duì)稱矩陣為對(duì)稱矩陣數(shù)數(shù)=線線性性代代nmnmmmnnnnmnijxaxaxayxaxaxayxaxaxayyyxxnjmia2211222212121212111111,), 1;, 1(10. 1的變

9、換到變量，稱變量為常數(shù)設(shè)線性變換與矩陣?yán)?81 ( 11()() )2211()()22TTTTTTTTAAAAAAAAC 而 C =即即C為反對(duì)稱矩陣為反對(duì)稱矩陣111111()()222222TTTTBCAAAAAAAAA且 1.2 矩陣的計(jì)算數(shù)數(shù)=線線性性代代1.2 矩陣的計(jì)算為由n維向量到m維向量的一個(gè)線性變換.利用矩陣乘法可將(1-8)式寫成mnnnmijmnmnnnmmmFyFxxxxxaAyyyyAxyxxxaaaaaaaaayyy,)(,21212121222121211121其中或數(shù)數(shù)=線線性性代代1.2 矩陣的計(jì)算FkFkTkTFTTTTFxAxxTFFnnnmn,),()

10、(,),()()(,)(,:)81 (滿足由此可知其定義為為記變換)()()()()()(kTkTTTAAAT同理有事實(shí)上數(shù)數(shù)=線線性性代代1.2 矩陣的計(jì)算矩陣為其中定義為矩陣為其中定義為設(shè)mpBFyByySFFSnmAFxAxxTFFTmpmnmn,)(:,)(:那么復(fù)合線性變換那么復(fù)合線性變換ST為為xBAAxBAxSxTSxST)()()()()(nFx ,故故ST的矩陣為的矩陣為BA數(shù)數(shù)=線線性性代代1.2 矩陣的計(jì)算例例1.11 線性方程組與矩陣線性方程組與矩陣mnmnmmnnnnnbxaxaxabxaxaxabxaxaxaxxxnmnm2211222221211121211121),(組成個(gè)未知量個(gè)方程、由線性方程組)121 ( mnmnmmnnbbbxxxaaaaaaaaa2121212222111211bAx )131 ( 或數(shù)數(shù)=線線性性代代1.2