matlab讲义4.ppt

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1、Week4DeterminantInnerProductsGivenA=[aij]WhereMijisthesubmatrixobtainedbydeletingrowiandcolumnj.Whycomputedet(A)?det(A)=0AissingularDeterminantdet(A)=ad-bcFor3-by-3matrixdet(A)=aei+dhc+gbf–ceg–fha–ibdNottrueforgeneraln-by-nmatrix.For2-by-2matrixIfn=20,n!=21018operations

2、.Assuming109operations/sec.2109sec.=100yr.Sohowdet(A)iscomputed?FinditbyMatlab!Wn≑nWn-1≑n•(n-1)•Wn-2≑n!W1=n!1=n!Ifworkofcomputingdet(An)=Wn,thenn=500;A=rand(n);%generatean-by-nmatrixtic;%settheclocktozerov=det(A);%computethedeterminanttoc%showtheelapsedtimeTodetermineth

3、etimingntimeratio0.1090---0.73506.7420005.84407.95400043.50007.44Conclusion:det(An)=O(n3)LetVbeavectorspace.AfunctionF:VVCisaninnerproductifInnerProductsF(u,v)=ut•vInMatlab:ut•v=u’*vordot(u,v)Example2:V=M2x3(R)=setofall2-by-3realmatrices.Isitaninnerproduct?Example1:V=Rn

4、.2.1.3.4.5.ButThusF(A,B)maynotbeaninnerproduct.Toshowthatitisnotaninnerproduct,weneedtofindA,B,C,M23(R),suchthatF(A+B,C)