科学计算科学计算 (12).pdf
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1、The mathematical definition of matrix eigenvalues.Find matrix eigenvalues and eigenvectorsGeometric significance of eigenvalues2 2.4 4 MMatrixatrix E Eigenvaluesigenvalues andand E EigenvectorsigenvectorsLet A be a square matrix with n rows and n columns.If there is a constant and an n-dimensional n
2、on-zero column vector x,which makes the equation Ax=x hold,then is called the eigenvalue of A,and x is the eigenvector corresponding to theeigenvalue。1 1Mathematical Mathematical D Definition of efinition of MMatrix atrix E Eigenvalueigenvalues sE=eig(A):finds all eigenvalues of matrix Ato form vect
3、or E.X,D=eig(A):finds all eigenvalues of matrix A to form diagonal matrix D,andgenerates matrix X.Each column of X is a corresponding eigenvector.2 2Find Find MMatrix atrix E Eigenvalues and igenvalues and E EigenvectorsigenvectorsIn MATLAB,the eig function is used to calculate matrix eigenvalues an
4、d eigenvectors.There are two commonly used call forms.A=1,1,0;1,0,5;1,10,2A=1 1 01 0 51 10 2 X,D=eig(A)X=0.0722 0.9751 0.08860.5234 -0.0750 -0.63560.8490 -0.2089 0.7669D=8.2493 0 00 0.9231 00 0 -6.1723 A*X(:,1)ans=0.59564.31747.0040 D(1)*X(:,1)ans=0.59564.31747.0040Example 1:Let=33322322Let be the e
5、igenvalue of R and the eigenvalue of S,=1,2,3 be the eigenvector of R corresponding to,j=1,2 be the eigenvector of S corresponding to,try to verify:(1),are eigenvalues of A.(2)=1,2,3,0,0 s the eigenvector of A corresponding to,j=0,0,0,1,2is the eigenvector of A corresponding to.R=-1,2,0;2,-4,1;1,1,-
6、6;S=1,2;2,3;A=R,zeros(3,2);zeros(2,3),S;X1,d1=eig(R)X2,d2=eig(S)X3,d3=eig(A)X3=0.8553 0.4517 0.1899 0 00.4703 -0.8395 -0.5111 0 00.2173 -0.3021 0.8383 0 00 0 0 -0.8507 -0.52570 0 0 0.5257 -0.8507d3=0.0996 0 0 0 00 -4.7165 0 0 00 0 -6.3832 0 00 0 0 -0.2361 00 0 0 0 4.2361X1=0.8553 0.4517 0.18990.4703
7、 -0.8395 -0.51110.2173 -0.3021 0.8383d1=0.0996 0 00 -4.7165 00 0 -6.3832X2=-0.8507 0.52570.5257 0.8507d2=-0.2361 00 4.2361The eigenvalues of matrix A are made up of the eigenvalues of matrix R and matrix S.According to the eigenvector of each eigenvalue of matrix A,the first three elements of thefir
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