ISC 210 Computational Methods Eigenvalues and Eigenvectors Fall

ISC 210 Computational Methods Eigenvalues and Eigenvectors Fall 2010/2011 Dr. Jehad Al Dallal Eigenvalues Dr. Jehad Al Dallal

Eigenvalues n Eigen values of a square matrix A are values of such that n n Eigenvalues (A- I)X=[0] det(A- I)=0 Direct determinant expansion Indirect determinant expansion Dr. Jehad Al Dallal

Direct determinant expansion n n Given Q= Solution n Eigenvalues det(Q- I)= find the eigenvalues =0 2 -5 +4=0 (characteristic equation) ( -4)( -1)=0 -> =4 or 1 Dr. Jehad Al Dallal

Example 2 n n Eigenvalues Given Q= equation Solution find the characteristic n det(Q)= n |Q|= 3 -7 2+14 -8=0 Dr. Jehad Al Dallal

Indirect determinant expansion n Form the [Q] matrix for the eigenproblem n n n Eigenvalues [Q]=[A]- I Assume n+1 values of Evaluate the corresponding determinants of the [Q] matrix Form an equivalent set of linear algebraic equations. Substitute the determinant values and the assume values and solve the resulting system to fine the coefficients of the characteristic equation. Use the characteristic equation to find the eigenvalues. Dr. Jehad Al Dallal

Example 1 n n Determine the characteristic equation corresponding to the following matrix using the indirect procedure Solution n Eigenvalues |Q|= Dr. Jehad Al Dallal

Solution (cont’d) n Step 1 n n Step 2 n n n =3 1=2 -> f(2)=-3 2=3 -> f(3)=-3 f( 0)=a 0+a 1 0+a 2 02 -> 3=a 0 f( 1)=a 0+a 1 1+a 2 12 -> -3=a 0+2 a 1+4 a 2 f( 2)=a 0+a 1 2+a 2 22 -> -3=a 0+3 a 1+9 a 2 Step 4 n n Eigenvalues 0=0 -> f(0)= Step 3 n n Assume 0=0, 1=2, 2=3 a 0=3, a 1=-5, a 2=1 f( )= 2 -5 +3 Dr. Jehad Al Dallal

Example 1 (cont’d) n Try different initial values. n n n Try 0, 1, 4 f(0)=3, f(1)=-1, f(4)=-1 Equations n n n Eigenvalues 3=a 0 -1=a 0+a 1+a 2 -1=a 0+4 a 1+16 a 2 a 0=3, a 1=-5, a 2=1 f( )= 2 -5 +3 Dr. Jehad Al Dallal

Example 2 n Eigenvalues Determine the characteristic equation corresponding to the following matrix using the indirect procedure Dr. Jehad Al Dallal

Solution n Step 1 n n Step 2 n n a 0=-860, a 1=-212, a 2=160, a 3=-23, a 4=1 f( )= 4 -23 3+160 2 -212 -860 Try another initial values n Eigenvalues f(0)=a 0 … Step 4 n n f(0)=-860, f(5)=-170, f(6)=-44, f(10)=20, f(12)=628 Step 3 n n Initial values 0, 5, 6, 10, 12 0, 10, 20, 30, 40 Dr. Jehad Al Dallal

Eigenvectors n To find the eigenvectors of a matrix A apply: n n Eigenvalues (A- I)X=[0] Find the X vector for each value of Dr. Jehad Al Dallal

Example 1 n Find the eigenvectors for the matrix: n Solution: n n n Eigenvalues 2 -5 +4=0 (characteristic equation) ( -4)( -1)=0 -> =4 or 1 Solve Dr. Jehad Al Dallal

Example 2 n Find the eigenvectors for the matrix: n Solution n n Eigenvalues 2 -5 +3=0 -> =4. 3 or 0. 7 solve Dr. Jehad Al Dallal

Example 3 n Find the eigenvectors for the matrix: n Solution n n 3 -7 2+14 -8=0 -> = 4 or 2 or 1 Solve n n n Eigenvalues =4 -> x 1[1 -2 1]T =2 -> x 1[1 0 -1]T =1 -> x 1[1 1 1]T Dr. Jehad Al Dallal