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# soln3 - MATH 235/W08 Solutions for Assignment 3 Eigenvalues...

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MATH 235/W08 Solutions for Assignment 3 Eigenvalues & Eigenvectors Notes: 1) The trace of an n × n matrix A = [ a ij ], denoted tr( A ), is defined to be the sum of the diagonal elements, that is, tr( A ) = n i =1 a ii . 2) Our textbook’s definition of the characteristic polynomial of an n × n matrix A , p A ( λ ), is p A ( λ ) := det( A λI n ). Another commonly used definition is c A ( λ ) := det( λI n A ). These two polynomials are closely related through p A ( λ ) = ( 1) n c A ( λ ). 1. Consider a matrix of the form 2 a c a c 2 a + 2 c 2 a 2 b a 2 a + 2 b 2 a b c a c 2 a + b + 2 c (a) Verify that the two vectors ( 1 1 1 ) T and ( 1 0 1 ) T are eigen- vectors and find the corresponding eigenvalues. (b) Find a third eigenvalue and a corresponding eigenvector. BEGIN SOLUTION: (a) a=sym(’a’); b=sym(’b’); c=sym(’c’); A = [ 2*a-c a-c -2*a+2*c 2*a-2*b a -2*a+2*b 2*a-b-c a-c -2*a+b+2*c ]; v1=[1;1;1]; v2=[1;0;1]; Av1=A*v1; Av2=A*v2; disp(’the two eigenvalues are’) the two eigenvalues are l1=Av1(1) % since first component is 1 1

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l1 = a l2=Av2(1) % since first component is 1 l2 = c disp(’check the two eigenvalue-eigenvector equation’) check the two eigenvalue-eigenvector equation Av1-l1*v1 ans = 0 0 0 Av2-l2*v2 ans = 0 0 0 (b) disp(’using the trace we find the third eigenvalue’) using the trace we find the third eigenvalue l3=trace(A)-l1-l2 l3 = b disp(’now find the eigenspace of l3 from the nullspace of ’) now find the eigenspace of l3 from the nullspace of Al3I=A-l3*eye(3) Al3I = [ 2*a-b-c, a-c, -2*a+2*c] [ 2*a-2*b, a-b, -2*a+2*b] [ 2*a-b-c, a-c, -2*a+2*c] disp(’we could use rref’) 2
we could use rref rref(Al3I) ans = [ 1, 0, 0] [ 0, 1, -2] [ 0, 0, 0] disp(’or we could do the row elimination directly’) or we could do the row elimination directly Al3I(3,:)=Al3I(3,:)-Al3I(1,:) Al3I = [ 2*a-b-c, a-c, -2*a+2*c] [ 2*a-2*b, a-b, -2*a+2*b] [ 0, 0, 0] disp(’we could continue with the eliminations or by observation’) we could continue with the eliminations or by observation Al3I=rref(Al3I) Al3I = [ 1, 0, 0] [ 0, 1, -2] [ 0, 0, 0] v3= [-Al3I(1:2,3);1] v3 = 0 2 1 disp(’verify the third eigenvalue-eigenvector’) verify the third eigenvalue-eigenvector A*v3-l3*v3 ans = 0 0 0 3

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echo off END SOLUTION. 2. Let a, b R , b negationslash = 0, and let A = bracketleftbigg a b b a bracketrightbigg . Find the eigenvalues and eigenvectors of A . BEGIN SOLUTION: a=sym(’a’); b=sym(’b’); lam=sym(’lam’); A=[a -b;b a]; disp(’we could be lazy and use matlab’) we could be lazy and use matlab [v,d]=eig(A) v = [ i, -i] [ 1, 1] d = [ a+i*b, 0] [ 0, a-i*b] l1=d(1,1) l1 = a+i*b l2=d(2,2) l2 = 4
a-i*b disp(’charact. polynomial is’) charact. polynomial is p=det(A-lam*eye(2)) p = a^2-2*a*lam+lam^2+b^2 disp(’or ’) or q=poly(A) q = x^2-2*x*a+a^2+b^2 disp(’we could apply the quadratic formula to find the roots ’) we could apply the quadratic formula to find the roots orl1= (2*a + sqrt( 4*a^2 - 4* (a^2+b^2) ) )/2 orl1 = a+(-b^2)^(1/2) orl2= (2*a - sqrt( 4*a^2 - 4* (a^2+b^2) ) )/2 orl2 = a-(-b^2)^(1/2) disp(’find the eigenspaces from the nullspaces ’) find the eigenspaces from the nullspaces A-l1*eye(2) ans = [ -i*b, -b] [ b, -i*b] 5

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soln3 - MATH 235/W08 Solutions for Assignment 3 Eigenvalues...

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