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

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Unformatted text preview: 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 textbooks definition of the characteristic polynomial of an n n matrix A , p A ( ), is p A ( ) := det( A I n ). Another commonly used definition is c A ( ) := det( I n A ). These two polynomials are closely related through p A ( ) = ( 1) n c A ( ). 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 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 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 = Av2-l2*v2 ans = (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 = 2 1 disp(verify the third eigenvalue-eigenvector) verify the third eigenvalue-eigenvector A*v3-l3*v3 ans = 3 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...
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soln3 - MATH 235/W08 Solutions for Assignment 3 Eigenvalues...

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