Example find the eigenvalues and associated

Info icon This preview shows pages 3–5. Sign up to view the full content.

View Full Document Right Arrow Icon
Example : Find the eigenvalues and associated eigenvectors of the matrix A = 7 0 - 3 - 9 - 2 3 18 0 - 8 . First we compute det( A - λ I ) via a cofactor expansion along the second column: 7 - λ 0 - 3 - 9 - 2 - λ 3 18 0 - 8 - λ = ( - 2 - λ )( - 1) 4 7 - λ - 3 18 - 8 - λ = - (2 + λ )[(7 - λ )( - 8 - λ ) + 54] = - ( λ + 2)( λ 2 + λ - 2) = - ( λ + 2) 2 ( λ - 1) . Thus A has two distinct eigenvalues, λ 1 = - 2 and λ 3 = 1. (Note that we might say λ 2 = - 2, since, as a root, - 2 has multiplicity two. This is why we labelled the eigenvalue 1 as λ 3 .) Now, to find the associated eigenvectors, we solve the equation ( A - λ j I ) x = 0 for j = 1 , 2 , 3. Using the eigenvalue λ 3 = 1, we have ( A - I ) x = 6 x 1 - 3 x 3 - 9 x 1 - 3 x 2 + 3 x 3 18 x 1 - 9 x 3 = 0 0 0 x 3 = 2 x 1 and x 2 = x 3 - 3 x 1 x 3 = 2 x 1 and x 2 = - x 1 . 3
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
So the eigenvectors associated with λ 3 = 1 are all scalar multiples of u 3 = 1 - 1 2 . Now, to find eigenvectors associated with λ 1 = - 2 we solve ( A + 2 I ) x = 0 . We have ( A + 2 I ) x = 9 x 1 - 3 x 3 - 9 x 1 + 3 x 3 18 x 1 - 6 x 3 = 0 0 0 x 3 = 3 x 1 . Something different happened here in that we acquired no information about x 2 . In fact, we have found that x 2 can be chosen arbitrarily, and independently of x 1 and x 3 (whereas x 3 cannot be chosen independently of x 1 ). This allows us to choose two linearly independent eigenvectors associated with the eigenvalue λ = - 2, such as u 1 = (1 , 0 , 3) and u 2 = (1 , 1 , 3). It is a fact that all other eigenvectors associated with λ 2 = - 2 are in the span of these two; that is, all others can be written as linear combinations c 1 u 1 + c 2 u 2 using an appropriate choices of the constants c 1 and c 2 . Example : Find the eigenvalues and associated eigenvectors of the matrix A = - 1 2 0 - 1 . We compute det( A - λ I ) = - 1 - λ 2 0 - 1 - λ = ( λ + 1) 2 . Setting this equal to zero we get that λ = - 1 is a (repeated) eigenvalue. To find any associated eigenvectors we must solve for x = ( x 1 , x 2 ) so that ( A + I ) x = 0 ; that is, 0 2 0 0 x 1 x 2 = 2 x 2 0 = 0 0 x 2 = 0 . Thus, the eigenvectors corresponding to the eigenvalue λ = - 1 are the vectors whose second component is zero, which means that we are talking about all scalar multiples of u = (1 , 0). 4
Image of page 4
Notice that our work above shows that there are no eigenvectors associated with λ = - 1 which are linearly independent of u . This may go against your intuition based upon the results of the example before this one, where an eigenvalue of multiplicity two had two linearly independent associated eigenvectors. Nevertheless, it is a (somewhat disparaging)
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern