Linear Algebra with Applications (3rd Edition)

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Math 417 homework 10 Solutions Problem 1 (a) Show: if v 1 , v 2 are eigenvectors of a matrix A for the same eigenvalue λ , then their linear combinations α 1 v 1 + α 2 v 2 are also eigenvectors (except for those = 0) for that eigenvalue λ . Answer: A ( α 1 v 1 + α 2 v 2 ) = α 1 Av 1 + α 2 Av 2 = λα 1 v 1 + λα 2 v 2 = λ ( α 1 v 1 + α 2 v 2 ) . (b) Consider A = SDS - 1 where S = 1 0 2 0 - 1 0 2 0 0 , D = - 1 0 0 0 2 0 0 0 - 1 . (1) What are the eigenvalues of A ? What are the eigenvectors? [There is a hard way and an easy way to do this.] Answer: the eigenvalues are the diagonal entries of D ; - 1 is a double root and 2 is single. The eigenvectors are the columns of S . Column i is an eigenvector for the eigenvalue in column i row i of D . (2) Find an orthogonal diagonalization of A , i.e. an orthogonal 3 × 3 matrix U with A = UDU T . Answer: we already have a diagonalization, but S is not an orthogonal ma- trix. But part (a) tells us that the Gram-Schmidt algorithm can be used to
Image of page 1

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

View Full Document Right Arrow Icon
Image of page 2
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