{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lecture36 - Exercises December 3 2010 1 If A is an...

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Exercises December 3, 2010 1. If A is an invertible matrix and λ is an eigenvalue of A , then show that λ - 1 is an eigenvalue of A - 1 . Note that you must prove that λ 6 = 0 first. 2. What is the relationship between the eigenvalues of A and those of A T . 3. Let T : C n C n be a linear transformation, let λ be an eigenvalue of T , and let p be a polynomial. (a) Show that p ( λ ) is an eigenvalue of p ( T ). (b) Now suppose that μ is an eigenvalue of p ( T ), show that there is an eigenvalue λ of T such that p ( λ ) = μ . (c) Are these statements true for linear transformations on R n . 4. This exercise is about the generalization of the dot product to C n . To distinguish it from the usual dot product we will call it the inner product. If ~ z , ~w are vectors in C n , then we define the inner product by h ~ z, ~w i = z 1 w 1 + · · · + z n w n . Remmeber that a + ib = a - ib . For ~v, ~w, ~ z C n and α C , prove the following facts (a) h ~ z + ~w,~v i = h ~ z,~v i + h ~w,~v i (b) h α~ z, ~w i = α h ~ z, ~w i (c) h ~w, ~ z i = h ~v, ~ z i (d) h ~w, ~w i = 0 if and only if ~w = ~ 0.
Image of page 1
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