1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Linear Algebra Proof- Pleaseeeee help!

  1. Apr 29, 2010 #1
    1. The problem statement, all variables and given/known data

    Let A be an n x n matrix with eigenvalue [tex]\lambda[/tex]. Prove that [tex]\lambda[/tex]^2 is an eigenvalue of A^2 and that if v is an eigenvector of A, then v is also an eigenvector for A^2.

    2. Relevant equations


    3. The attempt at a solution
    (A*A)V=([tex]\lambda[/tex] * [tex]\lambda[/tex])v
    so then v will be an eigenvector to A^2 when [tex]\lambda[/tex]^2
  2. jcsd
  3. Apr 29, 2010 #2
    Seems you have the right track, although you could be a little more precise:

    Let v be an eigenvector of the nxn matrix A.

    We have Av = [tex]\lambda[/tex]v.
    Then A2v = A(Av) = A([tex]\lambda[/tex]v) = [tex]\lambda (\lambda[/tex]v). = [tex]\lambda^2[/tex]v.

    Hence [tex]\lambda^2[/tex] is an eigenvalue of A2 and v is an eigenvector corresponding to [tex]\lambda^2[/tex].
  4. Apr 29, 2010 #3
    thank you but how come A([tex]\lambda[/tex]v) can = [tex]\lambda[/tex]([tex]\lambda[/tex]v?
  5. Apr 29, 2010 #4
    Any nonzero scalar multiple of an eigenvector is also an eigenvector, and is associated with the same eigenvalue.
  6. Apr 30, 2010 #5


    User Avatar
    Homework Helper

    A(λv) = (by the linearity of A!) = λA(v) = λ λv = λ^2 v.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook