Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

True or false eigenvalue problem

  1. Aug 11, 2011 #1
    Q)Different eigenvectors corresponding to an eigenvalue of a matrix must be linearly dependant?
    Is the above statement true or false.Give reasons.
     
  2. jcsd
  3. Aug 11, 2011 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    What did you try already?? If we know what you tried then we'll know where to help??

    Begin by looking at some examples of matrices. Pick some arbitrary (easy) matrices and calculate its eigenvectors.
     
  4. Aug 12, 2011 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    I would NOT look at specific matrices. This can be done better more abstractly.

    Suppose u and v are non-zero eigenvectors, for linear operator A, corresponding to distinct eigenvalues [itex]\lambda_1[/itex] and [itex]\lambda_2[/itex] respectively. Then [itex]Au= \lambda_1 u[/itex] and [itex]Av= \lambda_2 v[/itex].

    Suppose they are not independent. Then [itex]v= \mu u[/itex] for some non-zero scalar [itex]\mu[/itex]. That gives [itex]Av= \mu Au[/itex] or [itex]\lambda_1 v= \mu \lambda_2 v[/itex] so that [itex] v= (\mu \lambda_2)/\lambda_1)u[/itex].

    But that means [itex](\mu \lambda_2)/\lambda_1= \mu[/itex] so that [itex]\lambda_2= \lambda_1[/itex], a contradiction.

    (Of course, this requires [itex]\lambda_1\ne 0[/itex] so we can divide by it. If [itex]\lambda_1= 0[/itex], just reverse the [itex]\lambda_1[/itex] and [itex]\lambda_2[/itex]. They cannot both be 0 because they are distinct.)

    Now, can you extend that to any number of independent eigenvectors?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: True or false eigenvalue problem
  1. Explain true of false (Replies: 1)

  2. True or False? (Replies: 27)

  3. True or false ? (Replies: 1)

Loading...