Results 1 to 3 of 3

Math Help - Proof of two similar matrices have same eigenvalue and a question

  1. #1
    Senior Member x3bnm's Avatar
    Joined
    Nov 2009
    Posts
    300
    Thanks
    16

    Proof of two similar matrices have same eigenvalue and a question

    Theorem:

    I've a question regarding the proof of a theorem.

    There is a theorem in my linear algebra book that states:

    "If A and B are similar n x n matrices, then they have the same eigenvalues."


    Proof of this theorem:

    Let A and B be similar matrices so there exist an invertible matrix P such that B = P^{-1} A P

    By the properties of determinant it follows that:

    \begin{align*} \lvert{\lambda I - B}\rvert = \lvert \lambda I - P^{-1} A P \rvert =& \lvert P^{-1} \lambda I P - P^{-1} A P \rvert \\ =& \lvert P^{-1} (\lambda I - A) P \rvert \\ =& \lvert P^{-1} \rvert \lvert \lambda I - A \rvert \lvert P \rvert \\ =& \lvert P^{-1} \rvert \lvert P \rvert \lvert \lambda I - A \rvert \\ =& \lvert P^{-1} P \rvert \lvert \lambda I - A \rvert \\ =& \lvert \lambda I - A \rvert \end{align*}


    My question:

    My question is: why is that

     \lambda I  =   P^{-1} \lambda I P

    How do you proof that left hand side of this statement is equal to the right hand side?


    What property of linear algebra makes this true? I can't figure out the answer for this problem. Is it possible help me finding the answer?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5

    Re: Proof of two similar matrices have same eigenvalue and a question

    \text{det}(P^{-1}AP-\lambda I)=\text{det}(P^{-1}AP-\lambda P^{-1}IP)

    =\text{det}(P^{-1}(A-\lambda I)P)=\text{det}(P^{-1})\text{det}(A-\lambda I)\text{det}(P)

    =\text{det}(P^{-1})\text{det}(P)\text{det}(A-\lambda I)=\text{det}(I)\text{det}(A-\lambda I)=\text{det}(A-\lambda I)

    What is P^{-1}IP???

    It is just I.
    Last edited by dwsmith; December 3rd 2011 at 04:57 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member x3bnm's Avatar
    Joined
    Nov 2009
    Posts
    300
    Thanks
    16

    Re: Proof of two similar matrices have same eigenvalue and a question

    Quote Originally Posted by dwsmith View Post
    \text{det}(P^{-1}AP-\lambda I)=\text{det}(P^{-1}AP-\lambda P^{-1}IP)

    =\text{det}(P^{-1}(A-\lambda I)P)=\text{det}(P^{-1})\text{det}(A-\lambda I)\text{det}(P)

    =\text{det}(P^{-1})\text{det}(P)\text{det}(A-\lambda I)=\text{det}(I)\text{det}(A-\lambda I)=\text{det}(A-\lambda I)
    Thank you so much. I was thinking \lambda as a matrix(I don't know why I was thinking that). That was the error on my part. Sorry about that. Thanks for clarifying that using constant multiplication property of matrices.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Eigenvalue and matrices
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 10th 2011, 10:10 AM
  2. proof of similar matrices
    Posted in the Advanced Algebra Forum
    Replies: 8
    Last Post: June 30th 2011, 08:52 PM
  3. Finding matrices with a given eigenvector/eigenvalue
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: April 23rd 2011, 11:23 PM
  4. Eigenvalue algebra proof question
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 24th 2011, 03:19 PM
  5. question about similar matrices
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: October 31st 2010, 10:48 PM

/mathhelpforum @mathhelpforum