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Math Help - Eigenvectors and eigenvalues.

  1. #1
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    Eigenvectors and eigenvalues.

    Okay there's a question I've been working on all day.

    First it said: If v is an eigenvector of A with corresponding eigenvalue λ, show that v is an eigenvectors of A^-1. What is its corresponding eigenvalue?

    The answer is 1/λ.

    Then the second part of the question: Suppose a matrix A has real eigenvalues λ1 > λ2 > ..... > λn. Then was is lim k -> ∞A^kx0 for most initial vectors x0?

    (The x0 is supposed to be "x knot" )

    If anyone can help me with that second part I will appreciate it SO MUCH! Thanks
    Last edited by mr fantastic; April 28th 2010 at 07:06 PM. Reason: Re-titled.
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  2. #2
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    Quote Originally Posted by nicolem1051 View Post
    Okay there's a question I've been working on all day.

    First it said: If v is an eigenvector of A with corresponding eigenvalue λ, show that v is an eigenvectors of A^-1. What is its corresponding eigenvalue?

    The answer is 1/λ.

    Then the second part of the question: Suppose a matrix A has real eigenvalues λ1 > λ2 > ..... > λn. Then was is lim k -> ∞A^kx0 for most initial vectors x0?

    (The x0 is supposed to be "x knot" )

    If anyone can help me with that second part I will appreciate it SO MUCH! Thanks
    By any chance, does \lambda_1=1?
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  3. #3
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    No it doesn't say that it does. This question is just confusing to me!
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  4. #4
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    This problem is dealing with Markov Chains.

    A^k=XD^{k}X^{-1}\mathbf{x}_0

    If \lambda_1=1, then D^k=\begin{bmatrix}<br />
1^k & \hdots &  &  & 0\\ <br />
\vdots &  &  &  & \\ <br />
 &  & \ddots &  & \\ <br />
 &  &  &  & \\ <br />
0 &  &  &  & \lambda_n^k<br />
\end{bmatrix}\rightarrow\begin{bmatrix}<br />
1^{\infty} & \hdots &  &  & 0\\ <br />
\vdots &  &  &  & \\ <br />
 &  & \ddots &  & \\ <br />
 &  &  &  & \\ <br />
0 &  &  &  & \lambda_n^{\infty}<br />
\end{bmatrix}\rightarrow\begin{bmatrix}<br />
1 & \hdots &  &  & 0\\ <br />
\vdots &  &  &  & \\ <br />
 &  & \ddots &  & \\ <br />
 &  &  &  & \\ <br />
0 &  &  &  & 0<br />
\end{bmatrix}.

    Everything then would zero out except for the first time of D^K

    Therefore, A^k=\begin{bmatrix}<br />
\mathbf{x}_{1,1}*X^{-1}_{1,1}*D^k_{1,1}*X_{1,1}\\ <br />
\vdots\\ <br />
\\ <br />
\\ <br />
\mathbf{x}_{1,1}*X^{-1}_{1,1}*D^k_{1,1}*X_{n,1}<br />
\end{bmatrix} You would then have a nx1 matrix where the first term is the only surviving term.
    Last edited by dwsmith; April 29th 2010 at 06:13 AM.
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