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Math Help - Eigenvalues

  1. #1
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    Eigenvalues

    Hello!

    Eigenvalues-eigenvalues.jpg
    Here is a problem that I have no idea how to solve. So can I ask you for help?

    And since I am a beginner, I would be grateful if you could give me a source, from which I can read more about solving such type of problems.

    Thank you very much in advance!
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  2. #2
    MHF Contributor

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    Re: Eigenvalues

    You have no idea how to solve this? Even if you are a beginner, if you are given a problem like this, you should know how to find eigenvalues- at least the definition of "eigenvalue".

    A number, \lambda, is said to be an "eigenvalue" of linear transformation A if and only if there exist a non-zero ("non-trivial") vector, v, satisfying Av= \lambda v. Obviously, v= 0 always satisfies that- the key here is "non-zero".

    That equation is the same as Av- \lambda v= (A- \lambda I)v= 0. Now, in terms of matrices, if A- \lambda I had an inverse, we could multiply on both sides by that inverse, (A- \lambda I)^{-1}(A- \lambda I)v= v= (A- \lambda I)0= 0 showing that v= 0, the "trivial" solution is the only solution.

    So \lambda is an eigenvalue of A if and only if A- \lambda I does not have an inverse- and that is true if and only if the determinant of A- \lambda I (written as a matrix) is 0. That is, any eigenvalue, \lambda must satisfy the "characteristic equation" \left|A- \lambda I\right|= 0

    Here, A= \begin{pmatrix}3 & 1 \\ -1 & -\mu\end{pmatrix} so A- \lambda I= \begin{pmatrix}3 & 1 \\ -1 & -\mu\end{pmatrix}- \lambda\begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}= \begin{pmatrix}3-\lambda & 1 \\ -1 & -\mu-\lambda\end{pmatrix}

    The "characteristic equation is
    \left|\begin{array}{cc}3-\lambda & 1 \\ -1 & \mu-\lambda\end{array}\right|= (3- \lambda)(\mu- \lambda)+ 1-= \lambda^2- (\mu+ 3)\lambda+ 1+ 3\mu= 0.

    That is a quadratic equation to be solved for \lambda. Using the quadratic formula to solve it will let you determine what values of \mu give two real, one real, or two complex solutions.
    Last edited by HallsofIvy; January 3rd 2012 at 11:30 AM.
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