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Math Help - Find the eigenvalues

  1. #1
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    Find the eigenvalues

    Find the eigenvalues for the following matrix:
    A=
    1 2 2 2
    2 1 2 2
    2 2 1 2
    2 2 2 1

    det(A-λI)=
    1-λ 2 2 2
    2 1-λ 2 2
    2 2 1-λ 2
    2 2 2 1-λ

    Is there a shortcut to computing this? Or do I need to take (1-λ)det(3x3 matrix)-(2)det(3x3 matrix)+(2)det(3x3 matrix)-(2)det(3x3 matrix)?
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  2. #2
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    Re: Find the eigenvalues

    convert it into a diagonal matrix. all the leading diagonal elements form the eigenvalues.
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  3. #3
    Senior Member DeMath's Avatar
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    Re: Find the eigenvalues

    Use this

    \det (A + x) = \begin{vmatrix}a_{1,1} + x&a_{1,2} + x& \ldots &a_{1,n} + x \\ a_{2,1} + x&a_{2,2} + x& \ldots &a_{2,n} + x \\ \vdots & \vdots & \ddots & \vdots  \\ a_{n,1}+ x&a_{n,2} + x& \ldots &a_{n,n}+ x \end{vmatrix}= \det A + x \cdot \sum_{i = 1}^n \sum_{j = 1}^n A_{i,j}, where A_{i,j} - cofactors of the matrix A.

    Let A+x = \begin{bmatrix}1 - \lambda&2&2&2 \\  2&1 - \lambda&2&2 \\  2&2&{1 - \lambda }&2 \\  2&2&2&1 - \lambda\end{bmatrix} , where x=2 and A=\begin{bmatrix}- 1 - \lambda&0&0&0 \\  0&{ - 1 - \lambda }&0&0 \\  0&0&{ - 1 - \lambda }&0 \\  0&0&0&{ - 1 - \lambda } \end{bmatrix}.

    Since \det A= (-1-\lambda )^4= (\lambda  + 1)^4 and A_{i,j}= \begin{cases}0,&i \ne j\\ (-1-\lambda)^3,&i = j\end{cases}, it follows that

    \det (A + x) = \det A + 2\sum\limits_{i = 1}^4 \sum\limits_{j = 1}^4 A_{i,j}= (\lambda  + 1)^4+2 \cdot 4( -1-\lambda)^3= (\lambda + 1)^4-8( \lambda+1)^3=(\lambda-7)(\lambda  + 1)^3
    Last edited by DeMath; May 10th 2012 at 06:48 AM.
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  4. #4
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    Re: Find the eigenvalues

    Quote Originally Posted by saravananbs View Post
    convert it into a diagonal matrix. all the leading diagonal elements form the eigenvalues.
    Great- uh, how do you convert to a diagonal matrix without finding the eigenvalues first?
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  5. #5
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    Re: Find the eigenvalues

    by elementary transformation. we will get the upper triangular matrix, whose diagonal elements form the eigenvalues.

    is anythink wrong?
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