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Math Help - problem in diagonalizing a matrix

  1. #1
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    problem in diagonalizing a matrix

    \begin{bmatrix}-1&-1&1\\0&-2&1\\0&0&-1\end{bmatrix}

    Eigenvalues of the matrix are -1,-1,-2.

    -1 with algebric multiplicity 2.

    Eigenvectors for -1 are \begin{bmatrix}1\\0\\0\end{bmatrix} and \begin{bmatrix}0\\1\\1\end{bmatrix}

    I am finding difficulty in getting eigenvectors for -2.

    \begin{bmatrix}1&-1&1\\0&0&1\\0&0&1\end{bmatrix}\begin{bmatrix}x\\y\  \z\end{bmatrix}= \begin{bmatrix}0\\0\\0\end{bmatrix}

    How to solve this.

    I am getting z=0,y=0,and x=y-z.

    Can anyone help me.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by kumaran5555 View Post
    I am getting z=0,y=0,and x=y-z.
    Why y=0 ?. We have:

    (x,y,z)=(y,y,0)=y(1,1,0)

    Regards.
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  3. #3
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    I am just lost !!
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  4. #4
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    You do NOT get y= 0.
    \begin{bmatrix}1&-1&1\\0&0&1\\0&0&1\end{bmatrix}\begin{bmatrix}x\\y\  \z\end{bmatrix}
    is equivalent to the equations x- y+ z= 0, z= 0, and z= 0. That is, the last two equations are both z= 0.

    With that, x- y+ 0= 0 so y= x. Any eigenvector is of the form <x, y, z>= <x, x, 0>= x<1, 1, 0>.
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