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Math Help - [SOLVED] Eigenvalues

  1. #1
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    [SOLVED] Eigenvalues

    Hi again

    Just after an answer check if that's OK

    The eigenvalues of Matrix A are -1,1 and 2. What are the eigenvalues of the Matrix (2A-3I)^-1

    I have got the eigenvalues of 3I to be 3,3,3 so in the equation and then the inverse gives me \frac{-1}{5}, -1, 1

    Is this correct?

    Thanks
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  2. #2
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    Hello
    Quote Originally Posted by Ian1779 View Post
    Hi again

    Just after an answer check if that's OK

    The eigenvalues of Matrix A are -1,1 and 2. What are the eigenvalues of the Matrix (2A-3I)^-1

    I have got the eigenvalues of 3I to be 3,3,3 so in the equation and then the inverse gives me \frac{-1}{5}, -1, 1

    Is this correct?

    Thanks
    Yes it is correct !!!
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Ian1779 View Post
    Hi again

    Just after an answer check if that's OK

    The eigenvalues of Matrix A are -1,1 and 2. What are the eigenvalues of the Matrix (2A-3I)^-1

    I have got the eigenvalues of 3I to be 3,3,3 so in the equation and then the inverse gives me \frac{-1}{5}, -1, 1

    Is this correct?

    Thanks
    I can't say anything about your shortcut, but after working out the matrices I get your result.

    -Dan
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  4. #4
    Moo
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    Quote Originally Posted by topsquark View Post
    I can't say anything about your shortcut, but after working out the matrices I get your result.

    -Dan
    The identity matrix is always diagonal in any basis.

    If a matrix A has these eigenvalues, then in a certain basis, the matrix A will be \begin{pmatrix}-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{pmatrix}

    Consider we're on this basis, we have :

    2A-3I=\begin{pmatrix} -5&0&0 \\ 0&-1&0 \\ 0&0&1 \end{pmatrix}

    And the inverse of a diagonal matrix is the diagonal matrix formed by the inverses of the original diagonal.
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