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Linear Algebra: How to represent this transformation as a matrix?

  1. Feb 16, 2010 #1
    1. The problem statement, all variables and given/known data

    Find the spectrum of the given linear operator T on V and find an eigenvector of T corresponding to each eigenvalue.


    V = R_{2x2}, T (\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22} \end{bmatrix}) = \begin{bmatrix}-2a_{11}-a_{12}&a_{11}\\a_{21}&2a_{22} \end{bmatrix}

    2. Relevant equations

    3. The attempt at a solution

    I'm confused about how to write the columns and rows of the transformation matrix.. do I do this:


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


    Or the transposed? Or something else?

    When I tried to get the spectrum (eigenvalues?) of this matrix I get {1, 2, -2} but the answer is {1,-1,2}, which doesn't work with the characteristic polynomial..
  2. jcsd
  3. Feb 16, 2010 #2


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    Science Advisor
    Homework Helper

    That's one way to write the matrix. Though to be absolutely clear you should say which columns correspond to which basis vector. But I get eigenvalues {1,-1,2} for your matrix. Maybe just check your eigenvalue calculation.
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