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## Homework Statement

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

[tex]

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}

[/tex]

## Homework Equations

## The Attempt at a Solution

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

[tex]

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

[/tex]

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..