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

[zeion]

Homework Statement

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

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

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:

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

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..

 

[Dick]

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.