1. ## Eigenvalue question

Q1: Let $\displaystyle T:M_{2,2}\rightarrow{M_{2,2}}$ be represented by

$\displaystyle T\left(\begin{bmatrix}a&b\\c&d\end{bmatrix}\right)$$\displaystyle = \begin{bmatrix}a-c+d&b+d\\-2a+2c-2d&2b+2d\end{bmatrix} . Find the eigenvaulues and eigenvectors of \displaystyle T relative to the standard basis \displaystyle B= \displaystyle \begin{bmatrix}1&0\\0&0\end{bmatrix}, \begin{bmatrix}0&1\\0&0\end{bmatrix}, \begin{bmatrix}0&0\\1&0\end{bmatrix}, \begin{bmatrix}0&0\\0&1\end{bmatrix} . A: I evaluated \displaystyle T at each vector in the basis... \displaystyle T\left(\begin{bmatrix}1&0\\0&0\end{bmatrix}\right)$$\displaystyle = a-c+d$
$\displaystyle T\left(\begin{bmatrix}0&1\\0&0\end{bmatrix}\right)$$\displaystyle = b+d \displaystyle T\left(\begin{bmatrix}0&0\\1&0\end{bmatrix}\right)$$\displaystyle = -2a+2c-2d$
$\displaystyle T\left(\begin{bmatrix}0&0\\0&1\end{bmatrix}\right)$$\displaystyle = 2b+2d$

Then, I created a matrix $\displaystyle A$ from the above information...

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

I then found my eigenvauls to be $\displaystyle \lambda=0$ and $\displaystyle \lambda=3$ which is in the back of the book.

But, when I evaluate $\displaystyle \lambda{I}-A$ for either value, I end up with $\displaystyle 4\times{1}$ column vector's. Both eigenvectors in the book are $\displaystyle 2\times{2}$ matrices.

Where did I mess up?

2. Remember that the basis $\displaystyle B$ consists of the four matrices $\displaystyle \begin{bmatrix}1&0\\0&0\end{bmatrix}, \begin{bmatrix}0&1\\0&0\end{bmatrix}, \begin{bmatrix}0&0\\1&0\end{bmatrix}, \begin{bmatrix}0&0\\0&1\end{bmatrix}$. So if your eigenvector calculation leads to a column vector with entries p,q,r,s, then those four numbers are actually the coefficients in a linear combination of the basis vectors. So the eigenvector is the matrix $\displaystyle p\begin{bmatrix}1&0\\0&0\end{bmatrix} +q\begin{bmatrix}0&1\\0&0\end{bmatrix} +r\begin{bmatrix}0&0\\1&0\end{bmatrix} +s\begin{bmatrix}0&0\\0&1\end{bmatrix} =\begin{bmatrix}p&q\\r&s\end{bmatrix}$.