Find the eigenvalues of the linear transformation L : R2x2 ! R2x2 defined by L(A) = A^T + A.
I'm having trouble with LaTex, but are the eigenvalues 2a and 2d?
It seems to be $\displaystyle L:M_2(\mathbb{R})\rightarrow M_2(\mathbb{R})\,,\,\,L(A):= A^t+A$ , which is clearly linear. Let us choose now the following basis of $\displaystyle M_2(\mathbb{R})$ :
$\displaystyle B=\left\{\begin{pmatrix}1&0\\0&0\end{pmatrix}\,,\, \begin{pmatrix}0&1\\0&0\end{pmatrix}\,,\,\begin{pm atrix}0&0\\1&0\end{pmatrix}\,,\,\begin{pmatrix}0&0 \\0&1\end{pmatrix}\right\}$ , $\displaystyle L\begin{pmatrix}1&0\\0&0\end{pmatrix}=\begin{pmatr ix}2&0\\0&0\end{pmatrix}\,,\,L\begin{pmatrix}0&1\\ 0&0\end{pmatrix}=\begin{pmatrix}0&1\\1&0\end{pmatr ix}\,,\,L\begin{pmatrix}0&0\\1&0\end{pmatrix}$ $\displaystyle =\begin{pmatrix}0&1\\1&0\end{pmatrix}\,,\,L\begin{ pmatrix}0&0\\0&1\end{pmatrix}=\begin{pmatrix}0&0\\ 0&2\end{pmatrix}$ , so
the matrix of $\displaystyle L\,\,\,wrt\,\,\,B$ is:
$\displaystyle [L]_B=\begin{pmatrix}2&0&0&0\\0&1&1&0\\0&1&1&0\\0&0&0 &2\end{pmatrix}$ , and this matrix's char. pol. is:
$\displaystyle p(x):=\det(xI-A)=\left|\begin{pmatrix}x-2&0&0&0\\0&x-1&-1&0\\0&-1&x-1&0\\0&0&0&x-2\end{pmatrix}\right|$ $\displaystyle =x(x-2)^3$
Tonio