# Thread: M 2x2 matrices of C with standard basis.

1. ## M 2x2 matrices of C with standard basis.

$T(A)=\begin{bmatrix}0&1\\1&0\end{bmatrix}\times A\times\begin{bmatrix}0&1\\1&0\end{bmatrix}$

The matrix of transformation is

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

Eigenvalues are

$\lambda=1,1,-1,-1$

So finding the eigenspace.

When $\lambda=1$,
$\begin{bmatrix}-1&0&0&1\\0&-1&1&0\\0&1&-1&0\\1&0&0&-1\end{bmatrix}\Rightarrow\begin{bmatrix}-1&0&0&1\\0&-1&1&0\\0&0&0&0\\0&0&0&0\end{bmatrix}$

So does this mean the Eigenspace is for lambda = 1 is

$\begin{bmatrix}1&0\\0&1\end{bmatrix}, \ \begin{bmatrix}0&1\\1&0\end{bmatrix}$

2. ## Re: M 2x2 matrices of C with standard basis.

Originally Posted by dwsmith
$T(A)=\begin{bmatrix}0&1\\1&0\end{bmatrix}\times A\times\begin{bmatrix}0&1\\1&0\end{bmatrix}$

... does this mean the Eigenspace is for lambda = 1 is

$\begin{bmatrix}1&0\\0&1\end{bmatrix}, \ \begin{bmatrix}0&1\\1&0\end{bmatrix}$
Yes it does (or rather, the eigenspace is the set of linear combinations of those two matrices). You can check that by taking A to be either of those matrices and verifying that T(A) = A.