Diagonalize $$$\left[\begin{array}{cc}0 & 1\\1 & 0\end{array}\right]$$$

Diagonalize $$$\left[\begin{array}{cc}0 & 1\\1 & 0\end{array}\right]$$$.

First, find the eigenvalues and eigenvectors (for steps, see eigenvalues and eigenvectors calculator).

Eigenvalue: $$$-1$$$, eigenvector: $$$\left[\begin{array}{c}-1\\1\end{array}\right]$$$.

Eigenvalue: $$$1$$$, eigenvector: $$$\left[\begin{array}{c}1\\1\end{array}\right]$$$.

Form the matrix $$$P$$$, whose column $$$i$$$ is eigenvector no. $$$i$$$: $$$P = \left[\begin{array}{cc}-1 & 1\\1 & 1\end{array}\right]$$$.

Form the diagonal matrix $$$D$$$ whose element at row $$$i$$$, column $$$i$$$ is eigenvalue no. $$$i$$$: $$$D = \left[\begin{array}{cc}-1 & 0\\0 & 1\end{array}\right]$$$.

The matrices $$$P$$$ and $$$D$$$ are such that the initial matrix $$$\left[\begin{array}{cc}0 & 1\\1 & 0\end{array}\right] = P D P^{-1}$$$.

$$$P^{-1} = \left[\begin{array}{cc}- \frac{1}{2} & \frac{1}{2}\\\frac{1}{2} & \frac{1}{2}\end{array}\right]$$$ (for steps, see inverse matrix calculator).

$$$P = \left[\begin{array}{cc}-1 & 1\\1 & 1\end{array}\right]$$$A

$$$D = \left[\begin{array}{cc}-1 & 0\\0 & 1\end{array}\right]$$$A

$$$P^{-1} = \left[\begin{array}{cc}- \frac{1}{2} & \frac{1}{2}\\\frac{1}{2} & \frac{1}{2}\end{array}\right] = \left[\begin{array}{cc}-0.5 & 0.5\\0.5 & 0.5\end{array}\right]$$$A