Diagonalize Matrix Calculator

Diagonalize $$$\left[\begin{array}{ccc}1 & 1 & 3\\1 & 5 & 1\\3 & 1 & 1\end{array}\right]$$$.

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

Eigenvalue: $$$6$$$, eigenvector: $$$\left[\begin{array}{c}1\\2\\1\end{array}\right]$$$.

Eigenvalue: $$$3$$$, eigenvector: $$$\left[\begin{array}{c}1\\-1\\1\end{array}\right]$$$.

Eigenvalue: $$$-2$$$, eigenvector: $$$\left[\begin{array}{c}-1\\0\\1\end{array}\right]$$$.

Form the matrix $$$P$$$, whose column $$$i$$$ is eigenvector no. $$$i$$$: $$$P = \left[\begin{array}{ccc}1 & 1 & -1\\2 & -1 & 0\\1 & 1 & 1\end{array}\right]$$$.

Form the diagonal matrix $$$D$$$ whose element at row $$$i$$$, column $$$i$$$ is eigenvalue no. $$$i$$$: $$$D = \left[\begin{array}{ccc}6 & 0 & 0\\0 & 3 & 0\\0 & 0 & -2\end{array}\right]$$$.

The matrices $$$P$$$ and $$$D$$$ are such that the initial matrix $$$\left[\begin{array}{ccc}1 & 1 & 3\\1 & 5 & 1\\3 & 1 & 1\end{array}\right] = P D P^{-1}$$$.

$$$P^{-1} = \left[\begin{array}{ccc}\frac{1}{6} & \frac{1}{3} & \frac{1}{6}\\\frac{1}{3} & - \frac{1}{3} & \frac{1}{3}\\- \frac{1}{2} & 0 & \frac{1}{2}\end{array}\right]$$$ (for steps, see inverse matrix calculator).

$$$P = \left[\begin{array}{ccc}1 & 1 & -1\\2 & -1 & 0\\1 & 1 & 1\end{array}\right]$$$A

$$$D = \left[\begin{array}{ccc}6 & 0 & 0\\0 & 3 & 0\\0 & 0 & -2\end{array}\right]$$$A

$$$P^{-1} = \left[\begin{array}{ccc}\frac{1}{6} & \frac{1}{3} & \frac{1}{6}\\\frac{1}{3} & - \frac{1}{3} & \frac{1}{3}\\- \frac{1}{2} & 0 & \frac{1}{2}\end{array}\right]\approx \left[\begin{array}{ccc}0.166666666666667 & 0.333333333333333 & 0.166666666666667\\0.333333333333333 & -0.333333333333333 & 0.333333333333333\\-0.5 & 0 & 0.5\end{array}\right]$$$A