Diagonalize Matrix Calculator

The calculator will diagonalize the given matrix, with steps shown.

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Your Input

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

Solution

First, the find 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 $$$\left[\begin{array}{ccc}1 & 1 & 3\\1 & 5 & 1\\3 & 1 & 1\end{array}\right] = P D P^{-1}$$$.

Answer

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

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