Diagonalizar calculadora matricial

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

Primeiro, encontre os autovalores e autovetores (para ver as etapas, consulte calculadora de autovalores e autovetores).

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

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

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

Forme a matriz $$$P$$$, cuja coluna $$$i$$$ é o autovetor no. $$$i$$$: $$$P = \left[\begin{array}{ccc}1 & 1 & -1\\2 & -1 & 0\\1 & 1 & 1\end{array}\right]$$$.

Forme a matriz diagonal $$$D$$$ cujo elemento na linha $$$i$$$, coluna $$$i$$$ é o autovalor nº. $$$i$$$: $$$D = \left[\begin{array}{ccc}6 & 0 & 0\\0 & 3 & 0\\0 & 0 & -2\end{array}\right]$$$.

As matrizes $$$P$$$ e $$$D$$$ são tais que a matriz inicial $$$\left[\begin{array}{ccc}1 & 1 & 3\\1 & 5 & 1\\3 & 1 & 1\end{array}\right] = P D P^{-1}$$$.

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

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