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

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

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

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

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

Forme a matriz $$$P$$$, cuja coluna $$$i$$$ é o autovetor número $$$i$$$: $$$P = \left[\begin{array}{cc}1 & 1\\1 & 0\end{array}\right]$$$.

Forme a matriz diagonal $$$D$$$ cujo elemento na linha $$$i$$$, coluna $$$i$$$ é o autovalor nº $$$i$$$: $$$D = \left[\begin{array}{cc}1 & 0\\0 & 0\end{array}\right]$$$.

As matrizes $$$P$$$ e $$$D$$$ são tais que a matriz inicial $$$\left[\begin{array}{cc}0 & 1\\0 & 1\end{array}\right] = P D P^{-1}$$$.

$$$P^{-1} = \left[\begin{array}{cc}0 & 1\\1 & -1\end{array}\right]$$$ (para as etapas, consulte calculadora de matriz inversa.)

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

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

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