Diagonalizar $$$\left[\begin{array}{cc}1 & 2\\3 & 4\end{array}\right]$$$

Diagonalize $$$\left[\begin{array}{cc}1 & 2\\3 & 4\end{array}\right]$$$.

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

Autovalor: $$$- \frac{-5 + \sqrt{33}}{2}$$$, autovetor: $$$\left[\begin{array}{c}- \frac{3 + \sqrt{33}}{6}\\1\end{array}\right]$$$.

Autovalor: $$$\frac{5 + \sqrt{33}}{2}$$$, autovetor: $$$\left[\begin{array}{c}\frac{-3 + \sqrt{33}}{6}\\1\end{array}\right]$$$.

Forme a matriz $$$P$$$, cuja coluna $$$i$$$ é o autovetor número $$$i$$$: $$$P = \left[\begin{array}{cc}- \frac{3 + \sqrt{33}}{6} & \frac{-3 + \sqrt{33}}{6}\\1 & 1\end{array}\right]$$$.

Forme a matriz diagonal $$$D$$$ cujo elemento na linha $$$i$$$, coluna $$$i$$$ é o autovalor nº $$$i$$$: $$$D = \left[\begin{array}{cc}- \frac{-5 + \sqrt{33}}{2} & 0\\0 & \frac{5 + \sqrt{33}}{2}\end{array}\right]$$$.

As matrizes $$$P$$$ e $$$D$$$ são tais que a matriz inicial $$$\left[\begin{array}{cc}1 & 2\\3 & 4\end{array}\right] = P D P^{-1}$$$.

$$$P^{-1} = \left[\begin{array}{cc}- \frac{\sqrt{33}}{11} & - \frac{-11 + \sqrt{33}}{22}\\\frac{\sqrt{33}}{11} & \frac{\sqrt{33} + 11}{22}\end{array}\right]$$$ (para as etapas, consulte calculadora de matriz inversa.)

$$$P = \left[\begin{array}{cc}- \frac{3 + \sqrt{33}}{6} & \frac{-3 + \sqrt{33}}{6}\\1 & 1\end{array}\right]\approx \left[\begin{array}{cc}-1.457427107756338 & 0.457427107756338\\1 & 1\end{array}\right]$$$A

$$$D = \left[\begin{array}{cc}- \frac{-5 + \sqrt{33}}{2} & 0\\0 & \frac{5 + \sqrt{33}}{2}\end{array}\right]\approx \left[\begin{array}{cc}-0.372281323269014 & 0\\0 & 5.372281323269014\end{array}\right]$$$A

$$$P^{-1} = \left[\begin{array}{cc}- \frac{\sqrt{33}}{11} & - \frac{-11 + \sqrt{33}}{22}\\\frac{\sqrt{33}}{11} & \frac{\sqrt{33} + 11}{22}\end{array}\right]\approx \left[\begin{array}{cc}-0.522232967867094 & 0.238883516066453\\0.522232967867094 & 0.761116483933547\end{array}\right]$$$A