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

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

Pertama, cari nilai eigen dan vektor eigen (untuk langkah-langkahnya, lihat kalkulator nilai eigen dan vektor eigen).

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

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

Bentuk matriks $$$P$$$, di mana kolom ke-$$$i$$$ adalah vektor eigen ke-$$$i$$$: $$$P = \left[\begin{array}{cc}- \frac{3 + \sqrt{33}}{6} & \frac{-3 + \sqrt{33}}{6}\\1 & 1\end{array}\right]$$$.

Bentuk matriks diagonal $$$D$$$ yang elemennya pada baris $$$i$$$, kolom $$$i$$$ adalah nilai eigen ke-$$$i$$$: $$$D = \left[\begin{array}{cc}- \frac{-5 + \sqrt{33}}{2} & 0\\0 & \frac{5 + \sqrt{33}}{2}\end{array}\right]$$$.

Matriks $$$P$$$ dan $$$D$$$ sedemikian rupa sehingga matriks awal $$$\left[\begin{array}{cc}1 & 2\\3 & 4\end{array}\right] = P D P^{-1}$$$.

$$$P^{-1} = \left[\begin{array}{cc}- \frac{\sqrt{33}}{11} & - \frac{-11 + \sqrt{33}}{22}\\\frac{\sqrt{33}}{11} & \frac{\sqrt{33} + 11}{22}\end{array}\right]$$$ (untuk langkah-langkahnya, lihat kalkulator invers matriks).

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