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

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

Etsi ensin ominaisarvot ja ominaisvektorit (vaiheet: katso ominaisarvojen ja ominaisvektorien laskin).

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

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

Muodosta matriisi $$$P$$$, jonka sarake $$$i$$$ on ominaisvektori numero $$$i$$$: $$$P = \left[\begin{array}{cc}- \frac{3 + \sqrt{33}}{6} & \frac{-3 + \sqrt{33}}{6}\\1 & 1\end{array}\right]$$$.

Muodosta diagonaalimatriisi $$$D$$$, jonka rivin $$$i$$$, sarakkeen $$$i$$$ alkio on ominaisarvo numero $$$i$$$: $$$D = \left[\begin{array}{cc}- \frac{-5 + \sqrt{33}}{2} & 0\\0 & \frac{5 + \sqrt{33}}{2}\end{array}\right]$$$.

Matriisit $$$P$$$ ja $$$D$$$ ovat sellaiset, että alkuperäinen matriisi $$$\left[\begin{array}{cc}1 & 2\\3 & 4\end{array}\right] = P D P^{-1}$$$.

$$$P^{-1} = \left[\begin{array}{cc}- \frac{\sqrt{33}}{11} & - \frac{-11 + \sqrt{33}}{22}\\\frac{\sqrt{33}}{11} & \frac{\sqrt{33} + 11}{22}\end{array}\right]$$$ (vaiheet: katso käänteismatriisilaskin.)

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