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Sagot :
The eigen vectors are :
[tex]\left[\begin{array}{ccc}-4&8\\-8&-4\end{array}\right][/tex] = 4[tex]\sqrt{5}[/tex] [tex]\left[\begin{array}{ccc}cos(arctan(2)+π)&−sin(arctan(2)+π)\\sin(arctan(2)+π)&cos(arctan(2)+π)\end{array}\right][/tex]
given that
M = [tex]\left[\begin{array}{ccc}-4&8\\-8&-4\end{array}\right][/tex]
using the P[tex]D^{n}[/tex][tex]P^{-1}[/tex] method
λ1=−4+8i with the respective eigenvector which leads [tex]\left[\begin{array}{ccc}-i\\1\end{array}\right][/tex] factoring out the i out giving
[tex]\left[\begin{array}{ccc}0\\1\end{array}\right][/tex] + i [tex]\left[\begin{array}{ccc}-1\\0\end{array}\right][/tex] which means P = [tex]\left[\begin{array}{ccc}0&1\\-1&0\end{array}\right][/tex] and [tex]P^{-1}[/tex] = [tex]\left[\begin{array}{ccc}0&-1\\1&0\end{array}\right][/tex]
D = 4[tex]\sqrt{5}[/tex] [tex]\left[\begin{array}{ccc}cos(arctan(2)+π)&−sin(arctan(2)+π)\\sin(arctan(2)+π)&cos(arctan(2)+π)\end{array}\right][/tex]
I means [tex]\pi[/tex]
[tex]\left[\begin{array}{ccc}-4&8\\-8&-4\end{array}\right][/tex] = 4[tex]\sqrt{5}[/tex] [tex]\left[\begin{array}{ccc}cos(arctan(2)+π)&−sin(arctan(2)+π)\\sin(arctan(2)+π)&cos(arctan(2)+π)\end{array}\right][/tex]
To learn more about eigenvector :
https://brainly.com/question/13050052
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