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Differential Equations

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Eigenvalues and eigenvectors of 2x2 matrix

DIFFEQ-2BLQGX

Find the eigenvalues and corresponding eigenvectors of the following matrix:

$$A=\left[\begin{array}{rr}1&2\\\ -2&1\end{array}\right]$$

Express the eigenvectors as unit vectors.

A

$1+2i,\left(\cfrac{1}{\sqrt{2}},\cfrac{1}{\sqrt{2}}\right)$;$1-2i,\left(\frac{1}{\sqrt{2}},-\cfrac{1}{\sqrt{2}}\right)$

B

$1+2i,\left(\cfrac{1}{\sqrt{2}},\cfrac{i}{\sqrt{2}}\right)$;$1-2i,\left(\cfrac{1}{\sqrt{2}},-\cfrac{i}{\sqrt{2}}\right)$

C

$2i,\left(\cfrac{1}{\sqrt{2}},\cfrac{i}{\sqrt{2}}\right)$;$-2i,\left(\cfrac{1}{\sqrt{2}},-\cfrac{i}{\sqrt{2}}\right)$

D

$1,\left(\cfrac{1}{\sqrt{2}},\cfrac{1}{\sqrt{2}}\right)$;$-1,\left(\cfrac{1}{\sqrt{2}},-\cfrac{1}{\sqrt{2}}\right)$