Non-symmetric matrix has non-orthogonal but non-repearting eigenvectors, but positive eigenvalues? Can anyone explain to me why?

Question

Consider the following matrix:

$$ \left(\begin{matrix} 9 & 4 \\ 8 & 9 \\ \end{matrix} \right) $$ Its eigenvalues are given by [courtesy of http://www.bluebit.gr/matrix-calculator/]

(14.657, 0.000i)
( 3.343, 0.000i)

and eigenvectors by [courtesy of http://www.bluebit.gr/matrix-calculator/]

( 0.577, 0.000i) (-0.577, 0.000i)
( 0.816, 0.000i) ( 0.816, 0.000i)

Why are the eigenvectors nor orthogonal? Clearly V1*V1 = 0.577*0.577 + 0.816*0.816 = 0.998785 (aprrox 1), by V1*v2 = 0.577*-0.577 + 0.816*0.816 = 0.332927.

Answer 1

You don't need numeric calculations, it's a $2\times 2$ matrix.

The two eigenvalues are $9\pm 4\sqrt{2}$, and the two eigenvectors are

$$\frac{\sqrt{3}}{3}\begin{pmatrix}1\\\pm\sqrt 2\end{pmatrix}$$

and the two eigenvectors are not orthogonal.

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There is also no reason to expect that they will be orthogonal, because the matrix is not normal