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If not, what has to be done differently?

- Thread starter dmoney123
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- #1

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If not, what has to be done differently?

- #2

Deveno

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- #3

chiro

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Deveno

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the characteristic polynomial of:

[tex]\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}[/tex]

is:

[tex]x^2 - (2\cos\theta)x + 1[/tex]

which has real solutions only when:

[tex]4\cos^2\theta - 4 \geq 0 \implies \cos\theta = \pm 1[/tex]

for angles that aren't integer multiples of pi, this will lead to complex eigenvalues.

- #5

AlephZero

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... and the eigenvalues are [itex]\cos\theta \pm i \sin\theta[/itex]. Now, I wonder what that fact might have to do with "rotation"...the characteristic polynomial of:

[tex]\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}[/tex]

is:

[tex]x^2 - (2\cos\theta)x + 1[/tex]

- #6

chiro

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What has that got to do with what I said?the characteristic polynomial of:

[tex]\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}[/tex]

is:

[tex]x^2 - (2\cos\theta)x + 1[/tex]

which has real solutions only when:

[tex]4\cos^2\theta - 4 \geq 0 \implies \cos\theta = \pm 1[/tex]

for angles that aren't integer multiples of pi, this will lead to complex eigenvalues.

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