(adsbygoogle = window.adsbygoogle || []).push({}); Lin Alg - Eigenvector Existence proof (More of a proof ?, than eigenvector ?)

Here is the question from my book:

Show that If [itex]\theta \in \mathbb{R}[/itex], then the matrix

[tex]A = \left(\begin{array}{cc}cos \theta & sin \theta \\ sin \theta & -cos \theta \end{array}\right)[/tex]

always has an eigenvector in [itex]\mathbb{R}^2[/itex], and in fact that there exists a vector [itex]v_{1}[/itex] such that [itex]Av_{1} = v_{1}[/itex]

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Now I am a little caught up on one thing here:

Is this saying that for any value of [itex]\theta \in \mathbb{R}[/itex], A has an eigenvector in [itex]\mathbb{R}^2[/itex] ?? (imo it does)

Because I am working a simple proof by contradiction, and if I can just choose a value of [itex]\theta[/itex] I would be done. (Except for the second part, but imo, that is simple to do, as I would just need to show that there is some eigenvalue = 1.)

Thanks!

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# Homework Help: Lin Alg - Eigenvector Existence proof

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