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## Homework Statement

Show that the matrix

A = [cos θ -sin θ

sin θ cos θ]

will have complex eigenvalues if θ is not a multiple of π. Give a geometric interpretation of this result.

## Homework Equations

A

**x**= λ

**x**, so

det(A-λI) = 0

## The Attempt at a Solution

In this case,

A-λI = [(cos θ - λ) -sin θ

sin θ (cos θ - λ)]

so calculating the determinate, I have:

[itex]det(A-λI) = (cos θ - λ)(cos θ - λ) + sin^{2} θ [/itex]

[itex]= cos^{2} θ - 2λcos θ + λ^{2} + sin^{2} θ[/itex]

[itex]= λ^{2} - 2λcos θ + 1[/itex]

Setting this polynomial to zero, I have

[itex]

p(λ) = λ^{2} - 2λcos θ + 1 = 0

[/itex]

When cos θ is not a multiple of π, I need to use quadratic formula to solve this equation. Doing so, I get:

[itex]

λ = cos θ \pm sin θ

[/itex]

However, this value is not complex, is it? I'm a bit confused on what's going on.