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## Homework Statement

The problem amounts to finding the eigenvalues of the matrix

|0 1 0|

|0 0 1|

|1 0 0|

(I have no idea how to set up a matrix in the latex format, if anyone can tell me that'd be great)

## Homework Equations

The characteristic equation for this matrix is

[itex]\lambda^{3}=1[/itex]

## The Attempt at a Solution

The solution to this problem can be found on grephysics.net.

The characteristic equation can be solved by noting that

1=e[itex]^{2\pi i}[/itex]

Using this fact, the eigenvalues as noted in the solution are

[itex]\lambda_{n}=e^{\frac{2\pi i n}{3}}[/itex], (n=1,2,3)

What I don't understand, is how one goes from

[itex]\lambda^{3}=e^{2\pi i}[/itex]

to

[itex]\lambda_{n}=e^{\frac{2\pi i n}{3}}[/itex]

If [itex]\lambda^{3}=e^{2\pi i}[/itex] then we can take both sides to the power of [itex]\frac{1}{3}[/itex] to get [itex]\lambda=e^{\frac{2\pi i}{3}}[/itex]. But how can you just throw the n in the exponent and call these (n=1,2,3) the 3 eigenvalues?