- #1
CanIExplore
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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?