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Homework Statement
A square matrix ##n\times n##, A, that isn't the zero-matrix have powers ##A^{k-1}## that isn't the zero matrix. ##A^k## is the zero matrix. What are the possible values for ##k##?
Homework Equations
N/A
The Attempt at a Solution
I'm a bit lost here but I figure that maybe I could write the matrix as an eigenvalue decomposition.
##A = PDP^{-1}## and then I get ##A^k = PD^kP^{-1}##.
Since ##D^k## is diagonal we could say that it's weighting the columns of ##P##. So what happends when we get ##A^k = \mathbf 0## is that some rescaling of the columns in ##P## makes ##PP^{-1} = \mathbf{0}##.
That's the only thing I manage to reason out, as far as actually calculating ##k## I'm clueless where to even begin. Any hints on where to start?