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# Idempotent and nilpotent matrix proofs

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A matrix pxp matrix A is idempotent if A^2=A...

##### Solution Summary

This provides examples of proving several statements about an idempotent and a nilpotent matrix.

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Problem #3
If is idempotent, then . Suppose is an eigenvalue of and is the corresponding eigenvector, then we have . Then we have

This implies that or .
So each eigenvalue of an idempotent matrix is either or .
If is a nilpotent matrix, then for some positive integer . Suppose is an eigenvalue and is the corresponding eigenvector, then we have . Then we have

This implies ...

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