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    Idempotent and nilpotent matrix proofs

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

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    https://brainmass.com/math/linear-algebra/idempotent-nilpotent-matrix-proofs-218166

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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 ...

    Solution Summary

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

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