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A matrix pxp matrix A is idempotent if A^2=A...© BrainMass Inc. brainmass.com December 15, 2022, 7:24 pm ad1c9bdddf
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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 ...
This provides examples of proving several statements about an idempotent and a nilpotent matrix.