How could we prove that the "The trace of an idempotent matrix equals the rank of the matrix"?
This is another property that is used in my module without any proof, could anybody tell me how to prove this one?
2026-03-27 17:57:33.1774634253
Proving: "The trace of an idempotent matrix equals the rank of the matrix"
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An idempotent has two possible eigenvalues, zero and one, and the multiplicity of one as an eigenvalue is precisely the rank. Therefore the trace, being the sum of the eigenvalues, is the rank (assuming your field contains $\mathbb Q$...)