Is trace equal to rank for idempotent matrices?

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SUMMARY

For an idempotent matrix A, defined by the property A2 = A, it is established that trace(A) equals rank(A). This relationship arises because the eigenvalues of an idempotent matrix are exclusively 0 or 1, leading to a direct correlation between the trace, which sums the eigenvalues, and the rank, which counts the number of non-zero eigenvalues. Further exploration of identities linking trace and rank can be found in advanced matrix analysis literature.

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maverick280857
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Hi

Is it true that for an idempotent matrix A (satisfying A^2 = A), we have

trace(A) = rank(A)

Where can I find more general identities or rather, relationships between trace and rank? I did not encounter such things in my linear algebra course. I'm taking a course on regression analysis this semester and that's where I ran into it.

I'd appreciate if someone could point me to a book on matrix analysis or inference where these things would be mentioned in some detail. For some reason, the more "practically relevant" results were not covered in my freshman math courses.

Thanks in advance.
Cheers
Vivek.
 
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Hint 1: an idempotent matrix is diagonalizable.
Hint 2: the eigenvalues of an idempotent matrix are either 0 or 1.
 
radou said:
Hint 1: an idempotent matrix is diagonalizable.
Hint 2: the eigenvalues of an idempotent matrix are either 0 or 1.

Thanks..yes, I thought of the identity matrix and it all made sense.
 

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