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Rank of an idempotent matrix

  1. Jan 26, 2010 #1

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

    [tex]trace(A) = rank(A)[/tex]

    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.
  2. jcsd
  3. Jan 27, 2010 #2


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    Hint 1: an idempotent matrix is diagonalizable.
    Hint 2: the eigenvalues of an idempotent matrix are either 0 or 1.
  4. Jan 27, 2010 #3
    Thanks..yes, I thought of the identity matrix and it all made sense.
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