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Math Help - matrix trace and idempotent

  1. #1
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    matrix trace and idempotent

    X(X'X)^{-1}X'

    X is an n x K non-singular full rank matrix where n > K

    Is the result of this matrix operation an idempotent matrix? (It seems to be when I do it in stata.)

    Given the properties below, I am told it is possible to express the trace of the matrix that results from the above operation in terms of n and K. But I have been unable to figure out how. Can someone give me a clue?

    (A')^{-1}=(A^{-1})'

    tr(ABC)=tr(CAB)

    (where A,B, and C are non-singular square matrices)
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  2. #2
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    Quote Originally Posted by rainer View Post
    X(X'X)^{-1}X'

    X is an n x K non-singular full rank matrix where n > K

    Is the result of this matrix operation an idempotent matrix? (It seems to be when I do it in stata.)

    Given the properties below, I am told it is possible to express the trace of the matrix that results from the above operation in terms of n and K. But I have been unable to figure out how. Can someone give me a clue?

    (A')^{-1}=(A^{-1})'

    tr(ABC)=tr(CAB)

    (where A,B, and C are non-singular square matrices)


    What is X' , for a square matrix??

    Tonio
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    Quote Originally Posted by tonio View Post
    What is X' , for a square matrix??

    Tonio
    But X is not square. It is n-by-K, where n>K.
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  4. #4
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    Quote Originally Posted by rainer View Post
    But X is not square. It is n-by-K, where n>K.

    Whatever: what is X' , anyway?

    Tonio
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  5. #5
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    Quote Originally Posted by tonio View Post
    Whatever: what is X' , anyway?

    Tonio
    X' is the transpose of X.

    X'X yields a square matrix. I think that's part of the trick.
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