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MATHS----Matrix

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For a given square matrix A the predicted values of matrix B are:

predicted B=A(A'A)^(-1)A'B

why is the matrix C=A(A'A)^(-1)A' an idempotent and symmetric matrix? and is this matrix invertible?

  • MATHS----Matrix -

    Assuming (A'A) is invertible, then (A'A)-1 exists.

    A(A'A)-1A'
    By the property of inverse of product of matrices,
    (A'A)-1
    =A-1 A'-1

    Therefore
    C=A(A'A)-1A'
    =A(A-1 A'-1)A'
    =(A A-1) (A'-1A')
    = (I) (I)
    =I
    after application of associativity and the properties of inverse of matrices.
    Since I is idempotent and invertible, so is C.

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