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Math Help - Positive definite symmetric matrix has LDL^T decomposition with positive diagonals

  1. #1
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    Positive definite symmetric matrix has LDL^T decomposition with positive diagonals

    QUESTION:

    Let G be an n \times n matrix with a factorization G = LDL^{T}, where L is a unit lower triangular matrix and D is a diagonal matrix. Show that G is positive definite if and only if D_{ii} > 0, for i=1,2,\ldots,n.

    ATTEMPT:

    Assume D_{ii}>0 for i=1,2,\ldots,n. Then D = D^{\frac{1}{2}} D^{\frac{1}{2}} is possible and
    x^T G x = x^T L D L^T x = x^T L D^{\frac{1}{2}} D^{\frac{1}{2}} L^T x = x^T (LD^{\frac{1}{2}}) (L D^{\frac{1}{2}})^T x = ( (LD^{\frac{1}{2}})^T x)^T (L D^{\frac{1}{2}})^T x = y^T y = \sum^{n}_{i=1} y_i^2 \geq 0
    where y = (LD^{\frac{1}{2}})^T x.
    Because LD^{\frac{1}{2}} has n pivots it is nonsingular and hence y = 0 if and only if x = 0. Therefore x^T G x > 0 for all x \ne 0.

    Assume x^T G x = x^T LDL^T x is positive definite. How can I prove that D has positive diagonals?
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  2. #2
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    Re: Positive definite symmetric matrix has LDL^T decomposition with positive diagonal

    Hey math2011.

    Isn't one of the definitions of a positive define matrix having positive eigenvalues?

    Positive-definite matrix - Wikipedia, the free encyclopedia
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    Re: Positive definite symmetric matrix has LDL^T decomposition with positive diagonal

    Quote Originally Posted by chiro View Post
    Hey math2011.

    Isn't one of the definitions of a positive define matrix having positive eigenvalues?

    Positive-definite matrix - Wikipedia, the free encyclopedia
    Yes, but how can I show that the diagonal entries of D are the eigenvalues of G?
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  4. #4
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    Re: Positive definite symmetric matrix has LDL^T decomposition with positive diagonal

    Is your L matrix the matrix of eigenvectors and also is that matrix orthonormal? If so then L^T = L^(-1) and you have a eigen-decomposition of your matrix G which means that D is your matrix of eigenvalues.
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  5. #5
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    Re: Positive definite symmetric matrix has LDL^T decomposition with positive diagonal

    I don't know if L is the matrix of eigenvectors. The problem does not state that explicitly. Are you suggesting that L actually is the matrix of eigenvectors? How can I prove it?
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    Re: Positive definite symmetric matrix has LDL^T decomposition with positive diagonal

    My suggestion is to show that LDL^t is the diagonal form by showing that L is non-singular (since it is triangular with non-zero determinant) and then use this to show that L is also orthonormal.

    For the ortho-normal aspect, if you can show L*L^t = I then you're done.

    By showing that L^t = L^(-1) and by using the assumption that the diagonalization produces a unique decomposition, then you have shown that this is indeed the diagonalization.
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  7. #7
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    Re: Positive definite symmetric matrix has LDL^T decomposition with positive diagonal

    Thanks for explaining but I still cannot see how to do it. L^t is a unit upper triangular matrix and L is a unit lower triangular matrix, L^t L does not necessarily equal to the identity matrix unless the L is special. The problem does not state any other information about L that directly say L is orthogonal.
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