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Math Help - positive definite matrix

  1. #1
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    positive definite matrix

    Let A be the following symmetric matrix
    A =
    [4 2 -2 2
    2 2 1 1
    -2 1 14 -1
    2 1 -1 x]

    find the values of x , such that A is positive definite.
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  2. #2
    Junior Member mathemanyak's Avatar
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    answer

    a symmetric matrix is a square matrix, A, that is equal to its transpose and An n n real symmetric matrix M is positive definite if zTMz > 0 for all non-zero vectors z with real entries (i.e. zRn), where zT denotes the transpose of z.
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  3. #3
    MHF Contributor

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    Quote Originally Posted by tinng View Post
    Let A be the following symmetric matrix
    A =
    [4 2 -2 2
    2 2 1 1
    -2 1 14 -1
    2 1 -1 x]

    find the values of x , such that A is positive definite.
    There is a theorem telling that A=(a_{ij})_{1\leq i,j\leq n} (here, n=4) is positive definite if, and only if, for k=1,\ldots,n the determinant of A_k=(a_{ij})_{1\leq i,j\leq k} is positive.

    If you know this, then it is easy. The determinants of A_1, A_2, A_3 are positive, they do not depend on x, and the determinant of A_4=A is found to be 36(x-1), hence it is positive iff x>1. So A is positive definite iff x>1.
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