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All the ways to build positive definite matrices

  1. Dec 19, 2011 #1
    Often people asks how to obtain a positive definite matrix. I would like to make a list of all possible ways to generate positive definite matrices (I consider only square real matrices here). Please help me to complete it.

    Here M is any matrix, P any positive definite matrix and D any diagonal matrix.

    1) given a vector v, build a matrix with entries
    (T is transpose)
    2) given any matrix M, build
    3) any diagonal matrix with positive entries
    4) any diagonal dominant matrix with positive entries
    5) the product
    M D MT
    where D is a diagonal matrix with positive entries
    6) if P1,P2 are positive definite matrices then
    P1 P2
    P1 +P2
    are also positive definite
    7) if P is positive definite then also
    M P M-1
    are positive definite
    9) the matrix with blocks
    P1 M
    MT P2
    is positive definite
    10) the product
    of a positive scalar a times a positive definite matrix P
    11) any submatrix formed as the upper left square matrix of a positive definite matrix (principal minor) is also
    a positive definite matrix


    do you know other ways not trivially reconducible to one of the above?
    Last edited: Dec 19, 2011
  2. jcsd
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