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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
    vTivj
    (T is transpose)
    2) given any matrix M, build
    MMT
    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
    P-1
    are positive definite
    9) the matrix with blocks
    P1 M
    MT P2
    is positive definite
    10) the product
    aP
    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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