1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Concavity Of The Determinant

  1. Jul 11, 2012 #1
    1. The problem statement, all variables and given/known data

    How can I prove that given two [itex] nXn [/itex] positive semi-definite matrices [itex]A,B [/itex], then the following inequality holds:

    [itex] det(A+B)^\frac{1}{n} \geq det(A) ^\frac{1}{n} + det(B)^\frac{1}{n} [/itex]

    2. Relevant equations

    Brunn-Minkowski Inequality:
    http://en.wikipedia.org/wiki/Brunn–Minkowski_theorem

    3. The attempt at a solution
    I've tried proving that these kind of determinants represent volumes of n-dimensional compact bodies, but without any success.. Is there any algebraic/computational way of doing it?


    Thanks in advance !
     
  2. jcsd
  3. Jul 11, 2012 #2
    Maybe you can use that because [itex]A[/itex] and [itex]B[/itex] are positive semi-definite, they can be written as the Gram-matrix of certain vectors [itex]a_1, a_2, ..., a_n[/itex] and [itex]b_1, b_2, ..., b_n[/itex] respectively, and the determinant of a Gram-matrix is the content (or volume) squared of the [itex]n[/itex]-dimensional box spanned by the vectors of which it is composed? Then maybe the Brunn-Minkowski inequality? This seems intuitively right to me, but it's not a rigorous argument!
     
  4. Jul 12, 2012 #3

    chiro

    User Avatar
    Science Advisor

    Hey Combinatorics.

    On this page:

    http://en.wikipedia.org/wiki/Positive-definite_matrix#Further_properties

    It says on 11. the property of convexity for the matrices themselves. For the actual convexity condition, looking at this says that the set of all elements of the C*-algebra form a convex cone:

    http://en.wikipedia.org/wiki/Positive_operator.

    It looks like anything that is well-ordered in a sense is what is called a positive cone and my guess is that we establish the well-ordered property for the positive-definite matrices and then go from there to establish convexity.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Concavity Of The Determinant
  1. Concave Functions (Replies: 3)

Loading...