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Positive Definite Matrices

  1. Dec 8, 2013 #1
    1. The problem statement, all variables and given/known data

    0dVsrxc.png

    2. The attempt at a solution

    So part a. makes sense to me, it basically comes down to

    A1 =
    1 -1 -1
    -1 1 1
    -1 1 1

    A2 =
    1 -1 -1
    -1 2 -2
    -1 -2 11

    I'm not sure how to approach part b. because the question doesn't make much sense to me. It's asking to show that f1 is a single perfect square, so my guess is that from A1 I have to derive an expression for f1 that is a single perfect square.

    Haven't really looked to hard at part c, yet.
     
  2. jcsd
  3. Dec 8, 2013 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Use Chpolesky factorization, which works for any positive-semidefinite matrix. (Many discussions assume positive definiteness, but if you go through the material carefully you can show that it applies as well to the semidefinite case.) See, eg., http://en.wikipedia.org/wiki/Cholesky_decomposition .
    In fact, using the Cholesky algorithm is by far the easiest way to test positive definitness or semidefinitness---much easier than dealing with determinants, for example.
     
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