Positive Definiteness of a Real Matrix

  1. 1. The problem statement, all variables and given/known data

    Decide for or against the positive definiteness of

    [2 -1 -1
    -1 2 -1 = A
    -1 -1 2]

    [2 -1 -1
    -1 2 1 = B
    -1 1 2]

    [5 2 1
    2 2 2 = C
    1 2 5]

    2. Relevant equations

    Each of the following tests is a necessary and sufficient condition for the real symmetric matrix A to be positive definite:
    a) xTAx greater than 0 for all nonzero real vectors x.
    b) All the eigenvalues of A are greater than 0
    c) All the upper left submatrices of A have positive determinants
    d) All the pivots (without row exchanges) are greater than 0.

    3. The attempt at a solution

    For matrix A,
    I found that it fails tests b,c and d. I'm a bit confused because when I performed test a with vector x = [ 1 2 3 ] ^T the test passes, but with an x = [1 1 1]^T the test fails. Therefore, I said that it is not positive definite, but I'm unsure on this one.

    For matrices B and C, I said that they are both positive definite because they both pass test c. I'm assuming that if it passes one of the tests it is sufficient.

    Am I thinking correctly? Thanks!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. I like Serena

    I like Serena 6,194
    Homework Helper

    Hi tatianaiistb! :smile:

    Yes, it is sufficient if a matrix passes one of the tests.
    Each test is equivalent to each other test.

    Note that for test (a) the test has to pass for ALL nonzero real vectors.
    In other words, this is not a practical test.
     
  4. Ray Vickson

    Ray Vickson 6,170
    Science Advisor
    Homework Helper

    Note: (a) is the definition of positive-definiteness; it is not a test at all.
     
  5. So, if it fails one test it is sufficient to say that it is not positive definite, and viceversa? Thanks!!!!
     
  6. I like Serena

    I like Serena 6,194
    Homework Helper

    Yep!
     
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