Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Sequence of definite positive matrices

  1. Apr 10, 2010 #1
    if i have a sequence of definite positive matrices that converges, is it always that the limit matrix is always a definite positive matrix?
    if it's true, can someone please tell me why or link me to some proof?
    thank you.
  2. jcsd
  3. Apr 12, 2010 #2
    Hi jem05,

    in general I would say no. For instance let I be the identity matrix, which is positive definite, and let n be a natural number. The matrix I/n is positive definite, because for any vector x we have:

    x' (I/n) x = 1/n |x|^2

    which is positive if x is not null; however the limit of the sequence {I/n} is the null matrix, which is not positive definite, but it is at least positive semidefinite. And I feel like this is true in general, because if A[n] is a sequence of positive definite matrices that converges to A, then x'(A[n])x is a sequence of positive numbers if x is not null, and it cannot converge to a negative number. So we can write, since the limit of the product is the product of the limits:

    0 =< lim x'(A[n])x = x'(lim A[n])x = x'Ax

    Hence A is positive semidefinite.

    What do you think? Bye, hopefully that was helpful.

    Last edited: Apr 12, 2010
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook