Sequence of definite positive matrices

[jem05]

hello,
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.

 

[damix]

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.

Dario