# Weighted sums of positive semidefinite matrices are uniformly positive definite

Let

$$A_n = \sum_{k=1}^{n} {\bf x}_k{\bf x}_k^T$$

where ${\bf x}_k$ is a $p \times 1$ vector. Suppose that there is a $N \in \mathbb N$ such that the family of matrices $\{ A_n \}$ is uniformly positive definite for all $n > N$. Let $\{w_k\}$ be scalars uniformly bounded from above and away from zero. Is it true that we have

$$B_n = \sum_{k=1}^{n} w_k {\bf x}_k{\bf x}_k^T$$

also uniformly positive definite for $n > N$?

I have seen this result invoked in a paper but I've never seen it proven.

• @RodrigodeAzevedo consider a real vector $v$ then $v^Tx_1x_1^Tv=|x_1^Tv|\geq 0$ – Frank Moses May 31 '18 at 7:57
• is $w_k>0$ for all $k$? – Frank Moses May 31 '18 at 8:02
• @RodrigodeAzevedo I think he assumes them to be positive definite – Frank Moses May 31 '18 at 8:03
• @FrankMoses by $w_k$ bounded from above and away from zero I mean that there are a $L > 0$ and $U$ finite such that $L < w_k < U$ for all $k$. – Guillaume F. May 31 '18 at 8:05
• BTW which paper is that? – Frank Moses May 31 '18 at 8:10