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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.

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  • $\begingroup$ @RodrigodeAzevedo consider a real vector $v$ then $v^Tx_1x_1^Tv=|x_1^Tv|\geq 0$ $\endgroup$ – Frank Moses May 31 '18 at 7:57
  • $\begingroup$ is $w_k>0$ for all $k$? $\endgroup$ – Frank Moses May 31 '18 at 8:02
  • $\begingroup$ @RodrigodeAzevedo I think he assumes them to be positive definite $\endgroup$ – Frank Moses May 31 '18 at 8:03
  • $\begingroup$ @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 $k$. $\endgroup$ – Guillaume F. May 31 '18 at 8:05
  • $\begingroup$ BTW which paper is that? $\endgroup$ – Frank Moses May 31 '18 at 8:10

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