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I wrote the following statement:

$A(t) = \sum_{i \le t} r(i) r(i)^\top + \alpha I_N$

where $r(i) \in \mathbb{R}^N $.

As a sum of positive definite matrices $r(i)r(i)^\top$ and a positive definite matrix $\alpha I_N$, $A(t)$ is positive definite and therefore invertible.

My teacher highlighted the following parts and told me that my statement was wrong:

"As a sum of positive definite matrices $r(i)r(i)^\top$ and a positive definite matrix $\alpha I_N$, $A(t)$ is positive definite and therefore invertible."

What I found out from here and here are that $\alpha$ must be greater than zero which I did not mention and $r(i)r(i)^\top$ is positive semidefinite. However, due to his highlighting and writing a biiiig F next to the whole sentence, I feel like there is something else which is fundamentally wrong.

I'd appreciate any help! Thank you!

Lindaa
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1 Answers1

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Example: if $N=2$ and $r(i)=(1,1)^T$, then $r(i)r(i)^T$ is not positive definite.

Fred
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  • Thanks for your answer! In your case, $r(i)r(i)^\top$ would be positive semidefinite. Does it affect the fact that A(i) is invertible? – Lindaa Mar 15 '19 at 14:46