Timeline for Why is $\mathbf\Phi^{\top}\mathbf\Phi$ a positive definite matrix?
Current License: CC BY-SA 4.0
16 events
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| Nov 28, 2022 at 1:41 | history | edited | statmerkur | CC BY-SA 4.0 |
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| Oct 29, 2022 at 23:59 | history | edited | statmerkur | CC BY-SA 4.0 |
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| Oct 15, 2022 at 17:01 | comment | added | usεr11852 | Thank you for the edits. | |
| Oct 15, 2022 at 16:38 | comment | added | statmerkur | @User1865345 Thanks for the feedback. | |
| Oct 15, 2022 at 16:32 | comment | added | User1865345 | The post looks complete and concise, @statmerkur. | |
| Oct 15, 2022 at 16:30 | comment | added | User1865345 | @zzzhhh yes. They are linearly independent almost surely. | |
| Oct 15, 2022 at 16:21 | comment | added | statmerkur | @Sycorax made an edit. Hope it's clearer now. | |
| Oct 15, 2022 at 16:18 | history | edited | statmerkur | CC BY-SA 4.0 |
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| Oct 15, 2022 at 16:13 | comment | added | zzzhhh | Thank you for pointing out the condition that the columns of $\math\Phi$ should be linearly independent. This is true almost surely because the elements of the design matrix are (functions of) random variables. | |
| Oct 15, 2022 at 16:08 | vote | accept | zzzhhh | ||
| Oct 15, 2022 at 15:44 | comment | added | Sycorax♦ | This answer might be clearer if you state up front that the claim in the question is not true in general, unless we make an additional assumption about $\Phi$. | |
| Oct 15, 2022 at 15:42 | comment | added | statmerkur | @usεr11852 I do say that in my answer: "is positive definite if the columns of $\Phi$ are linearly independent" | |
| Oct 15, 2022 at 15:38 | comment | added | usεr11852 | You don't say that about $\Phi$ in your answer nor is mentioned in the OP's question. | |
| Oct 15, 2022 at 15:26 | comment | added | statmerkur | @usεr11852 No, positive definite. If $\Phi$ has full column rank then $\Phi v \neq 0_{ \mathbb{R}^N}$ and thus $v^\top \left(\Phi^\top\Phi\right) v = \|\Phi v \|_2^2 > 0$ for all $v \in \mathbb{R}^M \setminus \left\{ 0_{ \mathbb{R}^M} \right\}$ | |
| Oct 15, 2022 at 15:16 | comment | added | usεr11852 | semi-definite... | |
| Oct 15, 2022 at 14:56 | history | answered | statmerkur | CC BY-SA 4.0 |