#Definition positive semi definite as implication

11 messages · Page 1 of 1 (latest)

nocturne barn
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I have a question about the definition in the picture. How it is stated here it is basically defined only in one direction or? (I dont care if this holds in both directions but how it is defined here, it is stated only in one direction right?)

So basically we could rewrite this definition in the form of the following implication or am I wrong?

file-UiPyJcyZM5KF8fEQbzJ1ryat.png
obsidian sundialBOT
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nocturne barn
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$P\coloneq\forall c\in\mathbb{R}^{m\times 1}: c^{T}Kc=\sum_{i,j=1} c_i c_jK_{ij}\geq0$\
$Q\coloneq\text{K is positive semi definite}$ \
$P\rightarrow Q$: \
$\text{If } \forall c\in\mathbb{R}^{m\times 1}\text{ it holds }c^{T}Kc=\sum_{i,j=1} c_i c_jK_{ij}\geq0\text{ then K is positive semi definite}$

frozen isleBOT
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Mathmatix

373218904992120846.png
supple forum
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Short answer: yes, that's correct

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Long answer: the word "if" in a definition is imo inappropriate and should be obsolete

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One should probably use the word "when" in definitions

nocturne barn
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Awesome thanks so far!

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+close