[OS X TeX] Latex symbol for "define equal"
Markus Klyver
markusklyver at hotmail.com
Tue Aug 21 08:48:48 EDT 2018
Depends, literature tends to use the definitions interchangeably depending on what aspect is important to subject.
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Från: MacOSX-TeX <macosx-tex-bounces at email.esm.psu.edu> för Martin Berggren <martin.berggren at cs.umu.se>
Skickat: den 21 augusti 2018 13:59
Till: TeX on Mac OS X Mailing List
Ämne: Re: [OS X TeX] Latex symbol for "define equal"
On 21 Aug 2018, at 13:47, Markus Klyver <markusklyver at hotmail.com<mailto:markusklyver at hotmail.com>> wrote:
The thing is that you can have several different definitions, all equivalent. Consider the definition "a matrix A \in \mathbb R^{n \times n} is invertiable iff A have a multiplicative inverse". It turns out that this is equivalent to a lot of things, among det(A) !=0, A having full rang, A having n linearindependent eigenvectors, Ax=0 only having the trivial solution, Ax=b having a solution (which is unique) for every right-hand-side b, etc.
I would save that this is a theorem, not a definition. I think of a definition as a “macro”; that is, giving a short name to a mathematical property. Example: a matrix A is called positive semidefinite when x^T Ax \geq 0 for all vectors x. The point is that you in each instances when the name is used, it can be replaced by its definition.
Sincerely,
Martin Berggren
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