Definition talk:Determinant/Matrix

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Is it a good idea to define this concept (in a trivial manner, for now; not using exterior algebra and the like) for a linear transformation as well? It saves trouble in referring (one would need to invoke that $\det$ is invariant under change of basis of course). --Lord_Farin 08:05, 12 May 2012 (EDT)

I would say yes. --prime mover 08:43, 12 May 2012 (EDT)
Hmm, I just noticed some trouble appearing, in that one needs an inner product or so to define 'orthonormal basis' (or the determinant would be ill-defined). Of course, one can always pick an inner product on a finite dimensional vector space, but this is still allows for rescaling. So it will need to be defined with reference to a fixed inner product. --Lord_Farin 08:55, 12 May 2012 (EDT)

On the setting up of a disambiguation page

It's a suboptimal approach to create a disambiguation page before changing all the links. There are now a large number of outstanding pages with links to "determinant" which now all go to that disambiguation page. This is crappy.

If you feel strongly enough about the need to disambiguate a definition, then you'll feel strongly enough about it to do the hard work to change the links first. Just putting a fancy template on the page boasting gleefully that the disambiguation page has many incoming links is inadequate, especially when there hasn't even been a category defined for it yet. --prime mover (talk) 18:20, 8 January 2018 (EST)

This is no longer a disambiguation page, and has been set up the way it should have been. --prime mover (talk) 06:09, 21 April 2018 (EDT)

About the Underlying Structure of the Matrix

Should the page make explicit mention of the underlying structure of the matrix? For definition 2, addition and multiplication needs to make sense and, for definition 1, elements of the underlying structure need to be multipliable by $\map \sgn \lambda$ as well.

My current thinking is that to resolve definition 1, the underlying structure needs to be a ring with unity with $\map \sgn \lambda$ being $\map{\sgn_S}{\lambda}$ where:

  • $S$ is the underlying structure of the matrix
  • $\sgn_S$ is a function which returns $1_S$, $-1_S$, or $0_S$

Although I am not sure if $\sgn_S$ is a valid generalization of $\sgn$ or not.

Definition 2 just needs a ring, as far as I can tell (though I don't know what to do about the multiplication by $\paren{-1}^{n+1}$ at the end).

I should note that the current page, as it is, is meaningful, since the definition explicitly mentions that the underlying structure is one of the standard number systems if unspecified, but that does technically mean that the page defines the determinant only for them. (I am working under the assumption that the determinant is meaningful for arbitrary structures but that may not necessarily be the case - if that's not the case then this whole discussion is moot.) --Thuna (talk) 03:17, 29 March 2024 (UTC)

None of the source works I have seen so far construct such a determinant out of anything more general than numbers. Can you find a source work which addresses this? --prime mover (talk) 08:08, 29 March 2024 (UTC)
Just following the sources of Determinant of Matrix Product/Proof 1, I see that A Course in Group Theory, at page 236, takes the determinant of a matrix over an arbitrary field $F$. It doesn't define the determinant in such a case but it's at least an example of it being used, which, to me, legitimizes its existence. --Thuna (talk) 08:23, 29 March 2024 (UTC)