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Subject: Re: What is the determinant of [] ?
Date: Mon, 25 Oct 2010 17:54:04 +0000 (UTC)
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"Bruno Luong" <b.luong@fogale.findmycountry> wrote in message <ia3t2n$c7m$1@fred.mathworks.com>...
> Greg Heath <heath@alumni.brown.edu> wrote in message <99996f6f-471e-49b6-9c95-a6ff3efed38b@j2g2000yqf.googlegroups.com>...
> > I get
> > 
> > det([]) = 1
> > 
> > Is this reasonable?
> 
> Perfectly reasonable.
> 
> It's the same reason why prod([]) = 1. 1 is the neutral element of the (R,*). Determinant is the product of eigen-values. In zero-dimension space the eigen value is empty, then the most logical is returning the neutral element.
> 
> Bruno
- - - - - -
  I tend to side with Bruno (and MathWorks) on this question.  Besides his argument about eigenvalues, you can reason this way.  The determinant of a square n x n matrix is obtained as the sum, with appropriate signs, of n factorial products, each of which is composed of n factors.  If we let n be zero, this is zero factorial terms each composed of zero factors - that is to say one term composed of no factors and therefore possessing the value one.  Hence the sum must be one.

  Of course when stated in such a manner it sounds a bit absurd, but just as with the definition of zero factorial, it is convenient to define it in this manner because it hopefully constitutes the best extension of a notion that is well-defined for positive values of n.  I suppose in reality this question depends on how many contradictions one encounters such as Matt's in trying to make use of it.  In the case of factorial, the correctness of the extension to zero factorial was absolutely assured by the natural analytic extension furnished by the gamma function.  Perhaps there is a similar argument for an empty determinant.

Roger Stafford