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I am trying really hard to find a good definition of the determinant.

I have looked virtually every single resource online and everybody gives a different answer:

  1. sum of cofactors or minors https://mathworld.wolfram.com/Determinant.html, https://www.math.uchicago.edu/~may/VIGRE/VIGRE2007/REUPapers/FINALAPP/Peng.pdf
  2. signed volume https://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter04/section01.html
  3. a sum of permutation https://math.libretexts.org/Bookshelves/Linear_Algebra/Map%3A_Linear_Algebra_(Waldron_Cherney_and_Denton)/08%3A_Determinants/8.01%3A_The_Determinant_Formula, https://math.clarku.edu/~ma130/permutations.pdf
  4. a scalar function that satisfies certain properties: https://textbooks.math.gatech.edu/ila/determinants-definitions-properties.html
  5. a matrix that is defined with respect to the inverse of a matrix https://www.ncl.ac.uk/webtemplate/ask-assets/external/maths-resources/core-mathematics/pure-maths/matrices/matrix-determinant.html
  6. an object that is defined through a recursive relationship: https://services.math.duke.edu/~jdr/1617f-1553/materials/determinants.pdf
  7. an function that returns the product of all the eigenvalues of its matrix-valued argument https://archive-wp.epfl.ch/lions/wp-content/uploads/2019/01/lecture-1-2015.pdf
  8. an object whose formal definition is to be avoided https://www.hec.ca/en/cams/help/topics/Matrix_determinants.pdf

I understand all these definitions are more or less equivalent. But if a mathematician were to carefully choose one and potentially teach it to a group of students, which one would be the most correct one to choose?

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    $\begingroup$ This question seems impossible to answer without knowing what level the students are at. But there was some good discussion at matheducators.stackexchange.com/questions/1149 and math.stackexchange.com/questions/668 . $\endgroup$
    – David E Speyer
    Commented Jan 11 at 12:03
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    $\begingroup$ Is this question specifically about teaching? If so, this should be emphasised much more. What a mathematician uses for a definition of an object can be wholly inappropriate when teaching. $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented Jan 11 at 12:09
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    $\begingroup$ When it comes down to teaching, I like to give several flavors of the determinant (at least abstract, computational, and geometric). Different perspectives are useful in different ways, and I think explaining the determinant is a perfect occasion to illustrate this idea. $\endgroup$
    – Andrea Marino
    Commented Jan 11 at 12:42
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    $\begingroup$ There is no correct answer and all answers will be opinion based (and even my own opinion will be different for difference courses/audiences/…) - I vote to close as opinion based. $\endgroup$
    – Dirk
    Commented Jan 11 at 13:02
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    $\begingroup$ For teaching, I start with signed volumes; for myself, I use the top exterior power. $\endgroup$
    – Donu Arapura
    Commented Jan 11 at 13:10

1 Answer 1

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Not an answer, but too long for a comment. The most useful definition of the determinant will depend on the audience you are speaking to. The cleanest definition is probably in terms of the top part of the exterior algebra of a finite dimensional vector space. This is no good in teaching an introductory course on linear algebra, whether at undergraduate or graduate level. (This is the context in which I've spent the most time thinking about how to define a determinant.)

I've used the approach in item (4) of the question in classes. I've never been entirely happy with it. From this axiomatic starting point, the proof that the determinant is multiplicative is lengthy, and depends on rank deficient matrices having zero determinant and on the general linear group being generated by elementary matrices, etc. There's nothing obvious or elegant about it (in my opinion). I emphasise the multiplicative property of the determinant heavily in class. It simplifies computations and fits the determinant into the general philosophy of studying homomorphisms in algebra.

What I would like to do is begin with axioms for the determinant that make the multiplicative property fundamental and then derive all the usual properties from there. Presumably defining the determinant to be a function which

(1) is multiplicative

(2) is the product of diagonal entries for a diagonal matrix

should work. I haven't seen this done in any linear algebra textbook, so there's probably some stumbling block or reason that it isn't a good idea. I'd welcome people's thoughts on this.

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  • $\begingroup$ not sure if 1+2 implies the permutation and minors formulae easily. The latter are far more easy in practice for calculations of matrices of size $\le 4$. I agree that the mathematician's definition would be via the exterior algebra (which is really easy from a conceptual point of view). I guess (4) is the way of explaining the exterior approach without mentioning it. $\endgroup$
    – Andrea Marino
    Commented Jan 11 at 12:35
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    $\begingroup$ Oh this (1)-(2) definition is really cute! I haven't seen it anywhere before. But it is probably hard for beginners. Of course elementary operations are just multiplications by elementary matrices so it is sufficiently straightforward (though not easy for beginners) to show that this gives the usual determinant. (Handling elementary matrices is done by saying that matrices of transpositions are diagonalizable, and that elementary transformations that add a multiple of a row to another row can be written as commutators.) $\endgroup$
    – Vladimir Dotsenko
    Commented Jan 11 at 12:56
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    $\begingroup$ @GeoffRobinson If a definition works for algebraically closed fields, then just pretend that the entries of the matrix you are given are elements of the algebraic closure. This will give you the right value for the determinant, and you can prove later that the value of the determinant always lies in the base field. $\endgroup$
    – Timothy Chow
    Commented Jan 11 at 13:28
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    $\begingroup$ @GeoffRobinson, the product of the transposition with itself is the identity. So with the multiplicative property it’s determinant must be a square root of 1 and hence 1 in characteristic 2 $\endgroup$
    – Benjamin Steinberg
    Commented Jan 11 at 14:04
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    $\begingroup$ @VladimirDotsenko I’ve seen it before :-) That’s the definition used in popularization by J.-M. Souriau (who also wrote a few books on linear algebra). E.g. p. 184 of the book Grammaire de la Nature he was writing in his last years (but never published other than online; available as #125 here). Unfortunately you’ll find only statements there, no proofs. $\endgroup$
    – Francois Ziegler
    Commented Jan 11 at 18:23

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