Regarding the Laplace expansion more specifically:
I assume that a determinant is defined over matrices M : A x A -> R, where
A is a finite (unordered) set. Suppose we wish to perform cofactor
expansion along row i e. A. The j'th cofactor (where j e. A) is given by
the elements of M on (A\{i}) x (A\{j}) ,which is not again a square matrix,
so we must first compose with swap(i,j) : A\{j} -> A\{i}. This is the
natural way to restore squareness in this setting, but note that it is not
the usual way it is done over {1..n} where you just delete an element and
shift everything down. The difference between these two operations (if you
compose one with the inverse of the other) is a cycle of length |j - i|,
and the sign of this permutation is, fortuitously, (-1)^(j+i). Thus if you
state cofactor expansion using swap, you only need finite sets and you
don't need the sign term at all.
On Sat, Sep 5, 2020 at 3:53 AM Benoit <[email protected]> wrote:
> I do not have strong opinions on this issue and I will not be the one to
> do the work, but I still think "starting at 0" for matrices is more
> natural. Of course, the more important thing is to have as many results as
> possible stated for arbitrary finite sets (or even arbitrary sets for some
> results?). Actually, I think starting at 0 is preferable even more after
> reading Mario's arguments, and the shift from literature is not that
> important (I think it won't even confuse beginners, since this is such a
> minor variation.)
>
> As Norm said, it depends on the intended audience. I mentioned this
> question a few times when discussing other topics, and said it was bound to
> show up again and again, since many decisions depend on this question. Of
> course it's better to fulfill the needs of different audiences, but it's
> not always possible. You don't write a textbook the same way you write a
> reference treatise (admittedly, with electronic material, one might have
> more flexibility). Personally, I think it's time for Norm to realize the
> grandeur of what he created ;-) and that it's closer to Bourbaki's Elements
> than to your usual textbook.
>
> Benoît
>
> On Saturday, September 5, 2020 at 8:43:40 AM UTC+2 Alexander van der
> Vekens wrote:
>
>> As proposed by FL in
>> https://groups.google.com/g/metamath/c/n5g69qfwBmE/m/McJAgtdSAgAJ, the
>> additional assumption that the finite index sets are totally orderer should
>> be sufficient to express and prove the Laplace expansion. Regarding the
>> expression ` -1 ^ ( i + j ) ` , a special concept of parity must be defined
>> for (finite) totally ordered sets...
>>
>> On Wednesday, September 2, 2020 at 7:02:23 PM UTC+2 Thierry Arnoux wrote:
>>
>>> On 02/09/2020 23:25, 'fl' via Metamath wrote:
>>>
>>> The best to do is redefine the <" ...>" operator so that it takes (1 ...
>>> N) as its set of indices and then fix up all the proofs referring to the
>>> definition.
>>> You should have only one definition for matrices, tuples and words since
>>> all that is the same story. Or at most two: one with a abstract finite
>>> set of indices and another one with (1... N).
>>>
>>> Like mentioned by Norm in the original thread about index start for
>>> words, I'm afraid that would be a huge work, there are already hundreds of
>>> theorems making use of that range.
>>> That's why I would like to ask the same about matrices now, before
>>> writing more theorems within whichever convention.
>>>
>>> Of course we shall try to use arbitrary sets whenever possible... which
>>> leads me to the other point:
>>>
>>> On the other hand, does any one have any idea or suggestion about how I
>>>> could have expressed the Laplace expansion without integer indices, on any
>>>> finite set index?
>>>>
>>>
>>> The laplace expansion of the determinant (just one level of expansion,
>>> not recursive) requires a choice of row to expand over (which is an element
>>> of the index set), and then it's an unordered finite sum. No integers
>>> needed.
>>>
>>> Yes, but what is the factor you apply to each submatrix determinant ?
>>> The textbooks all have ` -1 ^ ( i + j ) ` , where i and j are *integer*
>>> indices. Of course, start index does not matter, but where do those numbers
>>> come from when working with arbitrary sets?
>>>
>>> _
>>> Thierry
>>>
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