Talk:Leibniz formula for determinants

Is the "physisists formula" usinge the Levi-Civita symbols correct? As I understand it, there is a 1/n! missing. —The preceding unsigned comment was added by 131.130.26.227 (talk • contribs) 14:03, 17 August 2006 (UTC)


 * So there was. I've now written the formula in a manner which obviates the need for the prefactor and makes the connection with the first formula more transparent. -- Fropuff 05:35, 11 February 2007 (UTC)

A part of the proof is missing
In the proof, the reason why the n-tuples are reduced to the permutations is missing. The reason why the ordered n-tuples $$(k_1, \dots, k_n)$$ are reduced to the permutations is because $$F$$ is alternating, and therefore $$F(E^{k_1}, \dots, E^{k_n})$$ is zero for all n-tuples $$(k_1, \dots, k_n)$$ that repeat indices. Jeff Wu 01:11, 6 December 2006 (UTC)

A second part of the proof is missing :)
The proof shows that if F is a function that satisfies the conditions, then F is equal to the Leibniz function. This shows the uniqueness of a solution. To prove the existence of a solution, you must show that the Leibniz function actually satisfies the conditions. Ceroklis 11:47, 27 September 2007 (UTC)

You are correct. The proof made the implicit assumption that by applying the 3 properties to an arbitrary function F that the final F would have these properties. While it is *possible* to very carefully apply such properties to an arbitrary function in such a way that you are guaranteed that the final function has them, it was not done so in this case. For example, the property of alternation was only applied to those F with the unit vectors in their domain. As such, I divided the proof into 2 sections, narrowing down the function space to a single function F, then showing that F has the 3 properties that determine a determinant. ~antares5245 —Preceding unsigned comment added by Antares5245 (talk • contribs) 15:54, 3 February 2009 (UTC)

part of proof
This way, it is easier to follow the proof:



\begin{align} F(A)& = F\left(\sum_{k_1 = 1}^n a_{k_1}^1 E^{k_1}, A^2, \dots, A^n\right)\\ & = \sum_{k_1 = 1}^n a_{k_1}^1 F\left(E^{k_1}, A^2, \dots, A^n\right)\\ & = \sum_{k_1 = 1}^n a_{k_1}^1 \sum_{k_2 = 1}^n a_{k_2}^1 F\left(E^{k_1}, E^{k_2}, A^3, \dots, A^n\right)\\ & = \sum_{k_1, k_2 = 1}^n \prod_{i = 1}^2 a_{k_i}^i F\left(E^{k_1}, E^{k_2}, A^3, \dots, A^n\right)\\ & = \cdots\\ & = \sum_{k_1, \dots, k_n = 1}^n \prod_{i = 1}^n a_{k_i}^i F\left(E^{k_1}, \dots, E^{k_n}\right). \end{align} $$

Ave caesar, 145.97.205.169 23:11, 12 October 2007 (UTC)

Practical examples?
Should we here include practical examples of this formula in use? It would make it more easy to understand for the more uninitiated reader, who is not fully conversant with mathematical notation of this level. If this does not violate any policy, it would be a good move. Rumblethunder (talk) 10:29, 24 March 2009 (UTC)

Unfortunately, this formula is interesting but not practical. The summation is done over a large exponential number of indices (even for a 5 by 5 matrix the number of terms in the summation is 5! or 120). It is not a polynomial approach to finding the determinant like block inversion or reduced row echelon. Even so it might be useful for some to understand the use of permutations as summations for an example with the iterations expanded. Antares5245 (talk) 20:04, 4 April 2009 (UTC)

It doesn't really matter if it's practical or not, it'd still be nice with a concrete step-by-step example. /85.229.223.151 (talk) 13:50, 26 January 2011 (UTC)

Why is this called the Leibniz formula?
Some source should be supplied that supports calling this the Leibniz formula, and also something to explain why that name is attached to it. Leibniz may have introduced the concept of determinant (in Europe), but this formula seems a bit unlikely to be stated in this generality in that timeframe. Special cases of it, sure, but the general formula is more doubtful. Citation needed. 130.243.83.36 (talk) 15:40, 27 February 2014 (UTC)