# Leibniz formula for determinants

﻿
Leibniz formula for determinants

In algebra, the Leibniz formula expresses the determinant of a square matrix $A = \left(a_\left\{ij\right\}\right)_\left\{i,j = 1, dots, n\right\}$ in terms of permutations of the matrix' elements. Named in honor of Gottfried Leibniz, the formula is

:$det\left(A\right) = sum_\left\{sigma in S_n\right\} sgn\left(sigma\right) prod_\left\{i = 1\right\}^n a_\left\{i,sigma\left(i\right)\right\}$

for an "n"×"n" matrix, where sgn is the sign function of permutations in the permutation group "S""n", which returns +1 and –1 for even and odd permutations, respectively.

Another common notation used for the formula is in terms of the Levi-Civita symbol and makes use of the Einstein summation notation, where it becomes:$det\left(A\right)=epsilon^\left\{i_1cdots i_n\right\}\left\{A\right\}_\left\{1i_1\right\}cdots \left\{A\right\}_\left\{ni_n\right\},$which may be more familiar to physicists.

In the sequel, a proof of the equivalence of this formula to the conventional definition of the determinant in terms of expansion by minors is given.

Theorem.There exists exactly one function:$F : mathfrak M_n \left(mathbb K\right) longrightarrow mathbb K$which is alternate multilinear w.r.t. columns and such that $F\left(I\right) = 1$.

Proof.

Let $F$ be such a function, and let $A = \left(a_i^j\right)_\left\{i = 1, dots, n\right\}^\left\{j = 1, dots , n\right\}$ be an $n imes n$ matrix. Call $A^j$ the $j$-th column of $A$, i.e. $A^j = \left(a_i^j\right)_\left\{i = 1, dots , n\right\}$, so that $A = left\left(A^1, dots, A^n ight\right).$

Also, let $E^k$ denote the $k$-th column vector of the identity matrix.

Now one writes each of the $A^j$'s in terms of the $E^k$, i.e.

:$A^j = sum_\left\{k = 1\right\}^n a_k^j E^k$.

As $F$ is multilinear, one has

:

As the above sum takes into account all the possible choices of ordered $n$-tuples $left\left(k_1, dots , k_n ight\right)$, it can be expressed in terms of permutations as

:$sum_\left\{sigma in mathfrak S_n\right\} prod_\left\{i = 1\right\}^n a_\left\{sigma\left(i\right)\right\}^i F\left(\left(E^\left\{sigma\left(1\right)\right\}, dots , E^\left\{sigma\left(n\right)\right\}\right)\right).$

Now one rearranges the columns of $left\left(E^\left\{sigma\left(1\right)\right\}, dots, E^\left\{sigma\left(n\right)\right\} ight\right)$ so that it becomes the identity matrix; the number of columns that need to be exchanged is exactly $sgn\left(sigma\right)$. Hence, thanks to alternance, one finally gets

:

as $F\left(I\right)$ is required to be equal to $1$.

Hence the determinant can be defined as the only function

:$det : mathfrak M_n \left(mathbb K\right) longrightarrow mathbb K$

which is alternate multilinear w.r.t. columns and such that $det\left(I\right) = 1$.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Cauchy-Binet formula — In linear algebra, the Cauchy Binet formula generalizes the multiplicativity of the determinant (the fact that the determinant of a product of two square matrices is equal to the product of the two determinants) to non square matrices. Suppose A… …   Wikipedia

• List of mathematics articles (L) — NOTOC L L (complexity) L BFGS L² cohomology L function L game L notation L system L theory L Analyse des Infiniment Petits pour l Intelligence des Lignes Courbes L Hôpital s rule L(R) La Géométrie Labeled graph Labelled enumeration theorem Lack… …   Wikipedia

• Лейбниц, Готфрид Вильгельм — Готфрид Вильгельм Лейбниц Gottfried Wilhelm Leibniz …   Википедия

• Laplace expansion — This article is about the expansion of the determinant of a square matrix as a weighted sum of determinants of sub matrices. For the expansion of an 1/r potential using spherical harmonical functions, see Laplace expansion (potential). In linear… …   Wikipedia

• Список объектов, названных в честь Лейбница — Существует несколько математических и другого рода объектов, названных в честь Лейбница: Содержание 1 Теоремы 2 Формулы 3 Прочее 4 См …   Википедия

• Antisymmetrizer — In quantum mechanics, an antisymmetrizer mathcal{A} (also known as antisymmetrizing operator [P.A.M. Dirac, The Principles of Quantum Mechanics , 4th edition, Clarendon, Oxford UK, (1958) p. 248] ) is a linear operator that makes a wave function… …   Wikipedia

• Determinant — This article is about determinants in mathematics. For determinants in epidemiology, see Risk factor. In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific… …   Wikipedia

• Matrix (mathematics) — Specific elements of a matrix are often denoted by a variable with two subscripts. For instance, a2,1 represents the element at the second row and first column of a matrix A. In mathematics, a matrix (plural matrices, or less commonly matrixes)… …   Wikipedia

• Vandermonde matrix — In linear algebra, a Vandermonde matrix, named after Alexandre Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row, i.e., an m × n matrix or …   Wikipedia

• Differentiation under the integral sign — Topics in Calculus Fundamental theorem Limits of functions Continuity Mean value theorem Differential calculus  Derivative Change of variables Implicit differentiation Taylor s theorem Related rates …   Wikipedia