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In algebra, the Leibniz formula, named in honor of Gottfried Leibniz, expresses the determinant of a square matrix in terms of permutations of the matrix elements. If A is an n×n matrix, where ai,j is the entry in the ith row and jth column of A, the formula is

where sgn is the sign function of permutations in the permutation group Sn, 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

which may be more familiar to physicists.

Directly evaluating the Leibniz formula from the definition requires operations in general—that is, a number of operations asymptotically proportional to n factorial—because n! is the number of order-n permutations. This is impractically difficult for large n. Instead, the determinant can be evaluated in O(n3) operations by forming the LU decomposition (typically via Gaussian elimination or similar methods), in which case and the determinants of the triangular matrices L and U are simply the products of their diagonal entries. (In practical applications of numerical linear algebra, however, explicit computation of the determinant is rarely required.) See, for example, Trefethen & Bau (1997).

Formal statement and proofEdit

Theorem. There exists exactly one function

 

which is alternating multilinear w.r.t. columns and such that  .

Proof.

Uniqueness: Let   be such a function, and let   be an   matrix. Call   the  -th column of  , i.e.  , so that  

Also, let   denote the  -th column vector of the identity matrix.

Now one writes each of the  's in terms of the  , i.e.

 .

As   is multilinear, one has

 

From alternation it follows that any term with repeated indices is zero. The sum can therefore be restricted to tuples with non-repeating indices, i.e. permutations:

 

Because F is alternating, the columns   can be swapped until it becomes the identity. The sign function   is defined to count the number of swaps necessary and account for the resulting sign change. One finally gets:

 

as   is required to be equal to  .

Therefore no function besides the function defined by the Leibniz Formula can be a multilinear alternating function with  .

Existence: We now show that F, where F is the function defined by the Leibniz formula, has these three properties.

Multilinear:

 

Alternating:

 

For any   let   be the tuple equal to   with the   and   indices switched.

 

Thus if   then  .

Finally,  :

 

Thus the only alternating multilinear functions with   are restricted to the function defined by the Leibniz formula, and it in fact also has these three properties. Hence the determinant can be defined as the only function

 

with these three properties.

See alsoEdit

ReferencesEdit

  • Hazewinkel, Michiel, ed. (2001) [1994], "Determinant", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  • Trefethen, Lloyd N.; Bau, David (June 1, 1997). Numerical Linear Algebra. SIAM. ISBN 978-0898713619.