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.
is the sign function of permutations in the permutation group
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
is the number of order-
This is impractically difficult for even relatively small
Instead, the determinant can be evaluated in
operations by forming the LU decomposition
(typically via Gaussian elimination or similar methods), in which case
and the determinants of the triangular matrices
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).
The determinant can also be evaluated in fewer than
operations by reducing the problem to matrix multiplication, but most such algorithms are not practical.
-th column vector of the identity matrix.
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.
sgn ( σ )
is defined to count the number of swaps necessary and account for the resulting sign change.
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.
be the tuple equal to
: 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