Leibniz formula for determinants

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 a_i,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 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

which may be more familiar to physicists.

Directly evaluating the Leibniz formula from the definition requires ${\displaystyle \Omega (n!\cdot n)}$ 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 ${\displaystyle A=LU}$ (typically via Gaussian elimination or similar methods), in which case ${\displaystyle \det A=(\det L)(\det U)}$ 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).

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