MathLiveLeibniz formula for determinants
Expresses the determinant of a square matrix as a sum over all permutations of its indices, each term being the sign of the permutation times a product of matrix entries. It is the basic definition of the determinant and is used in theoretical proofs and small explicit calculations.
\det(A) = \sum_{\sigma \in S_n} \mathrm{sgn}(\sigma) \prod_{i=1}^{n} a_{\sigma(i), i}What the symbols mean
| a square matrix of size n \times n | |
| the size (number of rows and columns) of the matrix A | |
| a permutation belonging to the symmetric group S_n of all permutations of {1, ..., n} | |
| the sign of the permutation \sigma, equal to +1 for even permutations and -1 for odd ones | |
| the index running over 1, ..., n (the column index of the entry in the product) | |
| the entry of A in row i and column j |
About this formula
- FieldMathematics
- TopicLinear algebra
- LevelUniversity
- Also calleddeterminant permutation expansion, determinant sum over permutations, big formula for determinant, determinant definition, Leibniz determinant formula, signed permutation determinant
- SourceWikidata Q538557
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