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2016 Update

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Tina Müller 2016-12-05 22:15:40 +01:00
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Task/Matrix-arithmetic/00DESCRIPTION
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For a given matrix, return the [[wp:Determinant|determinant]] and the [[wp:Permanent|permanent]] of the matrix.
The determinant is given by
:<math>\det(A) = \sum_\sigma\sgn(\sigma)\prod_{i=1}^n M_{i,\sigma_i}</math>
:: <big><math>\det(A) = \sum_\sigma\sgn(\sigma)\prod_{i=1}^n M_{i,\sigma_i}</math></big>
while the permanent is given by
: <math> \operatorname{perm}(A)=\sum_\sigma\prod_{i=1}^n M_{i,\sigma_i}</math>
:: <big><math> \operatorname{perm}(A)=\sum_\sigma\prod_{i=1}^n M_{i,\sigma_i}</math></big>
In both cases the sum is over the permutations <math>\sigma</math> of the permutations of 1, 2, ..., ''n''. (A permutation's sign is 1 if there are an even number of inversions and -1 otherwise; see [[wp:Parity of a permutation|parity of a permutation]].)
More efficient algorithms for the determinant are known: [[LU decomposition]], see for example [[wp:LU decomposition#Computing the determinant]]. Efficient methods for calculating the permanent are not known.
;Cf.:
* [[Permutations by swapping]]
<br><br>