In matrix theory, Sylvester's determinant identity is an identity useful for evaluating certain types of determinants.
It is named after James Joseph Sylvester, who stated this identity without proof in 1851.
denote its determinant.
Choose a pair of m-element ordered subsets of
denote the (n−m)-by-(n−m) submatrix of
obtained by deleting the rows in
Define the auxiliary m-by-m matrix
whose elements are equal to the following determinants where
denote the m−1 element subsets of
obtained by deleting the elements
Then the following is Sylvester's determinantal identity (Sylvester, 1851): When m = 2, this is the Desnanot-Jacobi identity (Jacobi, 1851).
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