Sylvester's determinant identity
Identity in algebra useful for evaluating certain types of determinants / From Wikipedia, the free encyclopedia
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.[1]
Given an n-by-n matrix , let
denote its determinant. Choose a pair
of m-element ordered subsets of , where m ≤ n.
Let
denote the (n−m)-by-(n−m) submatrix of
obtained by deleting the rows in
and the columns in
.
Define the auxiliary m-by-m matrix
whose elements are equal to the following determinants
where ,
denote the m−1 element subsets of
and
obtained by deleting the elements
and
, respectively. Then the following is Sylvester's determinantal identity (Sylvester, 1851):
When m = 2, this is the Desnanot-Jacobi identity (Jacobi, 1851).