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 ${\displaystyle A}$, let ${\displaystyle \det(A)}$ denote its determinant. Choose a pair

${\displaystyle u=(u_{1},\dots ,u_{m}),v=(v_{1},\dots ,v_{m})\subset (1,\dots ,n)}$

of m-element ordered subsets of ${\displaystyle (1,\dots ,n)}$, where m ≤ n. Let ${\displaystyle A_{v}^{u}}$ denote the (n−m)-by-(n−m) submatrix of ${\displaystyle A}$ obtained by deleting the rows in ${\displaystyle u}$ and the columns in ${\displaystyle v}$. Define the auxiliary m-by-m matrix ${\displaystyle {\tilde {A}}_{v}^{u}}$ whose elements are equal to the following determinants

${\displaystyle ({\tilde {A}}_{v}^{u})_{ij}:=\det(A_{v[{\hat {v}}_{j}]}^{u[{\hat {u}}_{i}]}),}$

where ${\displaystyle u[{\hat {u_{i}}}]}$, ${\displaystyle v[{\hat {v_{j}}}]}$ denote the m−1 element subsets of ${\displaystyle u}$ and ${\displaystyle v}$ obtained by deleting the elements ${\displaystyle u_{i}}$ and ${\displaystyle v_{j}}$, respectively. Then the following is Sylvester's determinantal identity (Sylvester, 1851):

${\displaystyle \det(A)(\det(A_{v}^{u}))^{m-1}=\det({\tilde {A}}_{v}^{u}).}$

When m = 2, this is the Desnanot-Jacobi identity (Jacobi, 1851).

See also

• Weinstein–Aronszajn identity, which is sometimes attributed to Sylvester

References