Sylvester's determinant identity

Not to be confused with the Weinstein–Aronszajn identity, which is sometimes attributed to Sylvester.

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).

References

1. Sylvester, James Joseph (1851). "On the relation between the minor determinants of linearly equivalent quadratic functions". Philosophical Magazine. 1: 295–305.
   Cited in Akritas, A. G.; Akritas, E. K.; Malaschonok, G. I. (1996). "Various proofs of Sylvester's (determinant) identity". Mathematics and Computers in Simulation. 42 (4–6): 585. doi:10.1016/S0378-4754(96)00035-3.