Sylvester's determinant identity explained

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

\det(A)

denote its determinant. Choose a pair

u=(u_1,...,u_m),v=(v_1,...,v_m)\subset(1,...,n)

of m-element ordered subsets of

(1,...,n)

, where m ≤ n.Let

	u
A
	v

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

	u
\tilde{A}
	v

whose elements are equal to the following determinants

	u
(\tilde{A}
	v)

_ij:=

	u[\hat{u
\det(A
	i]}

_v[\hat{v_j]}),

where

u[\hat{u_i}]

v[\hat{v_j}]

denote the m−1 element subsets of

and

obtained by deleting the elements

u_i

and

v_j

, respectively. Then the following is Sylvester's determinantal identity (Sylvester, 1851):

	m-1
\det(A)(\det(A
	v))

	u
=\det(\tilde{A}
	v).

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

Notes and References

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

Sylvester's determinant identity explained

See also

Notes and References