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).
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Transcription
See also
- Weinstein–Aronszajn identity, which is sometimes attributed to Sylvester
References
- ^ 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.
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