Sylvester's determinant identity

{{short description|Identity in algebra useful for evaluating certain types of determinants}}

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.{{cite journal | last = Sylvester | first = James Joseph | title = On the relation between the minor determinants of linearly equivalent quadratic functions | journal = Philosophical Magazine | volume = 1 | year = 1851 | pages = 295–305}}
Cited in {{Cite journal | last1 = Akritas | first1 = A. G. | last2 = Akritas | first2 = E. K. | last3 = Malaschonok | first3 = G. I. | doi = 10.1016/S0378-4754(96)00035-3 | title = Various proofs of Sylvester's (determinant) identity | journal = Mathematics and Computers in Simulation | volume = 42 | issue = 4–6 | page = 585 | year = 1996 }}

Given an n-by-n matrix $A$, let $\backslash det(A)$ denote its determinant. Choose a pair

:$u\; =(u\_1,\; \backslash dots,\; u\_m),\; v\; =(v\_1,\; \backslash dots,\; v\_m)\; \backslash subset\; (1,\; \backslash dots,\; n)$

of m-element ordered subsets of $(1,\; \backslash dots,\; n)$, where m ≤ n.

Let $A^u\_v$ denote the (n−m)-by-(n−m) submatrix of $A$ obtained by deleting the rows in $u$ and the columns in $v$.

Define the auxiliary m-by-m matrix $\backslash tilde\{A\}^u\_v$ whose elements are equal to the following determinants

:$$

(\tilde{A}^u_v)_{ij} := \det(A^{u[\hat{u}_i]}_{v[\hat{v}_j]}),

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

:$\backslash det(A)(\backslash det(A^u\_v))^\{m-1\}=\backslash det(\backslash tilde\{A\}^u\_v).$

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