In
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
theory, Sylvester's determinant identity is an
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), ...
useful for evaluating certain types of
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
s. It is named after
James Joseph Sylvester
James Joseph Sylvester (3 September 1814 – 15 March 1897) was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory, and combinatorics. He played a leadership ro ...
, who stated this identity without proof in 1851.
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Cited in
Given an ''n''-by-''n'' matrix
, let
denote its determinant. Choose a pair
:
of ''m''-element ordered
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
s 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