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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
Sylvester's determinant theorem states that for A, an m × n matrix, and B, an n × m matrix (so that A and B have dimensions allowing them to be multiplied in either order forming a square matrix): (+) = (+),
The determinant of the Sylvester matrix of two polynomials is their resultant, which is zero when the two polynomials have a common root (in case of coefficients in a field) or a non-constant common divisor (in case of coefficients in an integral domain). Sylvester matrices are named after James Joseph Sylvester.
In mathematics, Sylvester’s criterion is a necessary and sufficient criterion to determine whether a Hermitian matrix is positive-definite. Sylvester's criterion states that a n × n Hermitian matrix M is positive-definite if and only if all the following matrices have a positive determinant: the upper left 1-by-1 corner of M,
The Sylvester–Gallai theorem, on the existence of a line with only two of n given points. Sylvester's determinant identity. Sylvester's matrix theorem, also called Sylvester's formula, for a matrix function in terms of eigenvalues. Sylvester's law of inertia, also called Sylvester's rigidity theorem, about the signature of a quadratic form.
This condition is known as Sylvester's criterion, and provides an efficient test of positive definiteness of a symmetric real matrix. Namely, the matrix is reduced to an upper triangular matrix by using elementary row operations , as in the first part of the Gaussian elimination method, taking care to preserve the sign of its determinant during ...
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 .
As the resultant is the determinant of the Sylvester matrix (and of the Bézout matrix), it may be computed by using any algorithm for computing determinants. This needs O ( n 3 ) {\displaystyle O(n^{3})} arithmetic operations.