In mathematics, a Sylvester matrix is a matrix associated to two univariate polynomials with coefficients in a field or a commutative ring. The entries of the Sylvester matrix of two polynomials are coefficients of the polynomials. 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, a Sylvestermatrix is a matrix associated to two univariate polynomials with coefficients in a field or a commutative ring. The entries...
James Joseph Sylvester FRS HonFRSE (3 September 1814 – 15 March 1897) was an English mathematician. He made fundamental contributions to matrix theory, invariant...
which is the determinant of the Sylvestermatrix of A and A′. The nonzero entries of the first column of the Sylvestermatrix are a n {\displaystyle a_{n}}...
matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries...
square matrix of dimension d + e, which is called the Sylvestermatrix of A and B (for many authors and in the article Sylvestermatrix, the Sylvester matrix...
In mathematics, in the field of control theory, a Sylvester equation is a matrix equation of the form: A X + X B = C . {\displaystyle AX+XB=C.} It is named...
construction of Sylvester's Hadamard matrix. First consider the matrix F n {\displaystyle F_{n}} , the n × 2 n {\displaystyle n\times 2^{n}} matrix whose columns...
The resultant of P and Q is the determinant of the Sylvestermatrix, which is the (square) matrix of φ 0 {\displaystyle \varphi _{0}} on the bases of...
square matrix. The determinant of a matrix A is commonly denoted det(A), det A, or |A|. Its value characterizes some properties of the matrix and the...
system, which is a successor of the preceding. Macaulay matrix, a generalization of Sylvestermatrix to n homogeneous polynomials in n variables. Macaulay...
columns to the right, then you obtain a transposed Sylvestermatrix. The determinant of a transfer matrix is essentially a resultant. More precisely: Let...
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems...
algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors...
r such that the matrix is a product of an m by r matrix and an r by n matrix. Sylvester (1884) showed that fields satisfy Sylvester's law of nullity and...
matrix Hankel matrix Hilbert matrix Persymmetric matrixSylvester's law of inertia Toeplitz matrix Transpositions matrix See also symmetry in mathematics...
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