In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a Sylvester matrix is a matrix

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

 associated to two polynomial

Polynomial

In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

s that provides information about those polynomials. It is named for James Joseph Sylvester

James Joseph Sylvester

James Joseph Sylvester was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory and combinatorics...

.

Definition

Formally, let p and q be two nonzero polynomials, respectively of degree m and n. Thus:
The Sylvester matrix associated to p and q is then the matrix obtained as follows:

• the first row is:
• the second row is the first row, shifted one column to the right; the first element of the row is zero.
• the following (n-2) rows are obtained the same way, still filling the first column with a zero.
• the (n+1)-st row is:
• the following rows are obtained the same way as before.

Thus, if m=4 and n=3, the matrix is:

Applications

These matrices are used in commutative algebra

Commutative algebra

Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...

, e.g. to test if two polynomials have a (non constant) common factor. In such a case, the determinant

Determinant

In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

 of the associated Sylvester matrix (which is named the resultant of the two polynomials) equals zero. The converse is also true.

The solutions of the simultaneous linear equations
where is a vector of size and has size , comprise the coefficient vectors of those and only those pairs of polynomials (of degrees and , respectively) which fulfill
(where polynomial multiplication and addition is used in this last line).
This means the kernel

Null space

In linear algebra, the kernel or null space of a matrix A is the set of all vectors x for which Ax = 0. The kernel of a matrix with n columns is a linear subspace of n-dimensional Euclidean space...

 of the transposed Sylvester matrix gives all solutions of the Bézout equation

Bézout's identity

In number theory, Bézout's identity for two integers a, b is an expressionwhere x and y are integers , such that d is a common divisor of a and b. Bézout's lemma states that such coefficients exist for every pair of nonzero integers...

 where and .

Consequently the rank

Rank (linear algebra)

The column rank of a matrix A is the maximum number of linearly independent column vectors of A. The row rank of a matrix A is the maximum number of linearly independent row vectors of A...

 of the Sylvester matrix determines the degree of the greatest common divisor

Greatest common divisor

In mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...

of p and q.