# Sylvester matrix

﻿
Sylvester matrix

In mathematics, a Sylvester matrix is a matrix associated to two polynomials that gives us some information about those polynomials. It is named for James Joseph Sylvester.

Definition

Formally, let "p" and "q" be two polynomials, respectively of degree "m" and "n". Thus::$p\left(z\right)=p_0+p_1 z+p_2 z^2+cdots+p_m z^m,;q\left(z\right)=q_0+q_1 z+q_2 z^2+cdots+q_n z^n.$The Sylvester matrix associated to "p" and "q" is then the $\left(n+m\right) imes\left(n+m\right)$ 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)-th row is::
* the following rows are obtained the same way as before.

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

Applications

Those matrices are used in commutative algebra, e.g. to test if two polynomials have a (non constant) common factor. Indeed, in such a case, the determinant of the associated Sylvester matrix (which is named the resultant of the two polynomials) equals zero. The converse is also true.

The solution of the simultaneous linear equations:where $x$ is a vector of size $n$ and $y$ has size $m$, comprises the coefficient vectors of those and only those pairs $x, y$ of polynomials (of degrees $n-1$ and $m-1$, respectively) which fulfill:$x cdot p + y cdot q = 1$(where polynomial multiplication and addition is used in this last line).This means the kernel of the transposed Sylvester matrix gives all solutions of the Bézout equation where $deg x < deg q$ and $deg y < deg p$.

Consequently the rank of the Sylvester matrix determines the degree of the greatest common divisor of $p$ and $q$.:$deg\left(gcd\left(p,q\right)\right) = m+n-mathrm\left\{rank\right\}~S_\left\{p,q\right\}$.

ee also

* Transfer matrix

References

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Matrix - получить на Академике рабочий купон на скидку Строительный Двор или выгодно matrix купить с бесплатной доставкой на распродаже в Строительный Двор

• Matrix theory — is a branch of mathematics which focuses on the study of matrices. Initially a sub branch of linear algebra, it has grown to cover subjects related to graph theory, algebra, combinatorics, and statistics as well.HistoryThe term matrix was first… …   Wikipedia

• Matrix (mathematics) — Specific elements of a matrix are often denoted by a variable with two subscripts. For instance, a2,1 represents the element at the second row and first column of a matrix A. In mathematics, a matrix (plural matrices, or less commonly matrixes)… …   Wikipedia

• Sylvester's formula — In matrix theory, Sylvester s formula, named after James Joseph Sylvester, expresses matrix functions in terms of the eigenvalues and eigenvectors of a matrix. It is only valid for diagonalizable matrices; an extension due to Buchheim covers the… …   Wikipedia

• Sylvester-Kriterium — Definitheit ist ein Begriff aus dem mathematischen Teilgebiet der linearen Algebra. Er beschreibt, welche Vorzeichen reelle quadratische Formen annehmen können, die durch Matrizen oder allgemeiner durch Bilinearformen erzeugt werden.… …   Deutsch Wikipedia

• Sylvester'scher Trägheitssatz — Der Trägheitssatz von Sylvester oder Sylvester scher Trägheitssatz benannt nach James Joseph Sylvester ist ein Resultat aus der linearen Algebra. Dieser Satz macht eine Aussage über Invarianten darstellender Matrizen von symmetrischen… …   Deutsch Wikipedia

• Sylvester’scher Trägheitssatz — Der Trägheitssatz von Sylvester oder Sylvester scher Trägheitssatz benannt nach James Joseph Sylvester ist ein Resultat aus der linearen Algebra. Dieser Satz macht eine Aussage über Invarianten darstellender Matrizen von symmetrischen… …   Deutsch Wikipedia

• Sylvester equation — In control theory, the Sylvester equation is the matrix equation of the form:A X + X B = C,where A,B,X,C are n imes n matrices.Existence and uniqueness of the solutionUsing the Kronecker product notation and the vectorization operator… …   Wikipedia

• matrix — /may triks, ma /, n., pl. matrices /may tri seez , ma /, matrixes. 1. something that constitutes the place or point from which something else originates, takes form, or develops: The Greco Roman world was the matrix for Western civilization. 2.… …   Universalium

• Sylvester's law of inertia — In linear algebra, Sylvester s law of inertia is a theorem describing a canonical representative for a real symmetric matrix under congruence transformations. It is named for J. J. Sylvester who stated and proved it in 1852.The theorem states… …   Wikipedia

• Matrix congruence — In mathematics, two matrices A and B over a field are called congruent if there exists an invertible matrix P over the same field such that PTAP = B where T denotes the matrix transpose. Matrix congruence is an equivalence relation. Matrix… …   Wikipedia