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(z)=p_0+p_1 z+p_2 z^2+cdots+p_m z^m,;q(z)=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 (n+m) imes(n+m) matrix obtained as follows:
* the first row is::egin{pmatrix} p_m & p_{m-1} & cdots & p_1 & p_0 & 0 & cdots & 0 end{pmatrix}.
* 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::egin{pmatrix} q_n & q_{n-1} & cdots & q_1 & q_0 & 0 & cdots & 0 end{pmatrix}.
* the following rows are obtained the same way as before.

Thus, if we put "m"=4 and "n"=3, the matrix is::S_{p,q}=egin{pmatrix} p_4 & p_3 & p_2 & p_1 & p_0 & 0 & 0 \0 & p_4 & p_3 & p_2 & p_1 & p_0 & 0 \0 & 0 & p_4 & p_3 & p_2 & p_1 & p_0 \q_3 & q_2 & q_1 & q_0 & 0 & 0 & 0 \0 & q_3 & q_2 & q_1 & q_0 & 0 & 0 \0 & 0 & q_3 & q_2 & q_1 & q_0 & 0 \0 & 0 & 0 & q_3 & q_2 & q_1 & q_0 \end{pmatrix}.

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:{S_{p,q^mathrm{T}cdotegin{pmatrix}x\yend{pmatrix} = egin{pmatrix}0\0end{pmatrix}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(gcd(p,q)) = m+n-mathrm{rank}~S_{p,q}.

ee also

* Transfer matrix

References

* [http://aix1.uottawa.ca/~jkhoury/elimination.htm Additional overview]


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