- Sylvester matrix
In
mathematics , a Sylvester matrix is a matrix associated to twopolynomial s that gives us some information about those polynomials. It is named forJames 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, thedeterminant of the associated Sylvester matrix (which is named theresultant 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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