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Sylvester matrix

Formally, let p and q be two nonzero polynomials, respectively of degree m and n. Thus:

${\displaystyle p($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 ${\displaystyle (n+m)\times (n+m)}$ matrix constructed as follows:

• ifn > 0, the first row is:

${\displaystyle {\begin{pmatrix}p_{m}&p_{m-1}&\cdots &p_{1}&p_{0}&0&\cdots &0\end{pmatrix}}.}$

• 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, shifting the coefficients one column to the right each time and setting the other entries in the row to be 0.
• ifm > 0 the (n + 1)th row is:

${\displaystyle {\begin{pmatrix}q_{n}&q_{n-1}&\cdots &q_{1}&q_{0}&0&\cdots &0\end{pmatrix}}.}$

• the following rows are obtained the same way as before.

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

${\displaystyle S_{p,q}={\begin{pmatrix}p_{4}&p_{3}&p_{2}&p_{1}&p_{0}&0&0\\0&p_{4}&p_{3}&p_{2}&p_{1}&p_{0}&0\\0&0&p_{4}&p_{3}&p_{2}&p_{1}&p_{0}\\q_{3}&q_{2}&q_{1}&q_{0}&0&0&0\\0&q_{3}&q_{2}&q_{1}&q_{0}&0&0\\0&0&q_{3}&q_{2}&q_{1}&q_{0}&0\\0&0&0&q_{3}&q_{2}&q_{1}&q_{0}\end{pmatrix}}.}$

If one of the degrees is zero (that is, the corresponding polynomial is a nonzero constant polynomial), then there are zero rows consisting of coefficients of the other polynomial, and the Sylvester matrix is a diagonal matrix of dimension the degree of the non-constant polynomial, with the all diagonal coefficients equal to the constant polynomial. If m = n = 0, then the Sylvester matrix is the empty matrix with zero rows and zero columns.

The above defined Sylvester matrix appears in a Sylvester paper of 1840. In a paper of 1853, Sylvester introduced the following matrix, which is, up to a permutation of the rows, the Sylvester matrix of p and q, which are both considered as having degree max(m, n).^[1] This is thus a ${\displaystyle 2\max(n,m)\times 2\max(n,m)}$-matrix containing ${\displaystyle \max(n,m)}$ pairs of rows. Assuming ${\displaystyle m>n,}$ it is obtained as follows:

• the first pair is:

${\displaystyle {\begin{pmatrix}p_{m}&p_{m-1}&\cdots &p_{n}&\cdots &p_{1}&p_{0}&0&\cdots &0\\0&\cdots &0&q_{n}&\cdots &q_{1}&q_{0}&0&\cdots &0\end{pmatrix}}.}$

• the second pair is the first pair, shifted one column to the right; the first elements in the two rows are zero.
• the remaining ${\displaystyle max(n,m)-2}$ pairs of rows are obtained the same way as above.

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

${\displaystyle {\begin{pmatrix}p_{4}&p_{3}&p_{2}&p_{1}&p_{0}&0&0&0\\0&q_{3}&q_{2}&q_{1}&q_{0}&0&0&0\\0&p_{4}&p_{3}&p_{2}&p_{1}&p_{0}&0&0\\0&0&q_{3}&q_{2}&q_{1}&q_{0}&0&0\\0&0&p_{4}&p_{3}&p_{2}&p_{1}&p_{0}&0\\0&0&0&q_{3}&q_{2}&q_{1}&q_{0}&0\\0&0&0&p_{4}&p_{3}&p_{2}&p_{1}&p_{0}\\0&0&0&0&q_{3}&q_{2}&q_{1}&q_{0}\\\end{pmatrix}}.}$

The determinant of the 1853 matrix is, up to sign, the product of the determinant of the Sylvester matrix (which is called the resultantofp and q) by ${\displaystyle p_{m}^{m-n}}$ (still supposing ${\displaystyle m\geq n}$).

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

The solutions of the simultaneous linear equations

${\displaystyle {S_{p,q}}^{\mathrm {T} }\cdot {\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}0\\0\end{pmatrix}}}$

where ${\displaystyle x}$ is a vector of size ${\displaystyle n}$ and ${\displaystyle y}$ has size ${\displaystyle m}$, comprise the coefficient vectors of those and only those pairs ${\displaystyle x,y}$ of polynomials (of degrees ${\displaystyle n-1}$ and ${\displaystyle m-1}$, respectively) which fulfill

${\displaystyle x($z)\cdot p(z)+y(z)\cdot q(z)=0,}

where polynomial multiplication and addition is used. This means the kernel of the transposed Sylvester matrix gives all solutions of the Bézout equation where ${\displaystyle \deg x<\deg q}$ and ${\displaystyle \deg y<\deg p}$.

Consequently the rank of the Sylvester matrix determines the degree of the greatest common divisorofp and q:

${\displaystyle \deg(\gcd(p,q))=m+n-\operatorname {rank} S_{p,q}.}$

Moreover, the coefficients of this greatest common divisor may be expressed as determinants of submatrices of the Sylvester matrix (see Subresultant).

• Transfer matrix
• Bézout matrix

• Weisstein, Eric W. "Sylvester Matrix". MathWorld.