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Concept
Sylvester matrix
Summary
In mathematics, a Sylvester matrix is a matrix associated to two univariate polynomials with coefficients in a field or a commutative ring. The entries of the Sylvester matrix of two polynomials are coefficients of the polynomials. The determinant of the Sylvester matrix of two polynomials is their resultant, which is zero when the two polynomials have a common root (in case of coefficients in a field) or a non-constant common divisor (in case of coefficients in an integral domain).
Sylvester matrices are named after James Joseph Sylvester.
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 constructed as follows:
if n > 0, 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, shifting the coefficients one column to the right each time and setting the other entries in the row to be 0.
if m > 0 the (n + 1)th row is:
the following rows are obtained the same way as before.
Thus, if m = 4 and n = 3, the matrix is:
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).
This is thus a -matrix containing pairs of rows. Assuming it is obtained as follows:
the first pair is:
the second pair is the first pair, shifted one column to the right; the first elements in the two rows are zero.
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