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Справка Scilab >> Polynomials > sylm

# sylm

Sylvester matrix of two polynomials

### Syntax

`S = sylm(a, b)`

### Arguments

a, b

Two polynomials with real or complex coefficients.

S

matrix of real or complex numbers

### Description

`sylm(a,b)` gives the Sylvester matrix associated to polynomials `a` and `b`, i.e. the matrix `S` such that:

`coeff( a*x + b*y )' = S * [coeff(x)';coeff(y)']`.

Dimension of `S` is equal to `degree(a)+degree(b)`.

If `a` and `b` are coprime polynomials then `rank(sylm(a,b))=degree(a)+degree(b))` and the instructions

```u = sylm(a,b) \ eye(na+nb,1)
x = poly(u(1:nb),'z','coeff')
y = poly(u(nb+1:na+nb),'z','coeff')
```

compute Bezout factors `x` and `y` of minimal degree such that `a*x+b*y = 1`

### Examples

```x = poly(0,"x");
y = poly ([1, 2, 3], "x", "coeff")
sylm(x, y)```
```--> x = poly(0, "x");
--> y = poly([1, 2, 3], "x","coeff")
y  =
1 +2x +3x²

--> sylm(x, y)
ans  =
0.   0.   1.
1.   0.   2.
0.   1.   3.
```

### See Also

• bezout — GCD of two polynomials or two integers, by the Bezout method
• diophant — Solves the diophantine (Bezout) equation p1*x1 + p2*x2 = b

### Comments

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