# CoefficientArrays

CoefficientArrays[polys,vars]

gives the arrays of coefficients of the variables vars in the polynomials polys.

# Details and Options • CoefficientArrays gives a list containing SparseArray objects, which can be converted to ordinary arrays using Normal.
• If CoefficientArrays[polys,vars] gives {m0,m1,m2,}, then polys can be reconstructed as m0+m1.vars+m2.vars.vars+. »
• Any element of polys of the form lhs==rhs is taken to correspond to the polynomial lhs-rhs. »
• CoefficientArrays[polys,{form1,form2,}] takes all expressions in polys that match any of the formi to be variables.
• CoefficientArrays[polys] is equivalent to CoefficientArrays[polys,Variables[polys]].
• The length of the list CoefficientArrays[polys,vars] is one more than the total degree of polys.
• The mi are sparse arrays with ranks .
• The first element m0 has the same length as the list polys.
• If polys is a single polynomial rather than a list, m0 is also not a list.
• For linear equations, the solution to Thread[polys==0] is given by LinearSolve[m1,-m0].
• For nonlinear equations, the mi are not unique. CoefficientArrays by default assigns nonzero coefficients only to monomials where the variables appear in the same order as vars.
• CoefficientArrays[polys,vars,"Symmetric"->True] makes all the mi symmetric in all their indices. The resulting arrays will generally be less sparse.

# Examples

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## Scope(4)

Linear polynomial:

List of linear polynomials:

Nonlinear polynomials:

Polynomial equations:

## Options(1)

### "Symmetric"(1)

The default coefficient of a quadratic form corresponding to y x is zero:

With "Symmetric"->True, CoefficientArrays gives symmetric arrays:

## Applications(1)

Convert linear equations to arrays suitable for LinearSolve:

## Properties & Relations(3)

The polynomial can be reconstructed by using Dot (.):

The symmetric coefficient arrays correspond to vector derivatives at the origin:

CoefficientArrays gives a list of arrays of polynomial coefficients ordered by total degrees:

CoefficientList gives a tensor of the coefficients:

Use Coefficient to find the coefficient of :

In ca the position of this coefficient is a+b+1 followed by a 1s and b 2s (1 and 2 indicate the first and second variables):

In cl the coefficient of x^a y^b is the element at position {a+1,b+1}: