PolynomialAsMatrix

PolynomialAsMatrix
Creates a compact fully symmetric matrix representation of a polynomial
Other toolboxes required	none
Related functions	PolynomialOptimize PolynomialSOS
Function category	Polynomial optimization
Usable within CVX?	no

PolynomialAsMatrix is a function that computes a compact form of a fully symmetric matrix representation of an even-degree homogeneous polynomial. More specifically, if p is a homogeneous polynomial of degree 2d then there is a unique fully symmetric matrix \(M\) with the property that \(p(x_1,x_2,\ldots,x_n) = (\mathbf{x}^{\otimes d})^T M(\mathbf{x}^{\otimes d})\), where \(\mathbf{x} = (x_1,x_2,\ldots,x_n)\) as a column vector. Here, "fully symmetric" means that \(M^T = M\), \(M\) is supported on the symmetric subspace (i.e., \(PMP = M\), where \(P\) is the symmetric projection), and \(M\) equals its own partial transpose (across any bipartition).

This function returns this fully symmetric matrix \(M\), in symmetric coordinates. That is, there is an isometry \(V\) from the symmetric subspace of \((\mathbb{C}^n)^{\otimes d}\) to \((\mathbb{C}^n)^{\otimes d}\) itself with the property that the output of this function equals \(V^*MV\).

Syntax

• Coming soon.

Argument descriptions

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Examples

Coming soon.

Source code

Click here to view this function's source code on github.

References

Coming soon.