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 <math>M</math> with the property that <math>p(x_1,x_2,\ldots,x_n) = (\mathbf{x}^{\otimes d})^T M(\mathbf{x}^{\otimes d})</math>, where <math>\mathbf{x} = (x_1,x_2,\ldots,x_n)</math> as a column vector. Here, "fully symmetric" means that <math>M^T = M</math>, <math>M</math> is supported on the symmetric subspace (i.e., <math>PMP = M</math>, where <math>P</math> is the SymmetricProjection), and <math>M</math> equals its own PartialTranspose (across any bipartition).

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

Syntax

• Coming soon.

Argument descriptions

• Coming soon.

Examples

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Source code

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

References

Coming soon.