PolynomialAsMatrix: Difference between revisions
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<tt>'''PolynomialAsMatrix'''</tt> is a [[List of functions|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 [[symmetric projection | <tt>'''PolynomialAsMatrix'''</tt> is a [[List of functions|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|symmetric projection]]), and <math>M</math> equals its own [[PartialTranspose|partial transpose]] (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>. | 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>. | ||
Revision as of 02:56, 1 August 2023
| 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 symmetric projection), and <math>M</math> equals its own partial transpose (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
Coming soon.
Source code
Click here to view this function's source code on github.
References
Coming soon.