# SymmetrizedArray

SymmetrizedArray[{pos1val1,pos2val2,},dims,sym]

yields an array of dimensions dims whose entries are given by those in the rules posivali or through the symmetry sym.

SymmetrizedArray[list]

yields a symmetrized array version of list.

# Details

• is converted to a structured array expression of the form SymmetrizedArray[StructuredData[dims,{comps,sym}]] that contains the independent components comps and the symmetry sym of the input.
• SymmetrizedArray takes unspecified elements to be 0.
• Normal[SymmetrizedArray[]] gives the ordinary array corresponding to a symmetrized array object.
• SymmetrizedArrayRules[SymmetrizedArray[]] gives the list of independent rules {pos1val1,pos2val2,}.
• ArrayRules[SymmetrizedArray[]] gives the list of both dependent and independent rules {pos1val1,pos2val2,}.
• The elements in a SymmetrizedArray object need not be numeric.
• The position specifications posi can contain patterns.
• With rules posivali, the vali are evaluated separately for each independent component that matches posi.
• SymmetrizedArray[list] requires that list be a full array, with all parts at a particular level being lists of the same length.
• The individual elements of a symmetrized array cannot themselves be lists.
• SymmetrizedArray[rules] yields a symmetrized array, with dimensions exactly large enough to include elements whose positions have been explicitly specified.
• SymmetrizedArray[structureddata] is treated as a raw object by functions like AtomQ and for purposes of pattern matching.

# Examples

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## Basic Examples(2)

Construct an antisymmetric array:

Convert it to an ordinary matrix:

Convert an array with symmetry into its symmetrized form:

These are its independent components:

## Scope(8)

Construct a symmetrized array from rules:

SymmetrizedArray form of an array with symmetry:

SymmetrizedArray form of an array with antisymmetry:

Construct a symmetrized array of larger dimensions:

Let the Wolfram Language choose the minimal dimensions:

Construct a random skew-symmetric array:

Specify an arbitrary symmetry using generators:

An empty list of generators represents no symmetry:

Use generators with any root of unity:

Symmetrized arrays include properties that give information about the array:

The "Summary" property gives a brief summary of information about the array:

The "StructuredAlgorithms" property gives a list of functions that have algorithms that use the structure of the representation:

## Applications(5)

Construct an antisymmetric array:

Only the independent components are stored:

Extract any component of the array:

Extract a subarray:

Construct an antisymmetric matrix:

Multiple tensor product of the matrix with itself:

The result contains only 15 independent components, all with different values:

The sparse and normal representations are larger:

Antisymmetric arrays of ranks 4 and 10 in dimension 15:

The wedge product of those antisymmetric arrays can be computed efficiently:

The result can be presented in a shorter form through its Hodge dual:

Construct the array of all sixth-order partial derivatives of a function of four variables:

It is a depth-6 array in dimension 4, and therefore it has 4096 entries:

It is also a fully symmetric array, due to commutation of the partial derivatives:

Most entries are repeated multiple times and therefore the array is large:

The SymmetrizedArray representation stores each of the independent entries only once:

The normal form of the array can be recovered using Normal:

Construct a 4-variate distribution, using a symmetric matrix Σ:

Define a function that computes a moment of that distribution for given variable indices:

Check a particular case:

Construct the array of moments of order 6:

Moment and cumulant arrays for a multivariate distribution are fully symmetric:

Use SymmetrizedArray to compute each independent moment only once:

Both representations are equivalent:

Compare sizes:

## Properties & Relations(5)

If the same entry is specified several times, the average of those values is used:

For a nonsymmetric array, the result is a projected symmetrized part:

SymmetrizedArray offers a very compact representation of antisymmetric arrays:

SymmetrizedArray allows a compact representation of symmetric arrays:

A symmetric matrix can be represented using SymmetrizedArray or SymmetricMatrix:

The two representations are equal, but support different algorithms:

SymmetrizedArray supports tensorial operations such as D, Flatten, Inner and Outer:

SymmetricMatrix supports matrix-specific operations such as KroneckerProduct:

HermitianMatrix bears an analogous relationship to SymmetrizedArray for Hermitian matrices:

## Possible Issues(1)

Some symmetry specifications are only compatible with the array having only vanishing entries:

## Neat Examples(1)

Antisymmetric rank-4 array projected down to a symmetric matrix in different dimensions:

Equal elements correspond to the same color. White elements are zeros:

Wolfram Research (2012), SymmetrizedArray, Wolfram Language function, https://reference.wolfram.com/language/ref/SymmetrizedArray.html (updated 2020).

#### Text

Wolfram Research (2012), SymmetrizedArray, Wolfram Language function, https://reference.wolfram.com/language/ref/SymmetrizedArray.html (updated 2020).

#### CMS

Wolfram Language. 2012. "SymmetrizedArray." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2020. https://reference.wolfram.com/language/ref/SymmetrizedArray.html.

#### APA

Wolfram Language. (2012). SymmetrizedArray. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SymmetrizedArray.html

#### BibTeX

@misc{reference.wolfram_2024_symmetrizedarray, author="Wolfram Research", title="{SymmetrizedArray}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/SymmetrizedArray.html}", note=[Accessed: 10-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_symmetrizedarray, organization={Wolfram Research}, title={SymmetrizedArray}, year={2020}, url={https://reference.wolfram.com/language/ref/SymmetrizedArray.html}, note=[Accessed: 10-September-2024 ]}