# AntisymmetricMatrixQ

gives True if m is explicitly antisymmetric, and False otherwise.

# Details and Options

• AntisymmetricMatrixQ is also known as skew-symmetric.
• A matrix m is antisymmetric if m-Transpose[m].
• AntisymmetricMatrixQ works for symbolic as well as numerical matrices.
• The following options can be given:
•  SameTest Automatic function to test equality of expressions Tolerance Automatic tolerance for approximate numbers
• For exact and symbolic matrices, the option SameTest->f indicates that two entries mij and mkl are taken to be equal if f[mij,mkl] gives True.
• For approximate matrices, the option Tolerance->t can be used to indicate that all entries Abs[mij]t are taken to be zero.
• For matrix entries Abs[mij]>t, equality comparison is done except for the last bits, where is \$MachineEpsilon for MachinePrecision matrices and for matrices of Precision .

# Examples

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## Basic Examples(1)

Test if a numeric matrix is explicitly antisymmetric:

## Scope(5)

Real matrix:

Complex matrix:

A complex antisymmetric matrix has antisymmetric real and imaginary parts:

Dense matrix:

Sparse matrix:

Approximate MachinePrecision matrix:

Approximate arbitrary-precision matrix:

Symbolic matrix:

The matrix becomes antisymmetric when c = -b and a = d = 0:

Structured matrix:

## Options(2)

### SameTest(1)

This matrix is antisymmetric for a positive real , but AntisymmetricMatrixQ gives False:

Use the option SameTest to get the correct answer:

### Tolerance(1)

Generate a real-valued antisymmetric matrix with some random perturbation of order :

Adjust the option Tolerance to accept this matrix as antisymmetric:

The norm of the difference between the matrix and its transpose with opposite sign:

## Applications(4)

Any matrix generated from an antisymmetric function is antisymmetric:

The function is antisymmetric:

Using Table generates an antisymmetric matrix:

SymmetrizedArray can generate matrices (and general arrays) with symmetries:

The cross product of two vectors can be expressed as a product of an antisymmetric matrix and a vector:

This proves that :

This proves :

Find the function satisfying this time-dependent 3D equation:

Since can be expressed as an antisymmetric matrix, an ODE can be solved:

Here is the 3D plot of x[t] and x'[t]:

## Properties & Relations(9)

A matrix is antisymmetric if m-Transpose[m]:

AntihermitianMatrixQ gives True for antisymmetric real-valued matrices:

Any matrix can be represented as the sum of its symmetric and antisymmetric parts:

Use SymmetricMatrixQ to test whether a matrix is symmetric:

An antisymmetric matrix is always a normal matrix:

Use NormalMatrixQ to test whether the matrix is normal:

Eigenvalues for a real antisymmetric matrix are on the imaginary axis:

Use Eigenvalues to find eigenvalues:

CharacteristicPolynomial for such a matrix contains even powers only:

And for an odd-dimensioned matrix it contains odd powers only:

Antisymmetric matrices have a complete set of eigenvectors:

Use Eigenvectors to find eigenvectors:

Det for a real antisymmetric matrix of odd order is zero:

For even order it is non-negative:

MatrixExp of an antisymmetric matrix is orthonormal:

An antisymmetric matrix is always diagonalizable as tested with DiagonalizableMatrixQ:

## Possible Issues(1)

A complex antisymmetric matrix is not antihermitian:

Wolfram Research (2014), AntisymmetricMatrixQ, Wolfram Language function, https://reference.wolfram.com/language/ref/AntisymmetricMatrixQ.html.

#### Text

Wolfram Research (2014), AntisymmetricMatrixQ, Wolfram Language function, https://reference.wolfram.com/language/ref/AntisymmetricMatrixQ.html.

#### BibTeX

@misc{reference.wolfram_2020_antisymmetricmatrixq, author="Wolfram Research", title="{AntisymmetricMatrixQ}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/AntisymmetricMatrixQ.html}", note=[Accessed: 23-January-2021 ]}

#### BibLaTeX

@online{reference.wolfram_2020_antisymmetricmatrixq, organization={Wolfram Research}, title={AntisymmetricMatrixQ}, year={2014}, url={https://reference.wolfram.com/language/ref/AntisymmetricMatrixQ.html}, note=[Accessed: 23-January-2021 ]}

#### CMS

Wolfram Language. 2014. "AntisymmetricMatrixQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/AntisymmetricMatrixQ.html.

#### APA

Wolfram Language. (2014). AntisymmetricMatrixQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AntisymmetricMatrixQ.html