AntisymmetricMatrixQ

AntisymmetricMatrixQ[m]

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:
  • SameTestAutomaticfunction to test equality of expressions
    ToleranceAutomatictolerance 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  (2)

Test if a numeric 2×2 matrix is explicitly antisymmetric:

Test if a symbolic 3×3 matrix is antisymmetric:

Scope  (10)

Basic Uses (6)

Test if a real machine-precision matrix is antisymmetric:

A real antisymmetric matrix is also antihermitian:

Test if a complex matrix is antisymmetric:

A complex antisymmetric matrix has antisymmetric real and imaginary parts:

Test if an exact matrix is antisymmetric:

Make the matrix antisymmetric:

Use AntisymmetricMatrixQ with an arbitrary-precision matrix:

A random matrix is typically not antisymmetric:

Use AntisymmetricMatrixQ with a symbolic matrix:

The matrix becomes antisymmetric when and :

AntisymmetricMatrixQ works efficiently with large numerical matrices:

Special Matrices (4)

Use AntisymmetricMatrixQ with sparse matrices:

Use AntisymmetricMatrixQ with structured matrices:

Use with a QuantityArray structured matrix:

The identity matrix is not antisymmetric:

HilbertMatrix is not antisymmetric:

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  (5)

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:

Consider the family of rotation matrices corresponding to rotation by in the plane:

The logarithmic derivative r^(').TemplateBox[{r}, Inverse] is antisymmetric:

This will be true of any 1-parameter family of rotations:

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:

Represent the cross product by means of multiplication by the antisymmetric matrix :

Compute the exponential and use it to define a solution to the equation:

Verify that satisfies the differential equation and initial condition:

The matrix is orthogonal for all values of :

Thus, the orbit of the solution is at a constant distance from the origin, in this case a circle:

Properties & Relations  (15)

AntiymmetricMatrixQ[x] trivially returns False for any x that is not a matrix:

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

An antisymmetric matrix must have zeros on the diagonal:

Use Diagonal to pick out the diagonal elements:

A real-valued antisymmetric matrix is antihermitian:

But a complex-valued antisymmetric matrix may not be:

Use Symmetrize with the symmetry Antisymmetric to compute the antisymmetric part of a matrix:

This equals the normalized difference between m and Transpose[m]:

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

Use SymmetricMatrixQ to test whether a matrix is symmetric:

If is an antisymmetric matrix with real entries, then is Hermitian:

MatrixExp[m] for real antisymmetric m is both orthogonal and unitary:

For complex antisymmetric m, the exponential is orthogonal but not, in general, unitary:

A real-valued antisymmetric matrix is always a normal matrix:

A complex-valued antisymmetric matrix need not be normal:

Real-valued antisymmetric matrices have pure imaginary eigenvalues:

Use Eigenvalues to find eigenvalues:

Note that a complex-valued antisymmetric matrix may have both real and complex eigenvalues:

Consider a real antisymmetric m of even dimensions:

CharacteristicPolynomial[m,x] contains only even powers of x:

For an odd-dimensioned m, the polynomial contains only odd powers:

Real-valued antisymmetric matrices have a complete set of eigenvectors:

As a consequence, they must be diagonalizable:

Use Eigenvectors to find the necessarily complex-valued eigenvectors:

Note that a complex-valued antisymmetric matrix need not have these properties:

Det[m] for antisymmetric m of odd dimensions is zero:

If m has even dimensions and its entries are real, its determinant is non-negative:

The inverse of an antisymmetric matrix is antisymmetric:

Odd powers of an antisymmetric matrix are antisymmetric:

Even powers are symmetric:

Possible Issues  (1)

AntisymmetricMatrixQ uses the definition TemplateBox[{m}, Transpose]=-m for both real- and complex-valued matrices:

These complex matrices need not be normal or possess many properties of skew-adjoint (real antisymmetric) matrices:

AntihermitianMatrixQ tests the condition TemplateBox[{m}, ConjugateTranspose]=-m for skew-adjoint matrices:

Alternatively, test if the entries are real to restrict to real symmetric matrices:

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.

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

BibTeX

@misc{reference.wolfram_2022_antisymmetricmatrixq, author="Wolfram Research", title="{AntisymmetricMatrixQ}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/AntisymmetricMatrixQ.html}", note=[Accessed: 02-July-2022 ]}

BibLaTeX

@online{reference.wolfram_2022_antisymmetricmatrixq, organization={Wolfram Research}, title={AntisymmetricMatrixQ}, year={2014}, url={https://reference.wolfram.com/language/ref/AntisymmetricMatrixQ.html}, note=[Accessed: 02-July-2022 ]}