# AntihermitianMatrixQ

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

# Details and Options • AntihermitianMatrixQ is also known as skew-Hermitian.
• A matrix m is antihermitian if m-ConjugateTranspose[m].
• AntihermitianMatrixQ 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 matrix is explicitly antihermitian:

## Scope(4)

Real matrix:

Complex matrix:

A complex antihermitian matrix has antisymmetric real part and symmetric imaginary part:

Dense matrix:

Sparse matrix:

Approximate MachinePrecision matrix:

Approximate arbitrary-precision matrix:

Symbolic matrix:

The matrix becomes antihermitian when c=-b and the real part of a and d is zero:

All symbolic quantities are assumed to be complex:

## Options(2)

### SameTest(1)

This matrix is antihermitian for a positive real , but AntihermitianMatrixQ gives False:

Use the option SameTest to get the correct answer:

### Tolerance(1)

A complex-valued antihermitian matrix with some random perturbation of order :

Adjust the option Tolerance to accept this matrix as antihermitian:

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

## Applications(2)

Any matrix generated from an antihermitian function is antihermitian:

The function is antihermitian:

Using Table generates an antihermitian matrix:

The of an antihermitian matrix is unitary. Define a matrix function through its differential equation with initial value , and show that the solution is unitary:

Solve and check that the resulting matrix is unitary at each time:

With default settings, you get approximately unitary matrices:

The matrix 2-norm of the solution is 1:

Plot the rows of the matrix:

## Properties & Relations(11)

A matrix is antihermitian if m==-ConjugateTranspose[m]:

AntisymmetricMatrixQ gives True for antihermitian real-valued matrices:

Any matrix can be represented as the sum of its Hermitian and antihermitian parts:

Use HermitianMatrixQ to test whether a matrix is Hermitian:

If is a Hermitian matrix, then is antihermitian:

An antihermitian matrix is always a normal matrix:

Use NormalMatrixQ to test whether the matrix is normal:

Antihermitian matrices have eigenvalues on the imaginary axis:

Use Eigenvalues to find eigenvalues:

CharacteristicPolynomial for such a matrix has alternating real and imaginary coefficients:

Antihermitian matrices have a complete set of eigenvectors:

Use Eigenvectors to find eigenvectors:

Det for a complex antihermitian matrix of odd order is imaginary:

For even order it is real:

MatrixExp of an antihermitian matrix is unitary:

For real-valued matrices, AntisymmetricMatrixQ and AntihermitianMatrixQ coincide:

For complex-valued matrices, they differ:

An antisymmetric matrix is always diagonalizable as tested with DiagonalizableMatrixQ:

## Possible Issues(1)

A complex antisymmetric matrix is not antihermitian: