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 .
Examplesopen allclose all
Approximate MachinePrecision matrix:
Adjust the option Tolerance to accept this matrix as antihermitian:
Using Table generates an antihermitian matrix:
Properties & Relations (11)
A matrix is antihermitian if m==-ConjugateTranspose[m]:
Use HermitianMatrixQ to test whether a matrix is Hermitian:
Use NormalMatrixQ to test whether the matrix is normal:
Use Eigenvalues to find eigenvalues:
CharacteristicPolynomial for such a matrix has alternating real and imaginary coefficients:
Use Eigenvectors to find eigenvectors:
Det for a complex antihermitian matrix of odd order is imaginary:
MatrixExp of an antihermitian matrix is unitary:
An antisymmetric matrix is always diagonalizable as tested with DiagonalizableMatrixQ:
Wolfram Research (2014), AntihermitianMatrixQ, Wolfram Language function, https://reference.wolfram.com/language/ref/AntihermitianMatrixQ.html.
Wolfram Language. 2014. "AntihermitianMatrixQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/AntihermitianMatrixQ.html.
Wolfram Language. (2014). AntihermitianMatrixQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AntihermitianMatrixQ.html