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
Basic Examples (2)
Basic Uses (6)
Use AntihermitianMatrixQ with an arbitrary-precision matrix:
Use AntihermitianMatrixQ with a symbolic matrix:
AntihermitianMatrixQ works efficiently with large numerical matrices:
Adjust the option Tolerance to accept this matrix as antihermitian:
Using Table generates an antihermitian matrix:
SymmetrizedArray can generate matrices (and general arrays) with symmetries:
Convert back to an ordinary matrix using Normal:
In quantum mechanics, time evolution is represented by a 1-parameter family of unitary matrices . times the logarithmic derivative of is a Hermitian matrix called the Hamiltonian or energy operator . Its eigenvalues represent the possible energies of the system. For the following time evolution, compute the Hamiltonian and possible energies:
The exponential MatrixExp[v] of an antihermitian matrix is unitary. Define a matrix function through its differential equation with initial value , and show that the solution is unitary:
Properties & Relations (15)
A matrix is antihermitian if m==-ConjugateTranspose[m]:
Use Diagonal to pick out the diagonal elements:
This equals the normalized difference between m and ConjugateTranspose[m]:
Use HermitianMatrixQ to test whether a matrix is Hermitian:
MatrixExp[m] for antihermitian m is unitary:
Use NormalMatrixQ to test whether the matrix is normal:
Use Eigenvalues to find eigenvalues:
CharacteristicPolynomial[m,x] for antihermitian m alternates real and imaginary coefficients:
Use Eigenvectors to find eigenvectors:
Det[m] for antisymmetric m of odd dimensions is imaginary:
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