Details and Options
- HermitianMatrixQ is also known as a self-adjoint.
- A matrix m is Hermitian if m==ConjugateTranspose[m].
- HermitianMatrixQ 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 HermitianMatrixQ with an arbitrary-precision matrix:
Use HermitianMatrixQ with a symbolic matrix:
HermitianMatrixQ works efficiently with large numerical matrices:
Adjust the option Tolerance to accept this matrix as Hermitian:
Sources of Hermitian Matrices (5)
By using Table, it generates a Hermitian matrix:
SymmetrizedArray can generate matrices (and general arrays) with symmetries:
Convert back to an ordinary matrix using Normal:
Several statistical measures of complex data are Hermitian matrices, including Covariance:
Matrices drawn from GaussianUnitaryMatrixDistribution are Hermitian:
Matrices drawn from GaussianSymplecticMatrixDistribution are Hermitian:
Uses of Hermitian Matrices (3)
In quantum mechanics, time evolution is represented by a 1-parameter family of unitary matrices . The 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:
Properties & Relations (15)
A matrix is Hermitian if m==ConjugateTranspose[m]:
Use Diagonal to pick out the diagonal elements:
This equals mean of m and ConjugateTranspose[m]:
Use AntihermitianMatrixQ to test whether a matrix is antihermitian:
Use NormalMatrixQ to test whether a matrix is normal:
Use Eigenvalues to find eigenvalues:
CharacteristicPolynomial[m,x] for Hermitian m has real coefficients:
Use Eigenvectors to find eigenvectors:
Use Det to compute the determinant:
Real-valued matrix functions of Hermitian matrices are Hermitian, including MatrixExp:
And any univariate function representable using MatrixFunction:
Wolfram Research (2007), HermitianMatrixQ, Wolfram Language function, https://reference.wolfram.com/language/ref/HermitianMatrixQ.html (updated 2014).
Wolfram Language. 2007. "HermitianMatrixQ." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/HermitianMatrixQ.html.
Wolfram Language. (2007). HermitianMatrixQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HermitianMatrixQ.html