HermitianMatrixQ [m]
gives True if m is explicitly Hermitian, and False otherwise.
HermitianMatrixQ
HermitianMatrixQ [m]
gives True if m is explicitly Hermitian, and False otherwise.
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:
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- 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
open all close allBasic Examples (2)
Test if a 2×2 numeric matrix is explicitly Hermitian:
Test if a 3×3 symbolic matrix is explicitly Hermitian:
Scope (10)
Basic Uses (6)
Test if a real machine-precision matrix is Hermitian:
A real Hermitian matrix is also symmetric:
Test if a complex matrix is Hermitian:
A complex Hermitian matrix has symmetric real part and antisymmetric imaginary part:
Test if an exact matrix is Hermitian:
Make the matrix Hermitian:
Use HermitianMatrixQ with an arbitrary-precision matrix:
A random matrix is typically not Hermitian:
Use HermitianMatrixQ with a symbolic matrix:
The matrix becomes Hermitian when c=TemplateBox[{b}, Conjugate] and diagonal entries are explicitly real-valued:
HermitianMatrixQ works efficiently with large numerical matrices:
Special Matrices (4)
Use HermitianMatrixQ with sparse matrices:
Use HermitianMatrixQ with structured matrices:
Use with a QuantityArray structured matrix:
The identity matrix is Hermitian:
HilbertMatrix is Hermitian:
Options (2)
SameTest (1)
This matrix is Hermitian for a positive real , but HermitianMatrixQ gives False :
Use the option SameTest to get the correct answer:
Tolerance (1)
Generate a complex-valued Hermitian matrix with some random perturbation of order 10-14:
Adjust the option Tolerance to accept this matrix as Hermitian:
The norm of the difference between the matrix and its conjugate transpose:
Applications (8)
Sources of Hermitian Matrices (5)
A matrix generated from a Hermitian function is Hermitian:
The function is Hermitian:
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 :
The Pauli matrices are Hermitian:
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)
A positive-definite, Hermitian matrix or metric defines an inner product by :
Verify that is in fact Hermitian and positive definite:
Orthogonalize the standard basis of TemplateBox[{}, Complexes]^n to find an orthonormal basis:
Confirm that this basis is orthonormal with respect to the inner product :
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:
First, verify the matrices are, in fact, unitary under the assumptions that and are real:
Compute the logarithmic derivative:
This matrix is antihermitian:
Define the Hamiltonian:
Verify that the matrix is Hermitian:
Its real eigenvalues represent the possible energies:
Use a different method for Hermitian matrices, with failover to a general method:
Construct complex-valued matrices for testing:
For the non-Hermitian matrix m, the function myLS just uses Gaussian elimination:
For the Hermitian indefinite matrix mh, try Cholesky and continue with Gaussian elimination:
For the Hermitian positive definite matrix mpd, try Cholesky, which succeeds:
Properties & Relations (16)
HermitianMatrixQ [x] trivially returns False for any x that is not a matrix:
A matrix is Hermitian if m==ConjugateTranspose [m]:
A Hermitian matrix must have real diagonal elements:
Use Diagonal to pick out the diagonal elements:
A real-valued symmetric matrix is Hermitian:
But a complex-valued symmetric matrix may not be:
Use Symmetrize with the symmetry Hermitian to compute the Hermitian part of a matrix:
This equals mean of m and ConjugateTranspose [m]:
Any matrix can be represented as the sum of its Hermitian and antihermitian parts:
Use AntihermitianMatrixQ to test whether a matrix is antihermitian:
If is a Hermitian matrix, then is antihermitian:
MatrixExp [I h] is unitary for any Hermitian matrix h:
A Hermitian matrix is always a normal matrix:
Use NormalMatrixQ to test whether a matrix is normal:
Hermitian matrices have all real eigenvalues:
Use Eigenvalues to find eigenvalues:
CharacteristicPolynomial [m,x] for Hermitian m has real coefficients:
Moreover, it can be factored into linear terms:
Hermitian matrices have a complete set of eigenvectors:
As a consequence, they must be diagonalizable:
Use Eigenvectors to find eigenvectors:
Hermitian matrices have a real-valued determinant:
Use Det to compute the determinant:
The inverse of a Hermitian matrix is Hermitian:
Real-valued matrix functions of Hermitian matrices are Hermitian, including MatrixExp :
And any univariate analytic function representable using MatrixFunction :
Note that while integer matrix powers are Hermitian, noninteger powers are not:
HermitianMatrix can be used to explicitly construct Hermitian matrices:
These satisfy HermitianMatrixQ :
Possible Issues (1)
A complex symmetric matrix is not Hermitian:
Related Guides
Text
Wolfram Research (2007), HermitianMatrixQ, Wolfram Language function, https://reference.wolfram.com/language/ref/HermitianMatrixQ.html (updated 2014).
CMS
Wolfram Language. 2007. "HermitianMatrixQ." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/HermitianMatrixQ.html.
APA
Wolfram Language. (2007). HermitianMatrixQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HermitianMatrixQ.html
BibTeX
@misc{reference.wolfram_2025_hermitianmatrixq, author="Wolfram Research", title="{HermitianMatrixQ}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/HermitianMatrixQ.html}", note=[Accessed: 07-January-2026]}
BibLaTeX
@online{reference.wolfram_2025_hermitianmatrixq, organization={Wolfram Research}, title={HermitianMatrixQ}, year={2014}, url={https://reference.wolfram.com/language/ref/HermitianMatrixQ.html}, note=[Accessed: 07-January-2026]}