Test if a 2×2 numeric matrix is explicitly antihermitian:

Test if a 3×3 symbolic matrix is explicitly antihermitian:

Any matrix generated from an antihermitian function is antihermitian:

The function is antihermitian:

Using Table generates an antihermitian matrix:

SymmetrizedArray can generate matrices (and general arrays) with symmetries:

Convert back to an ordinary matrix using Normal:

Consider the family of rotation matrices corresponding to rotation by in the plane:

The logarithmic derivative is antihermitian:

This will be true of any 1-parameter family of rotations:

Stone's theorem says that any 1-parameter family of unitary matrices has an antihermitian logarithmic derivative. Verify the theorem for the following family of matrices:

First, confirm the matrices are unitary under the assumption that is real:

Compute the logarithmic derivative:

Verify the result is antihermitian:

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:

First, verify the matrices are, in fact, unitary:

Compute the logarithmic derivative:

This matrix is antihermitian:

Define the Hamiltonian:

Verify that the matrix is Hermitian:

Its real eigenvalues represent the 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:

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:

Each row lies on the unit sphere:

AntihermitianMatrixQ[x] trivially returns False for any x that is not a matrix:

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

An antihermitian matrix must have pure imaginary diagonal elements:

Use Diagonal to pick out the diagonal elements:

A real-valued antisymmetric matrix is antihermitian:

But a complex-valued symmetric matrix may not be:

Use Symmetrize with the symmetry Antihermitian to compute the antihermitian part of a matrix:

This equals the normalized difference between m and ConjugateTranspose[m]:

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:

MatrixExp[m] for antihermitian m is unitary:

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[m,x] for antihermitian m alternates real and imaginary coefficients:

Antihermitian matrices have a complete set of eigenvectors:

As a consequence, they must be diagonalizable:

Use Eigenvectors to find eigenvectors:

Det[m] for antisymmetric m of odd dimensions is imaginary:

If m has even dimensions, its determinant is real:

The inverse of an antihermitian matrix is antihermitian:

Odd powers of an antihermitian matrix are antihermitian:

Even powers are Hermitian:

A complex antisymmetric matrix is not antihermitian: