gftool.matrix.decompose_her¶

gftool.matrix.decompose_her(her_mat, check=True) → gftool.matrix.UDecomposition [source]¶

Decompose Hermitian matrix her_mat into eigenvalues and (right) eigenvectors.

Decompose the her_mat into rv, eig, rv_inv, with her_mat = (rv * eig) @ rv_inv. This is the unitary similarity transformation:

\[M = U Λ U^†, Λ = diag(λ₀, λ₁, …)\]

where \(λₗ\) are the eigenvalues and \(U\) the unitary matrix of right eigenvectors returned as rv with \(U^{-1} = U^†\). Internally, this is just a wrapper for numpy.linalg.eigh.

Parameters

her_mat(…, N, N) complex np.ndarray: matrix to be decomposed
checkbool, optional: If check, raise an error if her_mat is not Hermitian. (default: True)

Returns

Decomposition.rv(…, N, N) complex np.ndarray: The right eigenvectors \(U\)
Decomposition.eig(…, N) complex np.ndarray: The complex eigenvalues of her_mat
Decomposition.rv_inv(…, N, N) complex np.ndarray: The inverse of the right eigenvectors \(U\)

Raises

ValueError: If check=True and her_mat is not Hermitian.

Examples

Perform the eigendecomposition:

>>> matrix = np.random.random((10, 10)) + 1j*np.random.random((10, 10))
>>> her_mat = 0.5*(matrix + matrix.conj().T)
>>> rv, eig, rv_inv = gt.matrix.decompose_her(her_mat)
>>> np.allclose(her_mat, (rv * eig) @ rv_inv)
True
>>> np.allclose(rv @ rv.conj().T, np.eye(*her_mat.shape))
True

This can also be simplified using the Decomposition class

>>> dec = gt.matrix.decompose_her(her_mat)
>>> np.allclose(her_mat, dec.reconstruct())
True