gftool.matrix.Decomposition

class gftool.matrix.Decomposition(rv: ndarray, eig: ndarray, rv_inv: ndarray)[source]

Decomposition of a matrix into eigenvalues and eigenvectors.

\[M = P Λ P^{-1}, Λ = diag(λ₀, λ₁, …)\]

This class holds the eigenvalues and eigenvectors of the decomposition of a matrix and offers methods to reconstruct it. One intended use case is to use the Decomposition for the inversion of the Green’s function to calculate it from the resolvent.

The order of the attributes is always rv, eig, rv_inv, as this gives the reconstruct of the matrix: mat = (rv * eig) @ rv_inv

Parameters:

rv(…, N, N) complex np.ndarray: The matrix of right eigenvectors.
eig(…, N) complex np.ndarray: The vector of eigenvalues.
rv_inv(…, N, N) complex np.ndarray: The inverse of rv.

Examples

Perform the eigendecomposition:

>>> matrix = np.random.random((10, 10))
>>> dec = gt.matrix.decompose_mat(matrix)
>>> np.allclose(matrix, dec.reconstruct())
True

Inversion of matrix

>>> matrix_inv = dec.reconstruct(eig=1.0/dec.eig)
>>> np.allclose(np.linalg.inv(matrix), matrix_inv)
True

__init__(rv: ndarray, eig: ndarray, rv_inv: ndarray) → None

Methods

`__init__`(rv, eig, rv_inv)
`count`(value)
`from_gf`(gf)	Decompose the inverse Green's function matrix.
`from_hamiltonian`(hamilton)	Decompose the Hamiltonian matrix.
`index`(value, [start, [stop]])	Raises ValueError if the value is not present.
`reconstruct`([eig, kind])	Get matrix back from `Decomposition`.

Attributes

`rv`	The matrix of right eigenvectors.
`eig`	The vector of eigenvalues.
`rv_inv`	The inverse of `rv`.