Numpylinalgeigh

numpylinalgeigh refers to the NumPy function numpy.linalg.eigh, used to compute the eigenvalues and eigenvectors of a Hermitian or real symmetric matrix. It is part of NumPy’s linear algebra module and provides a stable, specialized solver for this class of matrices.

The function accepts a square array a with shape (N, N). An optional UPLO parameter (default 'L') indicates which triangle of the matrix is stored and used, enabling optimizations that leverage symmetry. numpy.linalg.eigh returns two objects: w, a 1D array of length N containing the eigenvalues in ascending order, and v, a 2D array of shape (N, N) whose columns are the corresponding eigenvectors. For Hermitian inputs, the eigenvalues are real and the eigenvectors can be chosen to be orthonormal.

Numpylinalgeigh is contrasted with the more general numpy.linalg.eig, which solves for eigenvalues and eigenvectors of arbitrary square matrices and may yield complex results. eigh is specialized for Hermitian or real symmetric matrices, providing greater numerical stability and efficiency for those cases. It is common in applications involving quadratic forms, vibrational modes, or other problems where symmetry conditions apply.

Example usage: A = [[2, 1], [1, 2]] as a NumPy array; w, v = np.linalg.eigh(A). The eigenvalues are [1, 3], and the corresponding eigenvectors form the columns of v, normalized to unity length.