random_spd_matrix¶

probnum.problems.zoo.linalg.random_spd_matrix(rng, dim, spectrum=None)[source]¶

Random symmetric positive definite matrix.

Constructs a random symmetric positive definite matrix from a given spectrum. An orthogonal matrix \(Q\) with \(\operatorname{det}(Q)\) (a rotation) is sampled with respect to the Haar measure and the diagonal matrix containing the eigenvalues is rotated accordingly resulting in \(A=Q \operatorname{diag}(\lambda_1, \dots, \lambda_n)Q^\top\). If no spectrum is provided, one is randomly drawn from a Gamma distribution.

Parameters

rng (Generator) – Random number generator.
dim (IntLike) – Matrix dimension.
spectrum (Optional[Sequence]) – Eigenvalues of the matrix.

Return type

ndarray

See also

random_sparse_spd_matrix: Generate a random sparse symmetric positive definite matrix.

Examples

>>> import numpy as np
>>> from probnum.problems.zoo.linalg import random_spd_matrix
>>> rng = np.random.default_rng(1)
>>> mat = random_spd_matrix(rng, dim=5)
>>> mat
array([[10.24394619,  0.05484236,  0.39575826, -0.70032495, -0.75482692],
       [ 0.05484236, 11.31516868,  0.6968935 , -0.13877394,  0.52783063],
       [ 0.39575826,  0.6968935 , 11.5728974 ,  0.21214568,  1.07692458],
       [-0.70032495, -0.13877394,  0.21214568,  9.88674751, -1.09750511],
       [-0.75482692,  0.52783063,  1.07692458, -1.09750511, 10.193655  ]])

Check for symmetry and positive definiteness.

>>> np.all(mat == mat.T)
True
>>> np.linalg.eigvals(mat)
array([ 8.09147328, 12.7635956 , 10.84504988, 10.73086331, 10.78143272])