# probnum.utils.linalg¶

Utility functions that involve numerical linear algebra.

## Functions¶

 inner_product(v, w[, A]) Inner product $$\langle v, w \rangle_A := v^T A w$$. induced_norm(v[, A, axis]) Induced norm $$\lVert v \rVert_A := \sqrt{v^T A v}$$. cholesky_update(S1[, S2]) Compute Cholesky update/factorization $$L$$ such that $$L L^\top = S_1 S_1^\top + S_2 S_2^\top$$ holds. tril_to_positive_tril(tril_mat) Orthogonally transform a lower-triangular matrix into a lower-triangular matrix with positive diagonal. gram_schmidt(v, orthogonal_basis[, ...]) Orthogonalize a vector with respect to an orthogonal basis and inner product. modified_gram_schmidt(v, orthogonal_basis[, ...]) Stabilized Gram-Schmidt process. double_gram_schmidt(v, orthogonal_basis[, ...]) Perform the (modified) Gram-Schmidt process twice.