eigSymGeneralized

eigSymGeneralized(A, B, options?): object

Generalized symmetric eigenproblem A x = lambda B x, for symmetric A and symmetric positive (semi)definite B.

When B is positive definite this is the textbook Cholesky reduction: B = L L^T, eigendecompose C = L^-1 A L^-T, then x = L^-T y. The returned eigenvectors are B-orthonormal (x^T B x = 1), matching LAPACK/scipy’s eigh(A, B).

When B is only semidefinite the Cholesky factorization does not exist — scipy raises here — so the reduction falls back to B’s truncated inverse square root (see invSqrtSym). Note what this solves: with P the orthogonal projector onto range(B), the returned pairs satisfy

P A x = lambda B x

the eigenproblem of A restricted to range(B), which is the most that is defined when B is singular — off that range A x = lambda B x generally has no solution at all. Null directions come back as zero eigenvectors with zero eigenvalues, and the remaining vectors are scaled to unit euclidean length, since B-orthonormality is undefined once B is singular. definite reports which route ran, so callers needing scipy’s strictness can check it (or pass strict).

Parameters

A

number[][]

Symmetric matrix

B

number[][]

Symmetric positive (semi)definite matrix

options?

rcond?

number

Relative eigenvalue cutoff for the semidefinite fallback; default n * eps

strict?

boolean

Throw instead of falling back when B is not positive definite

Returns

object

values[i] descending; vectors’ column i is the eigenvector for values[i]

definite

definite: boolean

values

values: number[]

vectors

vectors: number[][]

Throws

When A or B is not symmetric, sizes disagree, or strict is set and B is not positive definite