Can I rely on eigenvectors_sym returning orthogonal eigenvectors?

Summary:

The function eigenvectors_sym returns the factorisation QDQ’ for a real symmetric matrix. The documentation describes it as a generic eigendecomposition without specifying that the eigenvectors are orthogonal. This factorisation also coincides with the SVD. This is useful information that I think could be added to the manual.

Description:

Copy-pasting from this post on discourse:

I need to factorise a real symmetric matrix A as QDQ’. Since A is symmetric, the eigendecomposition can return such factorisation (but it may in principle return a different one TDT⁻¹ such that TT’≠I). Neither the Stan manual for eigenvectors_sym nor the corresponding eigen function mention that the eigenvectors returned are orthogonal (or, equivalently, that the operation coincides with the SVD).
I looked into the eigen documentation and it looks like they use the symmetric QR algorithm that, as far as I know, does return the factorisation I want. Testing Stan with a few random matrices seems to confirm that the eigenvectors returned are indeed orthogonal.

Since this is not documented in Stan, can I rely on this behaviour or not? If so, should it be also written in the documentation?

I have opened a corresponding issue for Eigen.

Current Version:

v2.18.0

Read the eigenvectors_sym entry in the Stan manual and the linked Eigen SelfAdjointEigenSolver documentation, then review the linked Eigen issue about orthogonality. Done means the manual clearly states whether the returned eigenvectors can be relied on as orthogonal and explains the SVD relationship, if confirmed.