isposdef(E::Eigen) is incorrect

[sztal]

Hi,

The problem is as in the title. In short, isposdef method for Eigen factorization is incorrect. It does not check for hermiticity/symmetricity and only looks at the eigenvalues, which is of course not enough.

using LinearAlgebra
T = UpperTriangular(rand(3, 3) .+ 1)
isposdef(T)  # False
isposdef(eigen(T))  # True

Best!

Version info

Julia Version 1.11.3
Commit d63adeda50d (2025-01-21 19:42 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 16 × AMD Ryzen 9 5900HX with Radeon Graphics
  WORD_SIZE: 64
  LLVM: libLLVM-16.0.6 (ORCJIT, znver3)
Threads: 1 default, 0 interactive, 1 GC (on 16 virtual cores)
Environment:
  JULIA_PROJECT = @.
  JULIA_PKG_PRESERVE_TIERED_INSTALLED = true
  JULIA_REVISE = manual

[dkarrasch]

I suspect this is a matter of documentation, like saying that isposdef(E::Eigen) only checks for real and positive eigenvalues, assuming that E is the eigendecomposition of a self-adjoint matrix ("RealHermOrSymComplexHerm"). I don't know how one would check for symmetry: any such check would be subject to floating point inaccuracies, so likely V'V is not the identity, and/or V'Diagonal(evals)*V is not symmetric/Hermitian. I vaguely remember, however, that there was some effort to store that piece of information (whether E is the eigendecomposition of a self-adjoint matrix).

A quick search revealed that @RalphAS, @antoine-levitt @KlausC shared interest in this.