[LinearAlgebra] strange isposdef behavior

[utotch]

In my observation, isposdef sometimes return false for positive definite matrices.

I think isposdef in the code below should both return true.
C's eigval is all positive, thus C should be posdef, I think, but posdef(C) returns false.

Am I wrong?

using Random
using LinearAlgebra
Random.seed!(0)

R = rand(3,3)
A = R./eigvals(R)[1]
B = A'*(2.0*[1 0 0; 0 1 0; 0 0 1])*A
C = A'*(3.0*[1 0 0; 0 1 0; 0 0 1])*A
@show eigvals(B) # all positive
@show eigvals(C) # all positive
@show isposdef(B) # true
@show isposdef(C) # false!

versioninfo()

eigvals(B) = [0.00334175, 0.24934, 2.97084]
eigvals(C) = [4.45626, 0.374009, 0.00501262] <-- All Positive
isposdef(B) = true
isposdef(C) = false <- Why?
Julia Version 1.1.1
Commit 55e36cc (2019-05-16 04:10 UTC)
Platform Info:
OS: macOS (x86_64-apple-darwin15.6.0)
CPU: Intel(R) Core(TM) i5-5287U CPU @ 2.90GHz
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-6.0.1 (ORCJIT, broadwell)
Environment:
JULIA_DIR = /Applications/Julia-1.1.app/Contents/Resources/julia
JULIA_HOME = /Applications/Julia-1.1.app/Contents/Resources/julia/bin
JULIA_EDITOR = atom -a
JULIA_NUM_THREADS = 2

[dkarrasch]

This is due to ishermitian(C) gives false, for which the reason is that

julia> C-C'
3×3 Array{Float64,2}:
  0.0          0.0  5.55112e-17
  0.0          0.0  0.0        
 -5.55112e-17  0.0  0.0

So, I guess the question is whether one wants to allow for some tolerance in the asymmetry/non-Hermitianess.

julia> isposdef(Hermitian(C))
true

On the other hand, I suspect that one would like to enforce users to wrap their presumably Hermitian matrices as Hermitians explicitly.

[KristofferC]

See e.g. JuliaLang/LinearAlgebra.jl#182.