Isposdef() true with small negative eigenvalues?

General Usage
4

So basically your matrix is singular, which means that the tiniest deviation caused by floating point inaccuracy can change its character. What isposdef does is to try and compute a Cholesky decomposition and in fact this succeeds:

julia> cholesky(r'r)
Cholesky{Float64, Matrix{Float64}}
U factor:
9×9 UpperTriangular{Float64, Matrix{Float64}}:
 1.41421  -2.12132    0.707107    0.0      …  0.0       0.0        0.0
  ⋅        0.707107  -0.707107    0.0         0.0       0.0        0.0
  ⋅         ⋅         2.10734e-8  0.0         0.0       0.0        0.0
  ⋅         ⋅          ⋅          1.41421     0.0       0.0        0.0
  ⋅         ⋅          ⋅           ⋅          0.0       0.0        0.0
  ⋅         ⋅          ⋅           ⋅       …  0.0       0.0        0.0
  ⋅         ⋅          ⋅           ⋅          1.41421  -2.12132    0.707107
  ⋅         ⋅          ⋅           ⋅           ⋅        0.707107  -0.707107
  ⋅         ⋅          ⋅           ⋅           ⋅         ⋅         2.10734e-8

So from a numerical point of view your matrix might as well be positive definite. Probably the safest way to check for semi-definiteness is to also examine the eigenvalues/singular values and see whether they are large enough.

julia> diag(ans.L)
9-element Vector{Float64}:
 1.4142135623730951
 0.7071067811865481
 2.1073424255447017e-8
 1.4142135623730951
 0.7071067811865481
 2.1073424255447017e-8
 1.4142135623730951
 0.7071067811865481
 2.1073424255447017e-8

You see 3 values of order \sqrt{eps} which means that the original matrix might as well be singular (and it is in this case). Generally determining exactly whether a matrix is singular is a difficult task numerically and essentially impossible. Best you can do is compute some eigen-/singular values and use a cutoff to decide which should be zero. This is what rank does e.g.

julia> rank(r'r)
6

So to summarize: The title is not quite true. Your matrix is singular which is hard to detect. Perhaps the docstring of isposdef should warn about this more.

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