# Isposdef() true with small negative eigenvalues?

**URL:** <https://discourse.julialang.org/t/isposdef-true-with-small-negative-eigenvalues/111504>\
**Category:** General Usage\
**Tags:** linearalgebra\
**Created:** [March 12, 2024, 7:11am UTC](https://discourse.julialang.org/t/isposdef-true-with-small-negative-eigenvalues/111504 "2024-03-12T07:11:24Z")\
**Posts on this page:** 4\
**Page:** 1

<div class="post-metadata">

**Author:** ![sparrowhawk](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/sparrowhawk/32/9148_2.png) [@sparrowhawk](https://discourse.julialang.org/u/sparrowhawk)\
**Post date:** [March 12, 2024, 7:11am UTC](https://discourse.julialang.org/t/isposdef-true-with-small-negative-eigenvalues/111504/1 "2024-03-12T07:11:24Z")

</div>

Hi all, I’m puzzled by `eigen()` and `isposdef()` acting on `r'r` for this matrix `r`

```julia
using LinearAlgebra

  r = [
  0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
 -1.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
  1.0 -2.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0
  0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
  0.0 0.0 0.0 -1.0 1.0 0.0 0.0 0.0 0.0
  0.0 0.0 0.0 1.0 -2.0 1.0 0.0 0.0 0.0
  0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
  0.0 0.0 0.0 0.0 0.0 0.0 -1.0 1.0 0.0
  0.0 0.0 0.0 0.0 0.0 0.0 1.0 -2.0 1.0
  ]

```

now if we were to do `r'r`, this has very small negative eigenvalues

```julia
julia> λ = eigen(r'r).values
9-element Vector{Float64}:
 -1.969708422286297e-16
 -1.969708422286297e-16
 -1.969708422286297e-16
  0.39444872453601043
  0.39444872453601043
  0.39444872453601043
  7.605551275463979
  7.605551275463979
  7.605551275463979

```

but why then is `isposdef(r'r) == true` ?

```julia
julia> isposdef(r'r)
true

```

Shoudn’t those zero or negative eigenvalues make positive definiteness for `r'r` untrue? On further investigating, there are some small singular values in `r`,

```julia
julia> svd(r).S'
1×9 adjoint(::Vector{Float64}) with eltype Float64:
 2.75782 2.75782 2.75782 0.628052 0.628052 0.628052 3.73941e-17 3.73941e-17 3.73941e-17

```

and as expected for `r'r`

```julia
julia> svd(r'r).S'
1×9 adjoint(::Vector{Float64}) with eltype Float64:
 7.60555 7.60555 7.60555 0.394449 0.394449 0.394449 2.22776e-16 2.22776e-16 2.22776e-16

```

but what makes `isposdef` true? Any insight will be much appreciated.

---

<div class="post-metadata">

**Author:** ![abraemer](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/abraemer/32/51403_2.png) [@abraemer](https://discourse.julialang.org/u/abraemer)\
**Post date:** [March 12, 2024, 7:24am UTC](https://discourse.julialang.org/t/isposdef-true-with-small-negative-eigenvalues/111504/2 "2024-03-12T07:24:26Z")

</div>

`isposdef` and similar functions like `rank` ignore small values. So even though you see a couple of negative eigenvalues their magnitude is below the machine epsilon and thus they will be treated as 0. I don’t know where exactly the cutoff is though.

---

<div class="post-metadata">

**Author:** ![sparrowhawk](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/sparrowhawk/32/9148_2.png) [@sparrowhawk](https://discourse.julialang.org/u/sparrowhawk)\
**Post date:** [March 12, 2024, 7:31am UTC](https://discourse.julialang.org/t/isposdef-true-with-small-negative-eigenvalues/111504/3 "2024-03-12T07:31:12Z")

</div>

Thanks for your quick reply, but …

> Blockquote  
> is below the machine epsilon and thus they will be treated as 0

for positive definiteness, don’t the eigenvalues have to be greater than zero?

---

<div class="post-metadata">

**Author:** ![abraemer](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/abraemer/32/51403_2.png) [@abraemer](https://discourse.julialang.org/u/abraemer)\
**Post date:** [March 12, 2024, 7:51am UTC](https://discourse.julialang.org/t/isposdef-true-with-small-negative-eigenvalues/111504/4 "2024-03-12T07:51:58Z")

</div>

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-repl
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-repl
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-repl
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.