# All positive eigen values but false in isposdef?

**URL:** <https://discourse.julialang.org/t/all-positive-eigen-values-but-false-in-isposdef/107393>\
**Category:** Numerics\
**Tags:** linearalgebra\
**Created:** [December 11, 2023, 12:05am UTC](https://discourse.julialang.org/t/all-positive-eigen-values-but-false-in-isposdef/107393 "2023-12-11T00:05:13Z")\
**Posts on this page:** 1\
**Showing post:** 2

<div class="post-metadata">

**Author:** ![stevengj](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/stevengj/32/71_2.png) [@stevengj](https://discourse.julialang.org/u/stevengj)\
**Post date:** [December 11, 2023, 12:26am UTC](https://discourse.julialang.org/t/all-positive-eigen-values-but-false-in-isposdef/107393/2 "2023-12-11T00:26:58Z")

</div>

> [@Wooheon](#):
>
> How could this be possible? As far as I know, eigenvalues being positive is if and only if a relation with positive definiteness of the matrix if the matrix is symmetric.  
> Can someone help what’s going on here?

The problem is that `isposdef` also requires the matrix to be Hermitian. Your matrix `X’*X` is indeed Hermitian (real-symmetric), but because of roundoff errors its inverse is not:

```julia
julia> A = [1800 900 18900 8955 900 14400;
       900 900 13950 4410 450 7200;
       18900 13950 258300 92595 9450 151200;
       8955 4410 92595 63645 4440 71640;
       900 450 9450 4440 900 7200;
       14400 7200 151200 71640 7200 148800];

julia> isposdef(A)
true

julia> isposdef(A^-1)
false

julia> ishermitian(A^-1)
false

julia> A^-1 - (A^-1)'
6×6 Matrix{Float64}:
  0.0 7.37257e-18 -8.13152e-19 1.0842e-19 1.51788e-18 3.79471e-19
 -7.37257e-18 0.0 0.0 3.35425e-19 1.73938e-18 5.42101e-19
  8.13152e-19 0.0 0.0 -3.854e-20 -1.74515e-19 -5.42101e-20
 -1.0842e-19 -3.35425e-19 3.854e-20 0.0 -1.06726e-19 -1.33408e-20
 -1.51788e-18 -1.73938e-18 1.74515e-19 1.06726e-19 0.0 0.0
 -3.79471e-19 -5.42101e-19 5.42101e-20 1.33408e-20 0.0 0.0

```

You can fix this by telling Julia that the matrix is Hermitian, so that this property is enforced through matrix inversion etcetera, just by wrapping it in the [`Hermitian`](https://docs.julialang.org/en/v1/stdlib/LinearAlgebra/#LinearAlgebra.Hermitian) type.

```julia
julia> H = Hermitian(A);

julia> ishermitian(H^-1)
true

julia> isposdef(H^-1)
true

```

PS. The generic advice in numerical computation is to avoid computing matrix inverses explicitly, and use matrix factorizations instead — normally the explicit inverse is not needed, and computing it is wasteful. The [`\` (backslash)](https://docs.julialang.org/en/v1/base/math/#Base.:%5C%5C-Tuple%7BAny,%20Any%7D) operator does this for you.

PPS. If you are computing `inv(X'X)` in order to solve a least-square problem, beware that you have squared the condition number of X by doing so. It is safer and more accurate to solve least-squares problems by `X \ b` (which uses QR factorization) rather than by computing `inv(X'X) * X'b` or `(X'X) \ (X'b)`. See also [Efficient way of doing linear regression - #33 by stevengj](https://discourse.julialang.org/t/efficient-way-of-doing-linear-regression/31232/33)

PPPS. Quote your code: [PSA: how to quote code with backticks](https://discourse.julialang.org/t/psa-how-to-quote-code-with-backticks/7530)

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