It is actually a bit tricky. Julia doesn't know that the matrix is
symmetric and she doesn't check for symmetry to save the time it takes to
perform the check (but she checks if the matrix is triangular). Because she
doesn't know that the matrix is symmetric, the inverse is calculated by a
general factorization (the LU) and here the exact symmetry is lost because
of small floating point errors. The check for positive definiteness is the
joint test that the matrix is symmetric (or Hermitian) and can be
factorized by the Cholesky factorization and therefore it returns false for
the inverse.
One solution is to tell Julia that you know that the matrix is positive
definite before you calculate the inverse. Then she can exploit that
information and maintain the symmetry, e.g.
julia> A = [i==j?1:0.3 for i=1:5, j=1:5]
5x5 Array{Union(Float64,Int64),2}:
1 0.3 0.3 0.3 0.3
0.3 1 0.3 0.3 0.3
0.3 0.3 1 0.3 0.3
0.3 0.3 0.3 1 0.3
0.3 0.3 0.3 0.3 1
julia> isposdef(inv(cholfact(A)))
true
2014-11-12 20:43 GMT-05:00 Sibnarayan Guria <[email protected]>:
> v=[i==j?1:0.3 for i=1:5, j=1:5]
> isposdef(v) ### true
> isposdef(inv(v)) ### false
>
>
> ----------
> why?
>
