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This seems like a bug in PositiveDefiniteMatrixQ to me:

m = {
 { 2, -1}, 
 {-3,  2}
};

PositiveDefiniteMatrixQ[m]
(* => False *)

PositiveDefiniteMatrixQ @ SparseArray[m]
(* => True *)

This is for version 9 and above; older versions (6-8) don't seem to be affected.

share|improve this question
    
This matrix is not positive definite. It is positive-semidefinite: {1, 1}.m.{1, 1} == 0. Note that the matrix is not Hermitian. Positive definiteness is typically defined for Hermitian matrices, and for these it is sufficient to require the eigenvalues to be positive. This is not true for generalizations to non-Hermitian matrices. –  Szabolcs Jun 28 at 14:03
    
en.wikipedia.org/wiki/… –  Szabolcs Jun 28 at 14:05
1  
The SparseArray result looks like a bug which you should report to WRI. –  Szabolcs Jun 28 at 14:07
    
@Teake HermitianMatrixQ @ m returns False –  eldo Jun 28 at 14:22
    
@eldo Mathematica uses a generalized definition of positive definiteness though, e.g. {{3, -1}, {-3, 2}} is identified of positive definite despite not being symmetric. –  Szabolcs Jun 28 at 14:27

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