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NegativeDefiniteMatrixQ

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

NegativeDefiniteMatrixQ[m]
    gives True if m is explicitly negative definite, and False
    otherwise.

A matrix m is negative definite if Re[Conjugate[x] . m . x] < 0
for every nonzero vector x.  Equivalently, -m is positive
definite, i.e. the negated Hermitian part has only positive
eigenvalues.  The test is performed by attempting a Cholesky
factorisation of -(m + ConjugateTranspose[m]) / 2; on numeric
matrices this is dispatched to BLAS/LAPACK's dpotrf (real) or
zpotrf (complex) when available.  Returns False on non-numeric,
non-square, ragged, empty, or higher-rank tensor inputs.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= NegativeDefiniteMatrixQ[{{-5, 1}, {1, -4}}]
Out[1]= True

In[2]:= NegativeDefiniteMatrixQ[{{-2.3, -1.2}, {0.6, -3.7}}]
Out[2]= True

In[3]:= NegativeDefiniteMatrixQ[{{-1, 2 I}, {-I, -4}}]
Out[3]= True

In[4]:= NegativeDefiniteMatrixQ[{{Pi, -5, 2}, {E, -3, -3}, {5, Sqrt[2], 5}}]
Out[4]= False

In[5]:= NegativeDefiniteMatrixQ[{{-1, a}, {b, -2}}]
Out[5]= False

In[6]:= NegativeDefiniteMatrixQ[Table[-1/(i + j - 1), {i, 8}, {j, 8}]]
Out[6]= True

Implementation notes

Algorithm. builtin_negative_definite_matrix_q tests Re[Conjugate[x].m.x] < 0 for all nonzero x, equivalently that −m is positive definite. It mirrors PositiveDefiniteMatrixQ exactly but negates entries at load time: after the square-matrix shape gate and the (re, im)-double coercion (ndq_leaf_to_double), it loads −m into a column-major buffer, forms the Hermitian part of −m in the upper triangle, checks its diagonal is strictly positive (i.e. m's diagonal strictly negative), and runs Cholesky via LAPACK dpotrf/zpotrf (with an in-house fallback). info == 0 on −m's Hermitian part ⇔ m negative definite ⇒ True; non-numeric/non-coercible entries give False. Wrong arity emits NegativeDefiniteMatrixQ::argx.

  • Protected.
  • Equivalent to: -m is positive definite, i.e. the negated Hermitian

Attributes: Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= NegativeDefiniteMatrixQ[{{-2, 0}, {0, 3}}]
Out[1]= False

In[1]:= NegativeDefiniteMatrixQ[{{-3, 1, 0}, {1, -3, 1}, {0, 1, -3}}]
Out[1]= True

In[1]:= NegativeDefiniteMatrixQ[{{-2, I}, {-I, -2}}]
Out[1]= True

In[1]:= NegativeDefiniteMatrixQ[-Table[1/(i + j - 1), {i, 3}, {j, 3}]]
Out[1]= True

Notes

NegativeDefiniteMatrixQ[m] tests whether Re[Conjugate[x] . m . x] < 0 for every nonzero x, equivalently that -m is positive definite. The diagonal example fails because of the positive entry 3. The symmetric tridiagonal matrix is negative definite (eigenvalues all negative). The complex Hermitian matrix {{-2, I}, {-I, -2}} shows that the test uses the Hermitian part. The last case is the negated 3x3 Hilbert matrix — notoriously ill-conditioned yet still detected as negative definite. The check is performed via an attempted Cholesky factorisation of -(m + ConjugateTranspose[m])/2, dispatched to LAPACK on numeric input.