Built-in Mathematica Symbol

PositiveDefiniteMatrixQ

PositiveDefiniteMatrixQ[m]
tests whether m is a positive definite matrix.

MORE INFORMATION

PositiveDefiniteMatrixQ[m] gives True if m is explicitly positive definite, and gives False if it is a matrix that is not positive definite.

A Hermitian matrix m is considered positive definite if and only if all its eigenvalues are positive.

PositiveDefiniteMatrixQ works with SparseArray objects.

PositiveDefiniteMatrixQ works for symbolic as well as numerical matrices.

EXAMPLES

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Basic Examples (1)

Test if a matrix is explicitly positive definite:

This means that the quadratic form

for all vectors

:

Test if a matrix is explicitly positive definite:

In[1]:=

In[2]:=

Out[2]=

This means that the quadratic form

for all vectors

:

In[3]:=

Out[3]=

Scope (5)

Test a matrix of machine numbers:

Test a matrix of complex numbers:

Test a matrix of arbitrary-precision numbers:

Test a matrix of exact numeric quantities:

Test a sparse matrix:

Generalizations & Extensions (1)

Test a matrix with symbolic entries:

The test returns False unless it is true for all possible complex values of symbolic parameters:

Properties & Relations (4)

A Hermitian matrix is positive definite if and only if its eigenvalues are all positive:

The eigenvalues of

are all positive:

So

must be positive definite:

A matrix

is positive definite if and only if its Hermitian part,

, is positive definite:

The Hermitian part has positive eigenvalues, so is positive definite:

Therefore,

must be positive definite:

Note, this does not mean that the eigenvalues of m are necessarily positive:

A positive definite Hermitian matrix has a square root given by the CholeskyDecomposition:

A square root of

is a matrix

such that

:

A sufficient condition for a minimum is a zero gradient and positive definite Hessian:

Check the conditions for up to five variables:

Possible Issues (1)

If positive definiteness is not certain at the matrix precision, the test returns False:

Hilbert matrices are positive definite:

The smallest eigenvalue is too small to be certainly positive at machine precision:

At machine precision, the matrix does not test positive definite:

Using precision high enough to resolve positiveness will work: