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

CLOSE ALL

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 it is positive definite:

Therefore,

must be positive definite:

Note, this does not mean that the eigenvalues of

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: