Test if a real machine-precision matrix is explicitly positive definite:

Test if a complex matrix is positive definite:

Test if an exact matrix is positive definite:

Use PositiveDefiniteMatrixQ with an arbitrary-precision matrix:

A random matrix is typically not positive definite:

Use PositiveDefiniteMatrixQ with a symbolic matrix:

The matrix becomes positive definite when :

PositiveDefiniteMatrixQ works efficiently with large numerical matrices:

Use PositiveDefiniteMatrixQ with sparse matrices:

Use PositiveDefiniteMatrixQ with structured matrices:

The identity matrix is positive definite:

HilbertMatrix is positive definite:

Consider a real, positive definite 2×2 matrix and its associated real quadratic :

Because is positive definite, the level sets are ellipses:

The plot of will be an upward-facing elliptic paraboloid:

For a real, positive definite matrix, the level sets are -ellipsoids:

A Hermitian matrix defines a real-valued quadratic form by :

If is positive definite, is positive for all nonzero inputs:

Visualize for real-valued inputs:

For a real-valued matrix , only the symmetric part determines whether is positive definite. Write with symmetric and antisymmetric:

As is real and symmetric , meaning is purely real:

Similarly, as is real and antisymmetric , or is pure imaginary:

Thus, , so is positive definite if and only if is:

For a complex-valued matrix , only the Hermitian part determines whether is positive definite. Write with Hermitian and antihermitian:

As is Hermitian, , meaning is purely real:

Similarly, as is antihermitian, , or is pure imaginary:

Thus, , so is positive definite if and only if is:

A real nonsingular Covariance matrix is always symmetric and positive definite:

A complex one is Hermitian:

A Gram matrix is a symmetric matrix of dot products of vectors:

A Gram matrix is always positive definite if vectors are linearly independent:

Matrices drawn from WishartMatrixDistribution are real, symmetric and positive definite:

The squares of nonsingular Hermitian matrices are positive definite:

The Lehmer matrix is symmetric positive definite:

Its inverse is tridiagonal, which is also symmetric positive definite:

The matrix Min[i,j] is always symmetric positive definite:

Its inverse is a tridiagonal matrix, which is also symmetric positive definite:

A positive definite, real symmetric matrix or metric defines an inner product by :

Verify that is in fact symmetric and positive definite:

Orthogonalize the standard basis of to find an orthonormal basis:

Confirm that this basis is orthonormal with respect to the inner product :

The moment of inertia tensor is the equivalent of mass for rotational motion. For example, kinetic energy is , with taking the place of the mass and angular velocity taking the place of linear velocity in the formula . can be represented by a positive definite symmetric matrix. Compute the moment of inertia for a tetrahedron with endpoints at the origin and positive coordinate axes:

Verify that the matrix is symmetric and positive definite:

Compute the kinetic energy if its angular velocity is :

As is positive definite, the kinetic energy is positive as long as is nonzero:

The second derivative test classifies critical points of a function as local minima if the Hessian is positive definite, local maxima if the Hessian is negative definite and saddle points if the Hessian is indefinite (the test fails if the Hessian is not one of these three types). Find the critical points of a function of two variables:

Compute the Hessian matrix :

The last of the three critical points is a saddle point:

The first two points are local minima:

Visualize the function. The red and blue points are minima and the green point is a saddle point:

Find the critical points of a function of three variables:

Compute the Hessian matrix of f:

The first two critical points are local minima:

The last three are saddle points:

For this function, any three of the critical points are linearly dependent, so they all lie in a single plane:

Compute the normal to that plane:

Visualize the function, with the minima green and the non-extreme critical points red:

CholeskyDecomposition works only with positive definite Hermitian matrices:

An upper triangular decomposition of is a matrix such that :