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

Test if a complex matrix is negative definite:

Test if an exact matrix is negative definite:

Use NegativeDefiniteMatrixQ with an arbitrary-precision matrix:

A random matrix is typically not negative definite:

Use NegativeDefiniteMatrixQ with a symbolic matrix:

The matrix becomes negative definite when :

NegativeDefiniteMatrixQ works efficiently with large numerical matrices:

Use NegativeDefiniteMatrixQ with sparse matrices:

Use NegativeDefiniteMatrixQ with structured matrices:

The identity matrix is not negative definite:

HilbertMatrix is not negative definite:

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

Because is negative definite, the level sets are ellipses:

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

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

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

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

Visualize for real-valued inputs:

For a real-valued matrix , only the symmetric part determines whether is negative 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 negative definite if and only if is:

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

As is Hermitian, , meaning is purely real:

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

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

Two-dimensional rotation matrices with angles in the interval are negative definite:

This follows from the fact that in this case corresponds to the normal dot product and :

The squares of nonsingular antihermitian matrices are negative definite:

The negated Lehmer matrix is symmetric negative definite:

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

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

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

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 maxima:

Visualize the function. The red and blue points are maxima 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 maxima:

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 maxima green and the non-extreme critical points red:

It is possible to apply CholeskyDecomposition to a negative definite matrix by negating it:

The Cholesky decomposition works only with positive definite Hermitian matrices:

An upper triangular decomposition of is a matrix such that :