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

Test if a complex matrix is negative semidefinite:

Test if an exact matrix is negative semidefinite:

Use NegativeSemidefiniteMatrixQ with an arbitrary-precision matrix:

A random matrix is typically not negative semidefinite:

Use NegativeSemidefiniteMatrixQ with a symbolic matrix:

The matrix becomes negative semidefinite when :

NegativeSemidefiniteMatrixQ works efficiently with large numerical matrices:

Use NegativeSemidefiniteMatrixQ with sparse matrices:

Use NegativeSemidefiniteMatrixQ with structured matrices:

The identity matrix is not negative semidefinite:

HilbertMatrix is not negative semidefinite:

Generate a real-valued diagonal matrix with some random perturbation of order :

Adjust the option Tolerance to accept matrices as negative semidefinite:

Consider a real, negative semidefinite 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:

However, the ellipses can be degenerate, turning into lines:

The plot of is then a cylinder over a parabola:

In an even more extreme case, the level set can be the whole plane as :

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

In three dimensions, these can degenerate into cylinders over ellipses:

As well as planes:

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

If is negative semidefinite, is non-positive for all inputs:

Visualize for real-valued inputs:

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

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

As is Hermitian, , meaning is purely real:

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

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

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

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

Thus, for , the matrices are in fact negative definite:

At the endpoints they are negative semidefinite but not negative definite:

The squares of antihermitian matrices are negative definite:

Every antihermitian matrix is negative semidefinite:

The negated Lehmer matrix is symmetric negative semidefinite:

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

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

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