# NegativeDefiniteMatrixQ

gives True if m is explicitly negative definite, and False otherwise.

# Details # Examples

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## Basic Examples(1)

Test if a matrix is explicitly negative definite:

This means that the quadratic form for all vectors :

## Scope(5)

A real matrix:

A complex matrix:

A dense matrix:

A sparse matrix:

An approximate MachinePrecision matrix:

An approximate arbitrary-precision matrix:

A matrix of exact numeric entries:

A matrix with symbolic entries:

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

## Applications(3)

Find the level sets for a quadratic form for a negative definite matrix:

In 2D, the level sets are ellipses:

In 3D, the level sets are ellipsoids:

It is possible to apply CholeskyDecomposition to a negative definite matrix by changing its sign:

CholeskyDecomposition works only with positive definite symmetric or Hermitian matrices:

An upper triangular decomposition of is a matrix such that :

A sufficient condition for a maximum of a function f is a zero gradient and negative definite Hessian:

Check the conditions for up to five variables:

## Properties & Relations(14)

A symmetric matrix is negative definite if and only if its eigenvalues are all negative:

The eigenvalues of m are all negative:

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

The eigenvalues of m are all negative:

A real is negative definite if and only if its symmetric part, , is negative definite:

The condition Re[Conjugate[x].m.x]<0 is satisfied:

The symmetric part has negative eigenvalues:

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

A complex is negative definite if and only if its Hermitian part, , is negative definite:

The condition Re[Conjugate[x].m.x]<0 is satisfied:

The Hermitian part has negative eigenvalues:

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

A diagonal matrix is negative definite if the diagonal elements are negative:

A negative definite matrix is always negative semidefinite:

The determinant and trace of a symmetric negative definite matrix are negative:

A symmetric negative definite matrix is invertible:

The inverse matrix is negative definite:

A Hermitian negative definite matrix is invertible:

The inverse matrix is negative definite:

A symmetric negative definite matrix has a uniquely defined square root such that :

The Kronecker product of two symmetric negative definite matrices is symmetric positive definite:

The Kronecker product of a symmetric negative definite matrix and a symmetric positive definite matrix is symmetric negative definite:

If is negative definite, then there exists such that for any nonzero :

A negative definite real matrix has the general form with a diagonal negative definite d:

m is a nonsingular square matrix:

a is an antisymmetric matrix:

## Possible Issues(1)

The Hilbert matrix m is positive definite and -m is negative definite:

The smallest eigenvalue of m is too small to be certainly negative at machine precision:

At machine precision, the matrix -m does not test as negative definite:

Using precision high enough to compute negative eigenvalues will give the correct answer: