PositiveSemidefiniteMatrixQ

PositiveSemidefiniteMatrixQ[m]

gives True if m is explicitly positive semidefinite, and False otherwise.

Details and Options

A matrix m is positive semidefinite if Re[Conjugate[x].m.x]≥0 for all vectors x. »
PositiveSemidefiniteMatrixQ works for symbolic as well as numerical matrices.
For approximate matrices, the option Tolerance->t can be used to indicate that all eigenvalues λ satisfying λ≤t λ_max are taken to be zero where λ_max is an eigenvalue largest in magnitude.
The option Tolerance has Automatic as its default value.

Examples

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Basic Examples (2)

Test if a 2×2 real matrix is explicitly positive semidefinite:

This means that the quadratic form for all vectors :

Visualize the values of the quadratic form:

Test if a 3×3 Hermitian matrix is positive semidefinite:

Scope (10)

Basic Uses (6)

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

Test if a complex matrix is positive semidefinite:

Test if an exact matrix is positive semidefinite:

Use PositiveSemidefiniteMatrixQ with an arbitrary-precision matrix:

A random matrix is typically not positive semidefinite:

Use PositiveSemidefiniteMatrixQ with a symbolic matrix:

The matrix becomes positive semidefinite when :

PositiveSemidefiniteMatrixQ works efficiently with large numerical matrices:

Special Matrices (4)

Use PositiveSemidefiniteMatrixQ with sparse matrices:

Use PositiveSemidefiniteMatrixQ with structured matrices:

The identity matrix is positive semidefinite:

HilbertMatrix is positive semidefinite:

Options (1)

Tolerance (1)

Generate a real-valued diagonal matrix with some random perturbation of order 10^-12:

Adjust the option Tolerance to accept matrices as positive semidefinite:

Applications (13)

The Geometry and Algebra of Positive Semidefinite Matrices (5)

Consider a real, positive semidefinite 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:

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

Visualize for real-valued inputs:

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

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

As is Hermitian, , meaning is purely real:

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

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

Sources of Positive Definite Matrices (8)

A real Covariance matrix is always symmetric and positive semidefinite:

A complex one is Hermitian:

Compute SingularValueDecomposition[m] for a square matrix :

The matrix is positive semidefinite:

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

A Gram matrix is always positive semidefinite:

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

The squares of Hermitian matrices are positive semidefinite:

Every antihermitian matrix is positive semidefinite:

The Lehmer matrix is symmetric positive semidefinite:

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

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

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

Properties & Relations (13)

PositiveSemidefiniteMatrixQ[x] trivially returns False for any x that is not a matrix:

A matrix is positive semidefinite if for all vectors :

The sign of is irrelevant:

A real matrix is positive semidefinite if and only if its symmetric part is positive semidefinite:

In general, a matrix is positive semidefinite if and only if its Hermitian part is positive semidefinite:

A real symmetric matrix is positive semidefinite if and only if its eigenvalues are all non-negative:

The statement is true of Hermitian matrices more generally:

A general matrix can have all non-negative eigenvalues without being positive semidefinite:

Equally, a matrix can be positive semidefinite without having non-negative eigenvalues:

The failure is due to the eigenvalues being complex:

The real part of the eigenvalues of a positive semidefinite matrix must be non-negative:

A diagonal matrix is positive semidefinite if and only if diagonal elements have non-negative real parts:

A positive semidefinite matrix has the general form with a diagonal positive semidefinite :

Split into its Hermitian and antihermitian parts:

By the spectral theorem, can be unitarily diagonalized using JordanDecomposition:

The matrix is diagonal with non-negative diagonal entries:

The matrix is unitary:

Verify that :

A matrix is positive semidefinite if and only if is negative semidefinite:

A positive definite matrix is always positive semidefinite:

There are positive semidefinite matrices that are not positive definite:

A positive semidefinite matrix cannot be indefinite or negative semidefinite:

The determinant and trace of a real, symmetric, positive semidefinite matrix are non-negative:

This is also true of positive semidefinite Hermitian matrices:

A real symmetric positive semidefinite matrix has a uniquely defined square root such that :

The square root is positive semidefinite and real symmetric:

A Hermitian positive semidefinite matrix has a uniquely defined square root such that :

The square root is positive semidefinite and Hermitian:

The Kronecker product of two symmetric positive semidefinite matrices is symmetric and positive semidefinite:

Replacing one matrix in the product by a negative semidefinite one gives a negative semidefinite matrix:

Possible Issues (2)

CholeskyDecomposition does not work with symmetric or Hermitian positive semidefinite matrices that are singular:

PositiveSemidefiniteMatrixQ gives False unless it can prove a symbolic matrix is positive semidefinite:

Using a combination of Eigenvalues and Reduce can give more precise results:

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PositiveSemidefiniteMatrixQ

Details and Options

Examples

Basic Examples (2)