IndefiniteMatrixQ

IndefiniteMatrixQ[m]

gives True if m is explicitly indefinite, and False otherwise.

Details and Options

  • A matrix m is indefinite if its Hermitian part is neither a positive nor a negative semidefinite matrix.
  • IndefiniteMatrixQ 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 λt λmax are taken to be zero where λmax is an eigenvalue largest in magnitude.
  • The option Tolerance has Automatic as its default value.

Examples

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Basic Examples  (1)

Test if a matrix is explicitly indefinite:

The quadratic form has positive and negative values for different vectors :

Scope  (6)

A real matrix:

A complex matrix:

Test a sparse matrix:

An approximate MachinePrecision real matrix:

An approximate MachinePrecision complex matrix:

An approximate arbitrary-precision matrix:

A matrix with exact numeric entries:

A matrix with symbolic entries:

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

Options  (1)

Tolerance  (1)

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

If the element of order is a roundoff error, then the matrix is incorrectly considered as indefinite:

Adjust the option Tolerance to give a correct answer:

Applications  (2)

The quadratic form for an indefinite matrix has degenerate level sets:

In 3D, the level sets are degenerate ellipsoids, in this case an elliptic cylinder:

The Redheffer matrix is a 0-1 indefinite matrix:

Properties & Relations  (6)

A real symmetric matrix is indefinite if and only if it contains both non-positive and non-negative eigenvalues:

The matrix m contains both non-positive and non-negative eigenvalues:

A Hermitian matrix is indefinite if and only if it contains both non-positive and non-negative eigenvalues:

The matrix m contains both non-positive and non-negative eigenvalues:

A real matrix is indefinite if its symmetric part, , is indefinite:

The symmetric part contains both non-positive and non-negative eigenvalues:

Note that this does not mean that the eigenvalues of m are non-positive or non-negative, because they can be complex:

A complex matrix is indefinite if its Hermitian part, , is indefinite:

The Hermitian part contains both non-positive and non-negative eigenvalues:

Note that this does not mean that the eigenvalues of m are non-positive or non-negative, because they can be complex:

A diagonal matrix is indefinite if it contains both positive and negative elements on its main diagonal:

An indefinite matrix has the general form , with a diagonal indefinite :

is any nonsingular square matrix:

is any antisymmetric matrix:

Possible Issues  (1)

The function returns False for symbolic matrices having non-numeric eigenvalues that cannot be determined as non-positive or non-negative:

It is not possible to determine if the eigenvalues of m are non-positive or non-negative:

Wolfram Research (2014), IndefiniteMatrixQ, Wolfram Language function, https://reference.wolfram.com/language/ref/IndefiniteMatrixQ.html.

Text

Wolfram Research (2014), IndefiniteMatrixQ, Wolfram Language function, https://reference.wolfram.com/language/ref/IndefiniteMatrixQ.html.

BibTeX

@misc{reference.wolfram_2021_indefinitematrixq, author="Wolfram Research", title="{IndefiniteMatrixQ}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/IndefiniteMatrixQ.html}", note=[Accessed: 30-November-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_indefinitematrixq, organization={Wolfram Research}, title={IndefiniteMatrixQ}, year={2014}, url={https://reference.wolfram.com/language/ref/IndefiniteMatrixQ.html}, note=[Accessed: 30-November-2021 ]}

CMS

Wolfram Language. 2014. "IndefiniteMatrixQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/IndefiniteMatrixQ.html.

APA

Wolfram Language. (2014). IndefiniteMatrixQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/IndefiniteMatrixQ.html