Class PositiveDefiniteness

java.lang.Object

org.flag4j.linalg.PositiveDefiniteness

public final class PositiveDefiniteness
extends Object

Utility class for checking the positive-(semi-)definiteness of a real or complex matrix.

A matrix \( M \) is positive-definite iff \( \Re\left[x^{H}Mx\right] > 0 \) for any vector \( x \) or equivalently, if all eigenvalues are real and strictly greater than zero.

In the case where \( M \) is real, this simplifies to \( x^{T}Mx > 0 \).

Similarly, a matrix \( M \) is positive-semi-definite iff \( \Re\left[x^{H}Mx\right] >= 0 \) for any vector \( x \) or equivalently, if all eigenvalues are real and greater than or equal to than zero.

Method Summary

Modifier and Type	Method	Description
static boolean	isHermPosDef(CMatrix src)	Checks if the matrix is Hermitian positive-definite.
static boolean	isPosDef(CMatrix src)	Checks if the matrix is positive-definite.
static boolean	isPosDef(Matrix src)	Checks if the matrix is positive-definite.
static boolean	isPosSemiDef(CMatrix src)	Checks if the matrix is positive semi-definite.
static boolean	isPosSemiDef(Matrix src)	Checks if the matrix is positive semi-definite.
static boolean	isSymmPosDef(Matrix src)	Checks if the matrix is symmetric positive-definite.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Method Details

isPosDef

public static boolean isPosDef(Matrix src)

Checks if the matrix is positive-definite. A matrix \( M \) is positive-definite iff \( x^{T}Mx > 0 \) for any vector \( x \) or equivalently, if all eigenvalues are strictly greater than zero.

Parameters:

src - Matrix to check if it is positive-definite.

Returns:

true if the matrix is positive-definite; false otherwise.

See Also:

• isPosSemiDef(Matrix)

isPosDef

public static boolean isPosDef(CMatrix src)

Checks if the matrix is positive-definite. A matrix \( M \) is positive-definite iff \( \Re\left[x^{T}Mx\right] > 0 \) for any vector \( x \) or equivalently, if all eigenvalues are strictly greater than zero.

Parameters:

src - Matrix to check if it is positive-definite.

Returns:

true if the matrix is positive-definite; false otherwise.

See Also:

• isPosSemiDef(CMatrix)

isSymmPosDef

public static boolean isSymmPosDef(Matrix src)

Checks if the matrix is symmetric positive-definite. A matrix \( M \) is symmetric positive-definite iff the matrix is symmetric and \( x^{H}Mx > 0 \) for any vector \( x \) or equivalently, if all eigenvalues are real and strictly greater than zero.

Parameters:

src - Matrix to check if it is positive-definite.

Returns:

true if the matrix is positive-definite; false otherwise.

See Also:

• isPosSemiDef(Matrix)

isHermPosDef

public static boolean isHermPosDef(CMatrix src)

Checks if the matrix is Hermitian positive-definite. A matrix \( M \) is Hermitian positive-definite iff the matrix is Hermitian and \( \Re\left[x^{H}Mx\right] > 0 \) for any vector \( x \) or equivalently, if all eigenvalues are real and strictly greater than zero.

Parameters:

src - Matrix to check if it is positive-definite.

Returns:

true if the matrix is positive-definite; false otherwise.

See Also:

• isPosSemiDef(Matrix)

isPosSemiDef

public static boolean isPosSemiDef(Matrix src)

Checks if the matrix is positive semi-definite. A matrix \( M \) is positive-semi-definite iff \( x^{T}Mx >= 0 \) for any vector \( x \), or equivalently, if all eigenvalues are greater than or equal to zero.

Parameters:

src - Matrix to check if it is positive semi-definite.

Returns:

true if the matrix is positive semi-definite; false otherwise.

See Also:

• isPosSemiDef(Matrix)

isPosSemiDef

public static boolean isPosSemiDef(CMatrix src)

Checks if the matrix is positive semi-definite. A matrix \( M \) is positive-semi-definite iff \( \Re\left[x^{T}Mx\right] >= 0 \) for any vector \( x \), or equivalently, if all eigenvalues are greater than or equal to zero.

Parameters:

src - Matrix to check if it is positive semi-definite.

Returns:

true if the matrix is positive semi-definite; false otherwise.

See Also:

• isPosSemiDef(CMatrix)