Positive Semi-Definite Matrix: Difference between revisions
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** If a matrix <math>-A</math> is positive semi-definite then the matrix <math>A</math> is called [[Negative Semi-Definite Matrix]]. | ** If a matrix <math>-A</math> is positive semi-definite then the matrix <math>A</math> is called [[Negative Semi-Definite Matrix]]. | ||
* <B>Example(s):</B> | * <B>Example(s):</B> | ||
** <math>\begin{bmatrix} | ** <math>\begin{bmatrix} 2 & 2 & 2 \\ 2 & 6 & 2 \\ 2 & 2 & 2 \end{bmatrix}</math> is a positive semi-definite matrix with all nonnegative [[eigenvalues]] 8, 2 and 0. | ||
** a [[Kernel Matrix]]. | ** a [[Kernel Matrix]]. | ||
* <B>Counter-Example(s)<\U>:</B> | |||
** <math>\begin{bmatrix} -2 & -2 & -2 \\ -2 & -6 & -2 \\ -2 & -2 & -2 \end{bmatrix}</math> is not a positive semi-definite matrix. | |||
* <B>See:</B> [[Negative Semi-Definite Matrix]], [[Positive Definite Matrix]], [[Covariance Matrix]], [[Semipositive-Definite Matrix]], [[Negative Definite Matrix]], [[Negative Semidefinite Matrix]], [[Positive Eigenvalued Matrix]], [[Positive Matrix]], [[Semidefinite Program (SDP)]]. | * <B>See:</B> [[Negative Semi-Definite Matrix]], [[Positive Definite Matrix]], [[Covariance Matrix]], [[Semipositive-Definite Matrix]], [[Negative Definite Matrix]], [[Negative Semidefinite Matrix]], [[Positive Eigenvalued Matrix]], [[Positive Matrix]], [[Semidefinite Program (SDP)]]. | ||
Revision as of 13:16, 6 January 2016
A Positive Semi-Definite Matrix is a Hermitian matrix whose eigenvalues are nonnegative.
- Context:
- It can be mathematically stated as: A matrix [math]\displaystyle{ A }[/math] is said to be positive semi definite if [math]\displaystyle{ \overline{X}^TAX \geq 0 }[/math] for any complex vector [math]\displaystyle{ X }[/math].
- It can have all leading minors value nonnegative.
- If a matrix [math]\displaystyle{ -A }[/math] is positive semi-definite then the matrix [math]\displaystyle{ A }[/math] is called Negative Semi-Definite Matrix.
- Example(s):
- [math]\displaystyle{ \begin{bmatrix} 2 & 2 & 2 \\ 2 & 6 & 2 \\ 2 & 2 & 2 \end{bmatrix} }[/math] is a positive semi-definite matrix with all nonnegative eigenvalues 8, 2 and 0.
- a Kernel Matrix.
- Counter-Example(s)<\U>:
- [math]\displaystyle{ \begin{bmatrix} -2 & -2 & -2 \\ -2 & -6 & -2 \\ -2 & -2 & -2 \end{bmatrix} }[/math] is not a positive semi-definite matrix.
- See: Negative Semi-Definite Matrix, Positive Definite Matrix, Covariance Matrix, Semipositive-Definite Matrix, Negative Definite Matrix, Negative Semidefinite Matrix, Positive Eigenvalued Matrix, Positive Matrix, Semidefinite Program (SDP).
References
2013
- http://en.wikipedia.org/wiki/Positive-definite_matrix#Positive-semidefinite
- M is called semipositive-definite (or sometimes nonnegative-definite) if :[math]\displaystyle{ x^{*} M x \geq 0 }[/math] for all x in Cn (or, all x in Rn for the real matrix), where x* is the conjugate transpose of x.
A matrix M is positive-semidefinite if and only if it arises as the Gram matrix of some set of vectors. In contrast to the positive-definite case, these vectors need not be linearly independent.
For any matrix A, the matrix A*A is positive semidefinite, and rank(A) = rank(A*A). Conversely, any Hermitian positive semidefinite matrix M can be written as M = A*A; this is the Cholesky decomposition.
- M is called semipositive-definite (or sometimes nonnegative-definite) if :[math]\displaystyle{ x^{*} M x \geq 0 }[/math] for all x in Cn (or, all x in Rn for the real matrix), where x* is the conjugate transpose of x.
2012
- http://mathworld.wolfram.com/PositiveSemidefiniteMatrix.html
- QUOTE: A positive semidefinite matrix is a Hermitian matrix all of whose eigenvalues are nonnegative.