Positive Semi-Definite Matrix: Difference between revisions
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** It can be mathematically stated as: A matrix <math>A</math> is said to be positive semi definite if <math>\overline{X}^TAX \geq 0</math> for any [[complex vector]] <math>X</math>. | ** It can be mathematically stated as: A matrix <math>A</math> is said to be positive semi definite if <math>\overline{X}^TAX \geq 0</math> for any [[complex vector]] <math>X</math>. | ||
** It can have all [[leading minors]] value nonnegative. | ** It can have all [[leading minors]] value nonnegative. | ||
** If a matrix <math>-A</math> is positive semi-definite then the matrix <math>A</math> is called [[Negative Semi-Definite]]. | ** 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} 0 & 1 & 1 \\ \sqrt{2} & 2 & 0 \\ 0 & 1 & 1 \end{bmatrix}</math> is a sositive semi-definite matrix with all nonnegative [[eigenvalues]] 2.79,0.21 and 0. | ** <math>\begin{bmatrix} 0 & 1 & 1 \\ \sqrt{2} & 2 & 0 \\ 0 & 1 & 1 \end{bmatrix}</math> is a sositive semi-definite matrix with all nonnegative [[eigenvalues]] 2.79,0.21 and 0. |
Revision as of 13:07, 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} 0 & 1 & 1 \\ \sqrt{2} & 2 & 0 \\ 0 & 1 & 1 \end{bmatrix} }[/math] is a sositive semi-definite matrix with all nonnegative eigenvalues 2.79,0.21 and 0.
- a Kernel 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.