Negative Semi-Definite Matrix: Difference between revisions
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A [[Negative Semi-Definite Matrix]] is a [[semi-definite matrix]] | A [[Negative Semi-Definite Matrix]] is a symmetric matrix whose [[eigenvalues]] are nonpositive. | ||
* <B>Context</U>:</B> | |||
** It can also be stated as: A matrix <math>A</math> is called Negative Semi-Definite if <math>-A</math> is [[positive semi-definite matrix]]. | |||
* <B>Example(s)</U>:</B> | |||
** <math>\begin{bmatrix} -2 & -2 & -2 \\ -2 & -6 & -2 \\ -2 & -2 & -2 \end{bmatrix}</math> is a Negative Semi-Definite matrix with [[eigenvalues]] -8, -2 and 0. | |||
* <B>See:</B> [[Positive Definite Matrix]]. | * <B>See:</B> [[Positive Definite Matrix]]. | ||
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Revision as of 13:26, 6 January 2016
A Negative Semi-Definite Matrix is a symmetric matrix whose eigenvalues are nonpositive.
- Context:
- It can also be stated as: A matrix [math]\displaystyle{ A }[/math] is called Negative Semi-Definite if [math]\displaystyle{ -A }[/math] is positive semi-definite matrix.
- Example(s):
- [math]\displaystyle{ \begin{bmatrix} -2 & -2 & -2 \\ -2 & -6 & -2 \\ -2 & -2 & -2 \end{bmatrix} }[/math] is a Negative Semi-Definite matrix with eigenvalues -8, -2 and 0.
- See: Positive Definite Matrix.
References
2015
- http://en.wikipedia.org/wiki/Positive-definite_matrix#Negative-semidefinite
- QUOTE: It is called negative-semidefinite if :[math]\displaystyle{ x^{*} M x \leq 0 }[/math] for all x in Cn (or, all x in Rn for the real matrix).
2004
- (Lanckriet et al., 2004a) ⇒ Gert R. G. Lanckriet, Nello Cristianini, Peter Bartlett, Laurent El Ghaoui, and Michael I. Jordan. (2004). "Learning the Kernel Matrix with Semidefinite Programming." In: The Journal of Machine Learning Research, 5.
- QUOTE: A linear matrix inequality, abbreviated LMI, is a constraint of the form: [math]\displaystyle{ F (u): = F_0 + u_1F_1 + ... + u_qF_q \preceq 0: }[/math] Here, [math]\displaystyle{ u }[/math] is the vector of decision variables, and [math]\displaystyle{ F_0, ..., F_q }[/math] are given symmetric [math]\displaystyle{ p \times p }[/math] matrices. The notation [math]\displaystyle{ F(u) = 0 }[/math] means that the symmetric matrix [math]\displaystyle{ F }[/math] is negative semidefinite.