Negative Semi-Definite Matrix

A Negative Semi-Definite Matrix is a symmetric matrix whose eigenvalues are nonpositive.

• AKA: Negative Semidefinite Matrix.
• Context:
  • It can also be stated as: A matrix [math]\displaystyle{ A }[/math] is called Negative Semi-Definite if [math]\displaystyle{ -A }[/math] is a 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.
  • …
• Counter-Example(s):
  • a Positive Semi-Definite Matrix.
• See: Positive Definite Matrix.

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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.

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