Semidefinite programming: Difference between revisions

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Revision as of 07:31, 1 December 2021

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=== Equivalent formulations ===

An <math>n \times n</math> matrix <math>M</math> is said to be [[Positive-definite matrix#Positive-semidefinite|positive semidefinite]] if it is the [[GramianGram matrix]] of some vectors (i.e. if there exist vectors <math>x^1, \ldots, x^n</math> such that <math>m_{i,j}=x^i \cdot x^j</math> for all <math>i,j</math>). If this is the case, we denote this as <math>M \succeq 0</math>. Note that there are several other equivalent definitions of being positive semidefinite, for example, positive semidefinite matrices are [[self-adjoint]] matrices that have only non-negative [[Eigenvalues and eigenvectors|eigenvalues]].

Denote by <math>\mathbb{S}^n</math> the space of all <math>n \times n</math> real symmetric matrices. The space is equipped with the [[Inner product space|inner product]] (where <math>{\rm tr}</math> denotes the [[Trace (linear algebra)|trace]])