What is matrix inequality such as $A>0$ or $A\succ 0$?

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I am trying to gather here different meanings of the same symbol, inequality symbol or the succ symbol. I find many other use them so many different ways.

Sometimes, $A>0$ means $\bar x^T A \bar x >0$. Sometimes, $A>0$ means element-wise i.e. $a_{i,j}>0$ for all $i,j$.

How do I know which definition of inequality people are meaning?

Examples

1. [Solved] Meaning of this? Why is it $A\succ 0$? LMI and example from Boyd's book. $\succ$ means positive-definiteness: check PD with Cholenksy decomposition, positive eigen-value check or Sylvester criteria. Chat.

2. Strict matrix inequality defined here requiring symmetric matrices.

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2 Answers

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The curved symbol (in my experience) means that the difference is positive definite (so, with zero it means that the matrix is positive definite). The $>$ depends on the context.

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For squared matrix $A$, $A>0$ means that $A$ is positive definite. Also, the real parts of all of $A$s eigenvalues are positive.

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