[Math] When does one use ‘succeeds’ and when does one use ‘greater than’

notation

I am reading a text on convex optimisation, and there is a line:

$f_i(\tilde{x})\leq0$ and $h_i(\tilde{x})=0$, and $\lambda \succeq 0$

and I was just wondering why for one term, $\leq$ is used and for the other, $\succeq$ is used.

I have a computer science background and for some reason we never were taught much formal mathematical notation.

Best Answer

$\succeq$ is used typically in the context of matrices and vectors.

  • If used in the context of vectors, it typically means that all elements of the vectors are non-negative, i.e., $\vec{\lambda} \succeq 0$, if $\lambda_i \geq 0$ for all $i$.

  • If used in the context of matrices, it typically means that the matrix is non-negative definite, i.e., $A \succeq 0$, if $x^TAx \geq 0$ for all $x \in \mathbb{R}^{n}$, where $A \in \mathbb{R}^{n \times n}$. However, on extremely rare occasions, this symbol could also mean that all entries in a matrix are non-negative, i.e., $A \succeq 0$, if $A(i,j) \geq 0$ for all $i,j$.