Matrix succeeds equal to 0

Question

I'm trying a reproduce the result of a paper. The paper contains a formula:

X is a matrix. And this matrix should be "succeed equal" to 0.

What does this mean for the formula and for argmax? What does this compare operator mean in general?

Answer 1

We usually write the curly comparison $\succeq$ for more general comparison operations. For vectors $x \in \mathbb{R}^n$ we would write $x \succeq 0$ to mean that $x_1 \ge 0, ..., x_n \ge 0$. For matrices, we write $X \succeq 0$ to mean that $X$ is positive semi-definite. That is, all of it's eigenvalues are greater than or equal to zero. (In convex analysis we generalize further, it is common to define a general comparison operator $\succeq_K$ with respect to some cone $K$.)

The operation $\underset{X \succeq 0}{\text{argmax }} f(X)$ returns the matrix $Z$ such that $Z \succeq 0$ and $f(Z) \ge f(Y)$ for any other $Y \succeq 0$. That is, we optimize $f$ over the space of positive semidefinite matrices.

Answer 2

It usually means that $X$ is positive semidefinite.