The "positive semi-definiteness" definition of a Matrix $K$ may be formulated as follows:
$$\sum_{i,j=1}^n c_i c_j K_{i,j} \ge 0 \equiv c^T K c \ge 0$$
for any $c_1, ... , c_n \in \Bbb{R}$
But I can't figure out what do the coefficients $c_1, ..., c_n$ add to the definition. One told me they are used for the $=$ par of the $\ge$ when $c_1, ..., c_n$ are all equals to zero but without certainty.
For any matrix $K$, we define the quadratic form associated with $K$ to be the function $q:\Bbb R^n \to \Bbb R$ given by $$ q(c) = q(c_1,c_2,\dots,c_n) = \sum_{ij}K_{ij}c_ic_j $$ The vector $c$ here is an input to a function; it's just like the $x$ in $f(x) = x^2$. We say that $K$ is positive semidefinite if $q(c) \geq 0$ for every possible $c$.