I want to know the exact definition of full-dimensional. And what does "dimension" refer to, is that in the sense of algebraic variety?
I have read several writing announcing that the cone of semi-defined positive matrix is full-dimensional without giving a clear definition of what dimension mean.
It is written in G.Barker and D. Carlson. Cones of diagonally dominant matrices. that if C is a cone in a vector space X. C is full if C - C = X
In another Doctoral report to demonstrate that $ \mathcal{S}_+^n $ is of full-dimensional, they create a basis of $ \mathcal{S}^n$ formed of element of $ \mathcal{S}_+^n $
It gets clearer after this last result, but not fully clear, So thank you for more clarification.
It means that the cone generates the vector space, equivalently, it contains a basis of the vector space. Since the cone is invariant under addition and multiplications by scalars from $[0, \infty)$ this is equivalent to $C - C = \text{ full space}$
For example, the cone of positive semi-definite matrices generates the space of symmetric matrices since every to symmetric matrix we can add a positive multiple of the identity and obtain a positive matrix, so the initial matrix is a difference of two positive semi-definite matrix.
A cone $C$ is not an algebraic variety. However, the algebraic closure of $C$ will be a linear subspace, in fact $C-C$. So in some sense it is the algebraic dimension, in the sense of maximal number of polynomial functions on $C$ that are algebraically independent.