Span of Density operators (Positive Semi-definite matrices of Trace one) While reading basic-mathematics of quantum mechanics I came across a statement - 

"For every complex euclidean space $\cal X$ there exists spanning sets
  of the space $L({\cal X})$ consisting of only density operators".

Here, 


*

*$\cal X$ is euclidean space, say ${\mathbb C}^n$; 

*$L({\cal X})$ correspond to linear operators over ${\cal X}$ which is just complex matrices of $n\times n$ size; and

*Density operators are just positive semi-definite matrices of trace one.


Now, I do not quite understand this that linear combination of positive semi-definite matrices should be at least a symmetric matrix, then how can span cover all the elements in $L({\cal X})$.
Any remarks will be of help. Thanks.

Given any $T\in L(\mathbb C^n)$, you can write 
$$
T=\frac{T+T^*}2+i\,\frac {T-T^*}{2i}
$$
so $T$ is a linear combination of selfadjoints. And for each selfadjoint you have the Spectral Theorem saying that they are a linear combination of rank-one projections (which are positive semidefinite of trace one). 
The result is even true when $\mathcal X$ is an infinite-dimensional Hilbert space, although it is not trivial in that case. 
