What are the characteristics of a matrix computed by taking the integral of the outer product of a vector with itself? Let $f(x) = [f_1(x), \dots, f_n(x)]^T$ such that $f(x): [0,1] \rightarrow \mathbb{R}^n$. Consider the following matrix
$A = \int\limits_{0}^1 f(x) \otimes f(x) dx$.
Is this operation to build matrix $A$ known? If so, what are the special characteristics of matrix $A$ that would make us interested in it? 
 A: It is straightforward to show that any $A$ produced in this manner will be a (symmetric and) positive semidefinite matrix with size $n \times n$. It's a bit trickier to show that any (symmetric) positive semidefinite matrix can be constructed in this fashion, so I will attempt to prove that here.  
I will work under the assumption that (possibly non-continuous) step-functions are allowed. For instance, we can allow $f$ to be any Riemann-integrable function.
To begin, we note that by taking $f$ to be a constant-function $f(x) = v$, we can produce any rank-1 positive semidefinite matrix $A = v\otimes v$.  
Now, we note that the set of matrices that can be produced with an integral formula is closed under addition.  In particular, suppose that
$$
A_1 = \int_0^1 f_1(x) \otimes f_1(x)\,dx, \quad 
A_2 = \int_0^1 f_2(x) \otimes f_2(x)\,dx.
$$
Now, define
$$
f(x) = \begin{cases}
2 f_1(2x) & 0 \leq x < 1/2\\
2 f_2(2x-1) & 1/2 \leq x \leq 1
\end{cases}
$$
We compute (by "$u$-substitution")
$$
\int_0^1 f(x) \otimes f(x)\,dx = 
\int_0^{1/2} 2f_1(2x) \otimes f_1(2x)\,dx 
+ \int_{1/2}^1 2f_2(2x-1) \otimes f_2(2x-1)\,dx\\
= 
\int_0^1 f_1(x) \otimes f_1(x)\,dx 
+ \int_{0}^1 f_2(x) \otimes f_2(x)\,dx
= A_1 + A_2.
$$
Since all positive semidefinite matrices can be written as a sum of (finitely many) rank 1 positive semidefinite matrices, we conclude that a suitable $f$ will produce any positive semidefinite matrix.
