Cell-wise integration of a psd matrix is still psd

Question

Let $A :\mathbb{R} \to \mathbb{R}^{n \times n}$ be a matrix valued function. Suppose $A(x)$ is positive semi-define for all $x$.

Let \begin{align} B= \int_0^a A(x) dx \end{align} where $a>0$ and where the integration is done cell-wise.

My question is $B$ still a positive semi-define?

Answer 1

Yes. Note that by definition, $A$ is positive semidefinite if and only if $v^TAv \geq 0$ for every vector $v$.

We note that for any vector $v$, we have $$ v^TBv = v^T\left( \int_0^a A(x)\,dx \right)v = \int_0^a \left[v^TA(x)v\right]\,dx \geq 0 $$ since the integral of any non-negative function is non-negative.