How does integration affect the spectrum of a vector outer product?

Start with a general eigenvalue problem. We have the following matrix: $$M(t) = a(t) a(t)^T$$
Where the notation $a(t)$ denotes that the vector $a$ is a function of the scalar variable $t$

For any $t$, the spectrum of $M(t)$ at any point $t$ is equal to ${ ||a(t)||, 0, dots,0}$, with the associated eigenvectors being $a(t)$ and any linearly independent set of vectors orthogonal to $a(t)$ (see this previous question)

Now suppose we take the integral of $M(t)$ over $t$, giving us a new matrix $Z$ with $$Z = int_a^b M(t) dt = int_a^b a(t)a(t)^T dt$$

Can we easily characterize the spectrum of this new matrix $Z$ based on our knowledge of the spectrum of $M(t)$?