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

Question

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 $M(t)$?

Answer 1

Take a simpler problem: if we know the spectra of $A$ and $B$, what can we say about the spectrum of $A+B$? Even in this simplest of cases the best one can do is provide some inequalities. For instance, consider the case where $A = B = \operatorname{diag}(1,2,3,\ldots, n)$. If we permute the diagonal entries of $A$, the spectrum stays the same, but the spectrum of the sum $A+B$ can change tremendously!

See also:

• https://mathoverflow.net/questions/4224/eigenvalues-of-matrix-sums
• https://terrytao.wordpress.com/tag/schur-horn-inequalities/