0

Given the spectral decompositions of a non-commuting collection of symmetric positive definite $N\times N$ matrices $$\left\{ K_{i}\right\} _{i=1}^{M}, U_{i}D_{i}U_{i}^{T}=K_{i},\quad i=1,\dots,M,$$ is there any $O(N^{2})$ method for computing the eigenvalues of a given convex combination$$\mathcal{D}^{T}=\mathcal{U}^{T}\left(\sum\alpha_{i}K_{i}\right)\mathcal{U}\text{ where }\sum\alpha_{i}=1\text{ and each }\alpha_{i}\geq0?$$

This answer to a related question suggests that there might some such methods but I have yet to find anything.

Edit: In the above linked question I'm referring not to Tao and Knutson's results but rather the additional comment: "Depending on what you want, there should be simpler results giving estimates on the eigenvalues of the sum".

Richard Border
  • 133

1 Answers1

4

No, there isn't such a method. It's difficult to find a reference that "proves a negative", but I can tell you that many people in my field (numerical linear algebra) would be very happy to know about one. :)

A partial argument to convince you could be: if this were possible, then you could write for any $2n\times 2n$ matrix $\begin{bmatrix} A &B \\ C & D\end{bmatrix} = \begin{bmatrix} A &B \\ 0 & 0\end{bmatrix} + \begin{bmatrix} 0 &0 \\ C & D\end{bmatrix}$, and solve arbitrary eigenvalue problems iteratively via "divide and conquer" in quadratic+logarithmic time. (Not really a valid argument as it is because I did not discuss how to compute eigenvector matrices).

What Knutson and Tao solved is a different, more abstract problem: given two (multi)sets of real numbers $\alpha_1,\dots,\alpha_n$, and $\beta_1,\dots,\beta_n$, for which choices of $\gamma_1,\dots,\gamma_n$ can one find Hermitian matrices $A$ and $B$ such that $A$ has spectrum $\alpha_1,\dots,\alpha_n$, $B$ has spectrum $\beta_1,\dots,\beta_n$, and $A+B$ has spectrum $\gamma_1,\dots,\gamma_n$?

Federico Poloni
  • 19,330
  • Thanks, I figured this was the case :). However, your counterexample involves sums of singular non-symmeteric matrices, rather than SPD matrices, unless I'm missing something? Also, I wasn't referring to the Knutson and Tao solution, but rather the other comment in the linked answer: "Depending on what you want, there should be simpler results giving estimates on the eigenvalues of the sum. A book like Bhatia's Matrix Analysis might have some helpful material." So I was hoping their might be some nice trick :( – Richard Border Aug 21 '18 at 15:05
  • So the fact that we are dealing with convex combinations instead of arbitrary linear combinations doesn't make any difference? – ViktorStein Jan 23 '23 at 15:03
  • @Ramanujan No: if you could solve the problem for convex combinations, it would be easy to rescale and add multiples of the identity matrix to treat the general case of a linear combination. – Federico Poloni Jan 23 '23 at 15:52