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Given two diagonal matrices $D_1,D_2$, matrices such that $\nabla(X):=D_1X-XD_2$ is low-rank are known in literature as Cauchy-like matrices. This includes your case, as $\operatorname{diag}(a_i^{-1}) …

They are not exactly what you are looking for, but I think you should take a look at two-parameter eigenvalue problems. For instance check http://www.math.technion.ac.il/iic/ela/ela-articles/articles/ …

This is a constrained nearest stable matrix problem. You can try using the technique in Guglielmi, Lubich Matrix Stabilization Using Differential Equations https://doi.org/10.1137/16M1105840 .

The solution of a Riccati equation can be found by determining the eigenvectors of the Hamiltonian; see e.g. here on Wikipedia. This is your best hope for a closed-form symbolic solution, in my view. …

Apart from very special cases (something commuting with something else), as far as I know there is no efficient algorithm for this kind of equations with more than two summands. (by "efficient" I mean …

There is no such bound. Let $u,v,w$ be orthonormal vectors. The matrices $A=[u,u+\varepsilon v]$ and $B=[u,u+\varepsilon w]$ have Q-factors $Q_A=[u,v]$ and $Q_B = [u,w]$ respectively, so $\|Q_A-Q_B\|$ …

The very boring answer, of course, is: write down the characteristic polynomial $p(x) = \det(A-xI)$ write down the Routh-Hurwitz criterion for $p$, expanding everything in terms of the matrix coeffic …

This is false; the result is not perfectly analogous because the definition of the spectral radius contains a maximum. Here is a counterexample. Take $$ A_1 = \begin{bmatrix}2 & 0\\ 0 & 1\end{bmatrix} …

$ACB$ ranges over all the matrices with rank smaller or equal to the rank of $C$, so this is equivalent to a problem with $C=I$ (and possibly with a smaller $k$). That said, it is not clear to me how …

That problem does not have a maximum. Unless $A$ or $k$ are zero, you can take $\alpha = Me_j$, where $e_j$ is a vector of the canonical basis, and then the objective function diverges when $M \to \in …

Up to an orthogonal change of basis you may assume $U = \begin{bmatrix}I_d\\0\end{bmatrix}$. In this basis, the problem becomes $$ \min_{X_{11} \succeq 0} \left\|\begin{bmatrix}X_{11} & X_{12}\\ X_{12 …

Converted from my comments. The first observation is that you can assume the $y_i$ to be nonnegative scalars: indeed, you can replace $y_i$ with $\|y_i\|$ without changing the problem. Then, the thes …

This is a more general case than the one of a switched dynamical system: it is obtained as the case in which $\alpha_i(t)$ are constant along discrete time steps of length $h$, and at each of these ti …

What you need is a definition of graph connectivity for undirected graphs. One can try to generalize the idea of Fiedler vector from directed graphs: if the second-to-last singular value of $I-P$ is z …

I don't think this works. Suppose you have a counterexample to the greedy heuristic, with a knapsack of size $S$ and all elements of density at most $D$ and weight at most $W$. Now add a large number …