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I am not sure if you are allowed to change your objective function, but a natural alternative for measuring "rank-1-ness" is $$ \frac{\lambda_1^2}{\lambda_1^2+\lambda_2^2+\dots+\lambda_n^2}. $$ This …

This subject has been studied in literature --- it turns out that for some class of matrices (such as M-matrices) you can obtain componentwise accurate solutions. A good starting point is the section …

That quantity $s = \frac{|u^Tv|}{\|u\|\|v\|}$ is the inverse of the eigenvalue condition number. The smaller it is, the more sensitive to perturbation the Perron value is. More precisely, any perturbe …

Computing a Cholesky factor is a much easier task than computing the eigendecomposition. For instance, you can do Cholesky while staying in the original field (modulo some square roots on the diagonal …

In numerical analysis lingo, you are more or less asking if matrix multiplication is backward stable. The answer seems to be no: see Section 3.5 of Higham, Accuracy and stability of numerical algorith …

If you take a Schur form $A=QTQ^T$, then $H=QT(I-T)^{-1}Q^T$, and you can ignore the orthogonal factors $Q$. You might also want to set $N=I-T$, so that $Q^THQ=N^{-1}-I$. Now the problem looks much si …

An interlacing result is in Theorem 4.3.4 of Horn, Johnson, Matrix Analysis (first edition - sorry, I don't have a copy of the second): for each Hermitian $B$, $$ \lambda_k(B) \leq \lambda_{k+1}(B\pm …

This seems a counterexample: $c_+=1$ (which you can assume without loss of generality), $n=2$, $A = \begin{bmatrix}2 & -\varepsilon \\\ -\varepsilon & 2\end{bmatrix}$, $x = \begin{bmatrix}1 \\ 1\end{b …

This is true for (EDIT: only the Euclidean norm) by the Bauer-Fike theorem, since for a symmetric / normal matrix the eigenvalue matrix is unitary and hence has condition number 1.

Not true in general, as noted by @SergeiIvanov, but true for (element-wise) nonnegative matrices. Note that if $\rho(S) < b$, then $b(bI-S)^{-1}=(I-\frac{S}{b})^{-1}=\sum_{i=0}^\infty \frac{S^i}{b^i} …

In general, there is no simple way to relate the eigenvalues of a sum to the eigenvalues of the original matrices. What you can do computationally is switching to iterative algorithms (Arnoldi method …

Just a random idea: The standard method for getting a small part of the spectrum in large and sparse symmetric problems is the restarted Lanczos method. Essentially, you run some iterations of the La …

EDIT: just a partial answer that does not settle the question completely. This is a variant of symplectic matrices. Your matrices $M_\mu$ are orthogonal wrt the indefinite scalar product induced by $$ …

[This is an answer to the first version of the question, which asked if $\min \sigma(S(x)) \leq 0 \leq \max \sigma(S(x)) \text{ a.e on } \mathbb{R}^d$ implies $\min \sigma \left( \int u S \right)\leq …

If you have a matrix with a rapidly growing diagonal, you can check alternatively if $DAD$ is diagonally dominant with a positive diagonal, where $D= \operatorname{diag}(1,\rho, \rho^2, \dots, \rho^{n …