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# Questions tagged [eigenvalues]

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21 views

### Shared basis of eigenvalues for two unitary transformations [migrated]

I have two unitary transformations: $T$ and $S$ in a unitary space, and I know $TS=ST$. I need to prove that $T$ and $S$ have a shared basis composed of eigenvectors of both. i.e if $B=\{u_1,...,u_n\}$...
66 views

### Energy of a symmetric matrix with $0$, $1$ or $-1$ entries

I have a symmetric matrix with entries $0$, $1$ or $-1$ which appeared in my works in graph theory (the diagonal entries are all zero). I need a good upper bound for the energy of this matrix; i.e. "...
39 views

### Is the largest eigenvalue that matches an eigenvector that spans $v$ is independent of basis? [migrated]

Let $W$ be a finate-dimensional inner product space over $\mathbb{R}$ (or $\mathbb{C}$) and let $M: W\to W$ be a self-adjoint operator. Then there exists an orthonormal basis $B=\{\phi_i\}$ consisting ...
155 views

### Eigenvalues of random graphs

At time $t=0$, let $G_n(V,E)$ be a graph with $n$ vertices and $m < n$ edges. Then there exists a unique symmetric adjacency matrix $A_n$ associated with $G_n(V,E)$, defined as follows: $a_{ij} = 1$...
180 views

### Is it faster to compute eigenvalues or coefficients of characteristic polynomials?

Given $A \in \mathsf{M}_n(\mathbb{C})$ (no special structure) is it (generally) faster to compute its eigenvalues or the coefficients of its characteristic polynomial? References/insights would be ...
27 views

### How to solve or analyse the smallest eigenvalue of 2 coupled 1st-order linear ODEs?

I am trying to solve the eigensystem of a 1st-order linear ODE system in the region $(-\infty,\infty)$ and with Dirichlet boundary condition at the infinities \begin{align} -\mathrm{i} u'(x) +f^*(x) ...
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Given a vector $u \in \Bbb R^n$, finding the value of the largest component of $u$ needs linear time in $n$. However, what if we additionally know that $u$ lies in some linear subspace $U \subset \Bbb ... 0answers 15 views ### Largest eigenvalue scaling in a certain Kac-Murdoch-Szegö matrix I'm looking at$N\times N$matrices$M_N$with elements $$M_N=\left( \rho^{|i-j|} \right)_{i,j=1}^N,$$ where$\rho$is a complex number of unit modulus. These matrices with$\rho\in\mathbb R$and$|\...
106 views

### The maximal eigenvalue of average of positive matrices

Let $A$ and $B$ be two square real positive (all entries are positive) matrices that differ only in the first row. Let $\lambda_A$ and $\lambda_B$ be the maximal real eigenvalues of $A$ and $B$, ...
36 views

### Calculate amount of FLOPs for an eigenvalue problem solver

I've got 2 complex, non symmetric, matrices $A_{1000x1000}$, $B_{1000x1000}$ and I am using Matlab to get it's eigenvalues (functions like eig or eigs). Both matrices are different - one is more dense ...
71 views

### Computing the support of the equilibrium measure in Johansson's 1999 paper “Shape fluctuations and random matrices” in detail?

I am trying to compute the equilibrium measure for the Meixner ensemble on page 19 (on the arxiv version). The "details" of the computation are in Section 6, where he finds the equilibrium measure is ...
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### Minimum and maximum eigenvalue

I don't know if this is the right place to post this question, but I find it interesting and have not gotten an answer elsewhere. If it violates any rules, I will gladly delete it. Let $\Lambda$ be ...

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