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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 $B=\{u_1,...,u_n\}$...
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        1answer
        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. "...
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        0answers
        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 ...
        2
        votes
        1answer
        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: $a_{ij} = 1$...
        2
        votes
        1answer
        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 ...
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        0answers
        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) ...
        6
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        0answers
        76 views

        Finding the maximal component of a vector in sublinear time

        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 ...
        2
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        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 $|\...
        2
        votes
        1answer
        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 $B$, ...
        1
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        0answers
        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 ...
        0
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        0answers
        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 ...
        1
        vote
        1answer
        68 views

        Sum of Square of the Eigenvalues of Wishart Matrix

        Let $A\in\mathbb{R}^{m\times d}$ matrix with iid standard normal entries, and $m\geqslant d$, and define $S=A^T A$. I want to have a tight upper bound for $\sum_{k=1}^d \lambda_k^2$, where $\...
        6
        votes
        0answers
        148 views

        Eigenvalues of cyclic stochastic matrices

        Let's consider the following $n \times n$ cyclic stochastic matrix $$ M= \begin{pmatrix} 0 & a_2 & & & &b_n \\\ b_1 & 0& a_3& &&& \\\ & b_2 & ...
        4
        votes
        0answers
        117 views

        Sum of eigenvalues is nonpositive

        Let $A$ be a symmetric positive semidefinite $n \times n$ matrix. How can I show that the sum of the largest $n-k+1$ eigenvalues of $A - k\cdot \textrm{diag}(A)$ is nonpositive, for any $k \in \{1, \...
        5
        votes
        0answers
        98 views

        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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