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        Questions tagged [eigenvalues]

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        2
        votes
        0answers
        43 views

        Convergence rate of the smallest eigenvalue of an integral of a multivariate squared Brownian Motion

        I am interested in deriving the convergence rate of the smallest eigenvalue of a sequence of random matrices with diverging dimension. More precisely, let $W_n(r)$ represent an $n$-dimensional ...
        2
        votes
        1answer
        109 views

        Jordan decomposition of a block matrix

        Assume $A$ is a block matrix of the form: $$A=\left[\begin{array}{cccc} A_{11}&A_{12}&\ldots&A_{1n}\\ A_{21}&A_{22}&\ldots&A_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ ...
        15
        votes
        2answers
        290 views

        Maximum dimension of space of matrices with a real eigenvalue

        Let $M_n(\mathbb{R})$ denote the space of all $n\times n$ real matrices. What is the maximum dimension $f(n)$ of a subspace $V$ of $M_n(\mathbb{R})$ such that every matrix in $V$ has at least one real ...
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        vote
        0answers
        61 views

        Solution of a manipulated equation vs the maximum eigenvalue and eigenvector of a non-negative matrix

        Lets assume we have the following equation: $AU=\lambda U \Rightarrow\left[ \begin{array}{c|c|c} 0 &A_{12}&A_{13}\\ \hline A_{21}& 0& A_{23}\\ \hline A_{31}&A_{32}&0 \end{...
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        votes
        0answers
        193 views

        Generalized eigen property of a matrix

        Given a $n \times n$ invertible matrix $A$, I am interested in the set $$ \mathcal{S}(A) = \{ D \textrm{ diagonal matrix } \mid \det(D - A) = 0 \}. $$ Thus, for all eigenvalues $\lambda_i$, we have $...
        6
        votes
        1answer
        247 views

        Monotonicity of Schrödinger Eigenvalues

        Let us consider the Schr?dinger operator $$ H_hf(x)=-\frac{d^2}{dx^2}f(x)+h(h\sin^2(x)-\cos(x))f(x) $$ on $L^2[-\pi,\pi]$ with Neumann boundary conditions $f^\prime(\pm\pi)=0$. Here, $h\geq 0$ is a ...
        6
        votes
        0answers
        137 views

        Can we approximate any eigenvalue of an infinite matrix via eigenvalues of some sequence of submatrices which approximates the matrix?

        Let $T:\ell^2\to\ell^2$ be a compact linear operator. Let $[T]=(a_{i,j})_{i,j=1}^{\infty}$ be the representing infinite matrix of $T$ with respect to the canonical base. Let $T_n$ be the finite rank ...
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        30 views

        A recap of regularity of singular values as a function over M_n

        So the core of the question is the study of the function $$ s :M_n(\mathbb R) \mapsto M_n(\mathbb R)$$ $$A \rightarrow s_n(A) $$ where $s_n(A)$ is the greatest singular value of A. I know there has ...
        2
        votes
        1answer
        112 views

        Eigenvalues Sturm-Liouville Operator

        Is the eigenvalue decomposition of the Sturm-Liouville operator $$ Lf(x)=-f''(x)+h\sin(x)f'(x),\quad h>0, $$ with Neumann boundary conditions $f'(-\pi)=f'(\pi)=0$ on the Hilbert space $L^2([-\pi,\...
        5
        votes
        1answer
        112 views

        eigenvalues of a symmetric matrix

        I have a special $N\times N$ matrix with the following form. It is symmetric and zero row (and column) sums. $$K=\begin{bmatrix} k_{11} & -1 & \frac{-1}{2} & \frac{-1}{3} & \frac{-1}{4}...
        5
        votes
        1answer
        164 views

        Perturbing a normal matrix

        Let $N$ be a normal matrix. Now I consider a perturbation of the matrix by another matrix $A.$ The perturbed matrix shall be called $M=N+A.$ Now assume there is a normalized vector $u$ such that $\...
        2
        votes
        1answer
        239 views

        An inequality on elementary symmetric polynomial of eigenvalues

        For $A \in \mathbb{R}^{n\times n}$, I am wondering if the following inequality holds, $$\text{tr}(A)^2 - \text{tr}(A^2) \leq ||A||_*^2 - ||A||_F^2$$ This is equivalent to the following inequality on ...
        4
        votes
        1answer
        132 views

        Spectrum of this block matrix

        Consider the following block matrix $$A = \left(\begin{matrix} B & T\\ T & 0 \end{matrix}?\right)$$ where all submatrices are square and matrix $B = \mbox{diag}\left(b_1 ,0,0,\dots,0,b_n \...
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        votes
        0answers
        46 views

        Smallest eigenvalues of block Kronecker product

        Let $D \in \mathbb{R}^{n \times n}$ defined as \begin{equation} D := \begin{pmatrix} 1 & 0 & \cdots & \cdots & 0 \\ -1 & 1 & \ddots & \ddots & 0 \\ \vdots & \ddots &...
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        vote
        0answers
        56 views

        Smallest eigenvalue for Gram matrix of unit norm matrices

        Given $n$ symmetric matrices $A_1, \dots, A_n \in \mathbb{R}^{k\times k}$, such that $\|A_i\| \leq 1$ for all $i$, we consider the matrix $M \in \mathbb{R}^{n\times n}$, where $M_{ij} = \langle A_i, ...

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