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        Questions tagged [eigenvalues]

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        31 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 ...
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        51 views

        Non-negative irreducible matrices with random (correlated or independent) non-zero entries

        Lets $M$ be a non-negative irreducible matrix. According to Perron-Frobenius Theorem, the maximum eigenvalue of $M$, $\lambda$, is positive and equal to its spectral radius $\rho(M)$. Now assume the ...
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        121 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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        29 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 ...
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        1answer
        105 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,\...
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        1answer
        105 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}...
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        1answer
        163 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 $\...
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        1answer
        127 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 ...
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        1answer
        128 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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        45 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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        46 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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        77 views

        What matrix has only negative or zero real part for all the eigenvalues?

        Say $X \in \mathbb{R}^{m\times m}$, Is it possible to have a constraint on $X$, such that all the eigenvalues has negative or zero real part? What I conjecture The following $X$ has only negative ...
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        2answers
        128 views

        Completely positive matrix with positive eigenvalue

        A matrix $A \in \mathbb{R}^{n \times n}$ is called completely positive if there exists an entrywise nonnegative matrix $B \in \mathbb{R}^{n \times r}$ such that $A = BB^{T}$. All eigenvalues of $A$ ...
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        1answer
        53 views

        Monotonicity/Scaling of Sturm-Liouville Eigenvalues

        Consider the regular Sturm-Liouville eigenvalue equation $$ \frac{d}{dx}(p_t(x)f^\prime(x))=\lambda_t f(x) $$ for $p_t\in\mathcal{C}^\infty([0,1])$ with Dirichlet boundary conditions on $[0,1]$. Here $...
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        0answers
        82 views

        Probabilistic characterization of first Neumann eigenvalue

        In this MO post, a question has been asked (and answered) about the probabilistic interpretation of the first Dirichlet eigenvalue of the Laplacian in terms of boundary hitting times. I wish to ask ...

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