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During my research, I encountered a problem involving symmetric tridiagonal matrices and their eigenvectors, which I have been attempting to solve since then. So far, I have only managed to obtain a ...
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Let $P \in \mathbb{R}^{d \times d}$ be an rank $p < d$ orthogonal projection matrix given by $P = VV^T$, $V \in \mathbb{R}^{d \times p}$. Do we have that $$ \underset{P^2 = P,\; \text{rank}(P) = p}{...
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This problem ‌stems from‌ a previous problem that has already been solved. Please refer to it. According to the solution by @Alex Gavrilov, the ‌similarity transforming matrix from ${\bf J}$ to ${\bf ...
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${\bf A} \in {\Bbb R}^{n \times n}$ is a real symmetric positive definite M-matrix, whose non-diagonal entries are non-positive. Defining a composite matrix ${\bf J}\in {\Bbb R}^{2n \times 2n}$ as $$\...
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I've numerically verified that the smallest eigenvalue of a specific matrix remains invariant when some parameters change. Looking for theoretical proof approaches. Appreciate any insights. ${\bf A} \...
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I am writing a UMAT code in Abaqus to simulate hyperelastic fracture. Towards that end, I have calculated the Second Piola-Kirchhoff tensor and the material consistent jacobian tensor. However, Abaqus ...
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Let $\Gamma \subset X$ be a finite connected $(q+1)$-regular Ramanujan multigraph, embedded as the 0,1-skeleta of a stratified (smooth away from $\Gamma$) 2-complex $X \subset \mathbb{R}^3$. A ...
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There is a class of results of the form "eigenvalues are continuous in square matrices" (e.g., this MSE question and its answers). An analogous question is whether the eigenspaces of an $n \...
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Consider a real-analytic, rank-one, matrix-valued function $M(t)\geq 0$ of single real variable $t$. Can one choose a symmetric factorization $M(t) = z(t) z^T(t)$ (where $z(t)$ is an eigenvector with ...
Bassam's user avatar
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When reading the following paper: Ed S. Coakley, Vladimir Rokhlin, A fast divide-and-conquer algorithm for computing the spectra of real symmetric tridiagonal matrices, Applied and Computational ...
William Casper's user avatar
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I want to solve $Ax=\lambda x$ in a rank-deficient non-standard inner product space $\mathcal{S}^r$. By "rank-deficient", I mean that although $x$ is represented by a vector of $\mathbb{R}^n$...
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Two SPD matrices admits eigen-decomposition $\Sigma_p=U_p S_p U_p^{\top}$ and $\Sigma_q=U_q S_q U_q^{\top}$, where $S_p$ and $S_q$ contain ordered eigenvalues that are distinct. Let $\Sigma_v=U_q^{\...
Jeff's user avatar
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Given a symmetric positive definite (SPD) matrix with eigendecomposition $A = U \Lambda U^{\top}$ follows $\operatorname{diag}(A)=\operatorname{diag}(\Lambda)$, is it necessary to have $U=I$ so that $...
Jeff's user avatar
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Let $S$ be a $2N\times 2N$ complex symplectic matrix ($\operatorname{Sp}(2N,{C})$) satisfying $S^\dagger JS=J$, where $$J=\begin{pmatrix}0&I_N\\-I_N&0\end{pmatrix}.$$ The symplectic inner ...
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I want to ask a couple of follow up questions to the question answered on the thread "Derivative of eigenvectors of a matrix with respect to its components". I noticed that in the accepted ...
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I would like to apologize for leaving things a bit vague. I think my question could be stated much more precisely but right now that is difficult for me to do. I nevertheless think it is an ...
Robert Wegner's user avatar
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Consider the Laplacian matrix of the path graph: $$ L = \begin{bmatrix} 1 & -1 & 0 & \cdots & 0 & 0\\ -1 & 2 & -1 & \cdots & 0 & 0\\ 0 & -1 & 2 & \...
user123's user avatar
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Let $\mathbb{X}, \mathbb{Y} \subset \mathbb{R}^d$ be finite sets. Suppose random vectors $X \in \mathbb{X}$ and $Y \in \mathbb{Y}$ are sampled according to a joint distribution $\mathbb{P}_{XY}$. ...
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Let $N$ be a natural number and let $w$ be a complex number. We define the $N\times N$ matrix $C_w=(a_{k,l})_{k,l=1}^N$ as follows, $$ a_{k,l}=\begin{cases}1 & l=k+1\\ w &...
ABB's user avatar
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Let $P$ be an $m \times N$ matrix of zeros and ones (think $N \gg m$), and let $\mathbf{u} \in \mathbb{R}^N$ be a unit vector satisfying $P^\top P\mathbf{u} = \lambda^2 \mathbf{u}$ for some $\lambda &...
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I want to find a 𝑁×𝑁 positive definite correlation matrix, which ensures that as 𝑁 goes to infinity, only a finite number of eigenvalues remain non-zero, while the rest eigenvalues approach zero. ...
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Suppose an $n\times n$ matrix has real entries and has $n$ real eigenvalues and its eigenvectors span $\mathbb R^n.$ Are there any interesting conditions under which $k$ of its eigenvectors are ...
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Let $\omega_1<\dots<\omega_n\in\mathbb{R}$. Then, define $G\in\mathbb{C}^{n\times n}$ such that $G_{k\ell}=\frac{1}{1-i(\omega_\ell-\omega_k)}$. For example, if $n=3$ we obtain $$ G=\begin{...
PIII's user avatar
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Consider the following $(2d+1)\times (2d+1)$ matrix: $$ A = \begin{pmatrix} 0 &\frac{2d}{2} & 0 &0 & \cdots &0 & 0 \\ \frac{1}{2} & 0 & \frac{2d-1}{2} &0& \...
Quokka's user avatar
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Wonder whether anyone has an idea on showing the following or to point out that it is not true: Let $A(t) \in \Re^{n \times n}$ be differentiable over an interval $I$, and it has a zero eigenvalue for ...
muddy's user avatar
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In their 1999 paper "Sturm-Liouville operators with singular potentials", Savchuk and Shkalikov prove the uniform resolvent convergence of an operator $L_\varepsilon \rightarrow L$ for $\...
builtdifferential's user avatar
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This question has the same flavor of this and this questions, but asks for something stronger. Assume that $A$ is a symmetric $n \times n$ matrix, $H$ is a $n \times n$ perturbation matrix. Moreover ...
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Let $X$ be a $n\times n$ symmetric matrix with iid zero-mean random entries on and above the diagonal. Denote by $v$ the eigenvector corresponding to the largest eigenvalue of $X$. Let $a$ be a fixed $...
legon's user avatar
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I am trying to understand the connection between the eigenspace of the continuous operator $$ H(x,y) = \frac{1}{x+y} $$ which is nothing but the square of the Laplace operator, and its discrete ...
knuth's user avatar
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Let $\{A_i, i\ge 3\}$ be the matrices whose columns represent numbers from $0$ to $2^i-1$ in binary form. For example, $A_3 = \begin{bmatrix} 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \...
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My question seems elementary, yet I could not find the solution after working on and searching for several days... I'd like to find the eigenfunctions of a simple integral kernel: \begin{equation} \...
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Let $H: \mathbf{R}^n \rightarrow \mathbf{R}$ be a bounded continuous function. Set $$\tag{1} \int_{\mathbf{R}^n}\left\{|\nabla \xi|^2+H(x) \xi^2\right\} \mathrm{d} x \geqslant 0, \quad \forall \xi \in ...
Elio Li's user avatar
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Let $\mathcal H$ be a Hilbert space and let $O$ be an operator. Obviously $M=O^\dagger O$ is a semi-positive definite operator and $v\in\ker M$ if and only if $v\in\ker O$. Therefore it seems to me ...
DeltaTron's user avatar
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I have a matrix $A$ $(n\times n)$ with eigenvalues $\lambda_i$, then I add another matrix to it as: $A+uv^\top$ where $u$ $(n\times 1)$ and $v$ $(n\times 1)$ are column vector. Also, $A=yy^\top$ with $...
brant's user avatar
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The following point is well-known in the literature. Theorem. Let $A$ be a non-negative matrix in $M_n(\mathbb{R})$. If $A$ is nil-potent, there is a permutation matrix $P$ such that $P^tAP$ is ...
ABB's user avatar
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Consider the following symmetric second order diffusion operator, defined, for $\phi \in \mathcal{C}^{2,1}_c\left(\mathbb{R}\times \mathbb{R}_+\right)$, by: $$L\phi := \lambda_1 \partial_{R_1}(R_1 \...
Greyearl's user avatar
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Let us consider the backward-shift matrix $B=(b_{ij})\in M_n(\mathbb{R})$ whose entries are given by $b_{k,k+1}=1$ and the other entries are all 0. We also consider $X=(x_{ij})\in M_n(\mathbb{R})$ ...
ABB's user avatar
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3 votes
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Computing the Maximum Likelihood Estimator of Gaussians in arbitrary finite Hilbert spaces seems no easy task and I must admit to lamentably fail at it. The classical theory most often relies on ...
hdeplaen's user avatar
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Given $n$ orthonormal vectors $v_1, \dots, v_n \in \mathbb R^d$, where $d > n$, we can show that there are many orthogonal matrices $X$ of size $d$ such that $v_1, \dots, v_n$ are eigenvectors of $...
SiXUlm's user avatar
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1 answer
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Let $G$ be a directed graph and let $P_i$ be its vertices. Let $A$ be the corresponding adjacency matrix of $G$, i.e. $a_{i,j}=1$ if and only if there is a directed edge from $P_i$ to $P_j$, ($a_{i,...
ABB's user avatar
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2 votes
1 answer
146 views

Suppose that $e_1, \cdots, e_n$ are the standard vectors of the Euclidean space $\mathbb{R}^n$. Let us consider the backward shift operator $T:\mathbb{R}^n\to \mathbb{R}^n$ given by $Te_k=e_{k-1}$ ...
ABB's user avatar
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2 votes
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Let $\mathbb{S}^2_+$ denote the closed upper hemisphere of the unit round sphere in $\mathbb{R}^3$. It is well known that the first positive eigenvalue of the Laplacian on the closed unit sphere is $2$...
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I am trying to write a QR algorithm in Python for eigenvectors and eigenvalues finding for large symmetric matrices, My initial thought was to use Householder transformation with a Wilkinson shift ...
Daniel Belaish's user avatar
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1 answer
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Consider the matrix $$A_2:= \begin{pmatrix} a & b_1 \\ b_2 & a\end{pmatrix}.$$ Let $\sigma_2 = \begin{pmatrix} 0 & i \\ -i & 0 \end{pmatrix}$, then $$\sigma_2 A_2 \sigma_2 = \begin{...
António Borges Santos's user avatar
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1 answer
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I have a matrix $A$ $(n\times n)$ with eigenvalues $\lambda_i$, then I add another matrix to it as: $A+xx^\top$ where $x$ $(n\times 1)$ is a column vector. and also $A=yy^\top$ with $y$ a $(n-1)$ rank ...
brant's user avatar
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1 vote
0 answers
404 views

Let $A$ be an adjacency matrix of undirected graph $G$, where $G$ is a connected graph. The normalized adjacency matrix is defined as $\hat{A}=D^{-1/2}AD^{-1/2}$, where $D$ is degree matrix of graph $...
MikeDean's user avatar
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Let $Q\subset \mathbb R^n$ be a bounded domain with boundary $\partial Q\in C^\infty$ and $\varphi_1,\varphi_2,\ldots$ are eigenfunctions of the Dirichlet problem for the Laplace equation in $Q$ ...
Andrew's user avatar
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I found a simple algorithm for simultaneous diagonalization of two commuting matrices (Nordgren - Simultaneous Diagonalization and SVD of Commuting Matrices), which seemed to be well-founded. For ...
TobiR's user avatar
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Suppose that $A=(a_{kl})_{k,l=1}^n$ is a symmetric tri-diagonal matix in $M_n(\mathbb{R})$ whose diagonal entries are $a_{kk}=\cos\frac{k\pi}{n+1}$ and $$a_{1,2}=\cdots=a_{n-1,n}=1$$ Any approach to ...
ABB's user avatar
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Let us consider the real symmetric tridiagonal matrix $T=(t_{kl})$ in $M_n(\mathbb{R})$ with $$t_{1,2}=t_{2,3}=\cdots=t_{n-1,n}=1$$ How can we compute the eigenvectors of $T$?
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