Questions tagged [eigenvector]
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305 questions
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Polynomial filter on symmetric tridiagonal matrices
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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Do top eigenvectors maximise both Tr$(P\Sigma)$ and Tr$(P\Sigma P\Sigma)$ for orthogonal projection matrices P?
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
$$
\underset{P^2 = P,\; \text{rank}(P) = p}{...
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Smallest eigenvalues comparison between two matrices: Seeking proof ideas
This problem stems from a previous problem that has already been solved. Please refer to it. According to the solution by @Alex Gavrilov, the similarity transforming matrix from ${\bf J}$ to ${\bf ...
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Can it be proved that the specific real symmetric matrix is positive definite? (Numerically confirmed)
${\bf A} \in {\Bbb R}^{n \times n}$ is a real symmetric positive definite M-matrix, whose non-diagonal entries are non-positive. Defining a composite matrix ${\bf J}\in {\Bbb R}^{2n \times 2n}$ as
$$\...
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Numerical tests show the smallest eigenvalue of a certain matrix remains invariant when some parameters vary – any proof ideas?
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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Inverse of a fourth order tensor and derivative of a tensor with respect to another using spectral decomposition
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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Transfer of spectral data from stratified surfaces to embedded graphs
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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Are eigenspaces continuous?
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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Choosing eigenvectors continuously for positive-semidefinite matrix function of rank one
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 ...
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Eigenvectors for specific eigenvalues
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 ...
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Diagonalizing a linear self-adjoint matrix in a rank-deficient non-standard inner product space
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 are identical?
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^{\...
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Let $A$ be a SPD matrix. Suppose diagonal $(A)_{ii}$ equals to its eigenvalue $\lambda_i$. Must $A$ be a diagonal matrix?
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 $...
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Symplectic inner products of eigenvectors of complex symplectic matrix
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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Can the derivative of eigenvectors with respect to its components be taken as zero if all eigenvalues are equal?
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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