# Questions tagged [generalized-eigenvector]

This tag is for questions relating to Generalized-eigenvector, a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector.

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### Effect of rank-one update on the smallest eigenvalue and its eigenvector

Suppose diagonal $D\in \mathbb R^{n\times n}$ with $D\succeq 0, v\in \mathbb R^n,$ and $\alpha>0$ are given. Can we $\textit{exactly}$ identify the smallest eigenvalue of $D+\alpha vv^T$ and its ...
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### Eigenvalues and eigenspaces of automorphisms

Consider $T,S: \mathfrak{g} \longrightarrow \mathfrak{g}$ automorphisms of the $n$-dimensional Lie algebra $\mathfrak{g}$. Is there any relation between the eigenvalues of $T$ and $S$ and the ...
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### On Orthogonality of Generalized Eigenspaces

I was reading a paper, and it made a claim that for some nilpotent matrix $A$, we can say that we can find a Jordan basis of $A$ that is orthonormal. I understand that what this means is that all sets ...
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### Are generalized eigenvectors orthogonal to eachother if the problem is symmetric?

Given two real symmetric matrices $A$ and $B$, are the eigenvectors of the generalized eigenvalue problem $Ax = \lambda B x$, $x^T B x = 1$ orthogonal? Put another way, if we denote by $P$ the matrix ...
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### What does it mean that a distribution is a generalized function and why are generalized eigenvectors distributions?

I am studying eigenvalues and eigenvectors of operators on infinite-dimensional spaces, and I am struggling to understand generalized eigenvectors. In order to explain myself better, I'm going to ...
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### Connection between generalized eigenvectors and Jordan normal form

I have the matrix A = $\begin{bmatrix} 3 & 4 & 3 \\ -1 & 0 & -1 \\ 1 & 2 & 3 \end{bmatrix}$ and want to find a matrix S such that $SAS^{-1}$ is an upper triangular matrix. ...
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### why $\frac{1}{2} \sum_{i, j} \mathbf{A}^{(t)}(i, j)\left\|\mathbf{y}_i-\mathbf{y}_j\right\|_2^2$ can boils down to a generalized eigen-problem [closed]

I'm reading a paper, it writes "$\frac{1}{2} \sum_{i, j} \mathbf{A}^{(t)}(i, j)\left\|\mathbf{y}_i-\mathbf{y}_j\right\|_2^2$ can boils down to a generalized eigen-problem" (snapshot: part1 ...
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### How to I solve Generalized Eigenvalue Problem with Cholesky Factorization if $A$ and $B$ are symmetrical?

Assume that we are going to solve generalized eigenvalue problem $$Av = \lambda B v$$ Where $A$ and $B$ are symmetrical matrices. Assume that we can only use the MATLAB routine ...
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### Representation of an operator on a generalized eigenfunction

I am looking for a particular result about generalized eigenfunctions which i am not sure exists, but i havent been able to find a counter example. I have read through (most of) the Gel'fand/Shilov ...
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### What are the eigenvalues of a squared matrix?

Suppose matrix $A$ has eigenvalues $\lambda_1$ and $\lambda_2$. Are the eigenvalues of $A^2$: $\lambda_1^2$ and $\lambda_2^2$? If so, can I prove this by simple diagnolization, where $T$ is the ...
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### Prove existance of orthonormal basis of $\mathbb{R}^3$ consisting of eigenvectors of generalized eigenvalue equation

Context I am studying normal modes oscillations and normal modes [1,2]. In an earlier post [3], I asked for a proof that the generalized eigenvalue equation in normal-mode analysis has positive ...
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### Linear System O.D.E.: 1<dim(Eigenspace)<eigenvalue multiplicity

Let’s be: the linear system $\boldsymbol{x}’=A\boldsymbol{x}$ where $A\in M_{n\times n}(\mathbb{R})$ is constant and an eigenvalue ($\lambda$) with multiplicity $m>1$. What happens if the ...
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### An element both in $V_\mu$ and in $V_\overline{\mu}$ is zero?

I was doing an exercise with following notations: $V$ is a $\mathbb{R}$-vectorspace $\alpha \in$Hom$_\mathbb{R} (V,V)$ and $\alpha_\mathbb{C}\in$Hom$_\mathbb{C} (V_\mathbb{C},V_\mathbb{C})$ its ...
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### How to solve a non-linear eigenvalue problem based on experimental data

I have a matrix $A(\operatorname{Im}(\lambda))$ whos terms $a_{ij}$ depends on the imaginary part of $\lambda$ with a relationship that is experimentally assessed (to give a bit more context, I'm ...
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Introduction: I have a mass-spring-damper system with $q$ degrees of freedom. We model it by using a 2nd-order vectorial ODE \eqref{1} on variable $\left\{X\right\}$: \left[M\right] \cdot \left\{\... • 1,302 2 votes 2 answers 60 views ### What is the proof for the formula used in calculating generalized eigenvectors? Definitionally, a generalized eigenvector for matrix A is a vector \textbf{x} such that \begin{align} (A - \lambda I)\textbf{x} \neq \textbf{0} \\ (A - \lambda I)^m\textbf{x} = \textbf{0} \end{... 0 votes 1 answer 36 views ### Normalize projection matrix - Generalized eigenvalue problem In mechanical dynamics (vibration), I have generalized eigenvalue problem with two symmetric matrix M (mass) and K (stiffness) K \cdot v = \lambda M \cdot v $$using ... • 1,302 3 votes 1 answer 89 views ### About the consecutive dimensions of k-eigenspaces If \lambda is an eigenvalue of matrix A, is it possible that for some positive integer n, dim(N(A-\lambda I)^{n+1})-dim(N(A-\lambda I)^{n})>1. I am studying generalised eigenspaces and ... • 2,721 0 votes 1 answer 396 views ### Rewrite generalized eigenvalue problem as standard eigenvalue problem I have two matrix \mathbf{A} and \mathbf{B} and I want to find the values of \lambda such that$$ \mathbf{A} \cdot \mathbf{v} = \lambda \cdot \mathbf{B} \cdot \mathbf{v} \mathbf{A} and \... • 1,302 2 votes 2 answers 134 views ### Maximization of tr[U^HAU(U^HBU)^{-1}] I am looking for the solution of \begin{align*} \max_{{U}\in\mathbb{C}^{M\times P},\ {U}^H{U}={I}_P } &tr\left[ {U}^H{A}{U}\left({U}^H{B}{U}\right)^{-1}\right], \end{align*} where P\leq M, {A} ... • 185 1 vote 1 answer 97 views ### Why can't columns of a generalized modal matrix for the same Jordan block be interchanged? Suppose we have the matrixM=\begin{pmatrix} 1 & 0 & 1\\ 1 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}$$Then we can find a Jordan normal form J and generalized modal matrix P such ... 1 vote 0 answers 41 views ### Chains of generalised eigenvectors of an operator pencil I am reading a paper "Finding Eigenvalues of Holomorphic Fredholm Operator Pencils using boundary value problems and Contour integrals" of Beyn, Latushkin, Rottmann-Matthes and in subsection ... • 654 0 votes 0 answers 27 views ### Eigen decomposition of two matrix equation I have fallowing eigen equation:$$ C_{\tau}r_i = C_0r_i\lambda_i $$where C_{\tau}r_i and C_0 are non-positive definitive matrices and r_i and \lambda_i are eigenvectors and eigenvalues. ... 1 vote 0 answers 71 views ### What gurantees the existence of generalized eigenvectors of rank m? Let A be a n\times n matrix. If \lambda is an eigenvalue of A with multiplicity m>1. Then there could be two cases: There are m linearly independent eigenvectors of A corresponding to \... • 11 1 vote 0 answers 62 views ### Proving the relationship between left and right eigenvalues for a complex generalized eigenproblem A generalized right eigenproblem is defined as: A q = \lambda B q where A and B are complex matrices in general and q represents the right eigenvector, which is a column vector, and \lambda ... • 43 0 votes 0 answers 54 views ### Generalised Eigenfunctions Given \mathbf{J}, the Jacobian matrix of a function \mathbf{F}:\mathbb{R}^n\to\mathbb{R}^n, the system$$ \mathbf{J}\mathbf{v}=\lambda \mathbf{v}  has solutions corresponding to eigen-pairs $(\... • 3,427 5 votes 0 answers 63 views ### Finding a unit vector$v\$ that makes only one quadratic form vanish

I was reading a proof on the non-convexity (even locally) of loss landscape in high-dimensional neural networks. Specifically, in the paper, it seems like the proof of proposition 2 at some point uses ...