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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generalized eigenspace is the same as the eigenspace of the semi-simple part.

Let $f: V\rightarrow V$ be an endomorphism of complex vector spaces and $ f=f_S+f_N$ it's decomposition into semi-simple and nilpotent parts. Then apparently we have that the generalized eigenspace $V^...
Adronic's user avatar
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generalized implicitly restarted lanczos method

I am looking for references on how to solve the generalized eigen-problem : $$Ax = \lambda Bx \tag{1}$$ Where $A$ is a symmetric matrix and $B$ is symmetric positive definite. I know a standard ...
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Generalized Eigenspaces of Composed Operators

I am looking for a proof check, as well as a bit of help. Let $X$ be a finite-dimensional vector space over $\mathbb{C}$. Let $R$ and $S$ be linear operators on $X$. (a) Prove that $SR$ and $RS$ have ...
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Why does increasing the block size in the block Lancsoz algorithm help me find all the eigenvalues I'm looking for?

I'm dealing with a non-properly constrained mechanical model. I have to find the vibration properties of the system hence I'm dealing with a generalized eigenvalue problem of the following form: $ (K-\...
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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 ...
T.L's user avatar
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Can multiple generalized eigenvectors of higher rank get mapped to the same generalized eigenvector of lower rank?

I'm reading the wikipedia page for generalized eigenvectors, and I'm stuck on one part: They show in the canonical basis section that (assuming a fixed eigenvalue $\lambda$ for the whole post), $\...
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Generalized Eigenvectors Lemma from Artin's "Algebra"

This is from Artin's "Algebra" (2nd Edition), Proposition 4.7.1 and Lemma 4.7.2.. Let $x$ be a generalized eigenvector of a linear operator $T$, with eigenvalue $\lambda$ and exponent $d$. ...
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Pose quadratic equation with matrix coefficients in eigenvalue form

I'm working on a problem that involves an equation in the form $$ x^2[A][v] + x[B][v] + [C][v] = [0] $$ where $[A], [B], [C]$ are some known full rank $R^{n\times n}$ square matrices, $[v]$ is some ...
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Which LAPACK routine should I use if I only want to compute the eigenvectors?

Assume that you are having two types of matrices: $A$ and $B$. They both are symmetric. Which fastest LAPACK routine should I use if I only want to compute the eigenvectors $v$ from the generalized ...
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What's the difference between using the inverse trick VS cholesky trick for solving generalized eigenvalue problem?

My math professor said once that solving the generalized eigenvalue problem $$A\lambda = \lambda B v$$ There are two methods to use: Method #1 Use cholesky decomposition $$B = LL^T$$ Solve $Y$ $$AY = ...
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Find the eigenvectors , generalized eigenvectors and the minimal polynomial of a certain matrix

Given $A=$ $ \begin{pmatrix} -3 & 1 &-1 \\ -7 & 5 & -1 \\ -6 & 6 & -2 \\ \end{pmatrix} $ and also the characteristic polynomial $P_A(x)=(x+2)^2 \cdot (x-4)$ Find the ...
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Finding the matrices $J$ (Jordan normal form) and $T$ so that $J=T^{-1}AT$

To solve an automation engineering exercise, I need to find the Jordan normal form $J$ and a matrix $T$ so that $J=T^{-1}AT$, where $A$ is the initial matrix (given by the exercise). For example: $$A =...
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Why for generalized eigensubspace $V( \lambda )$ to be non-trivial $\lambda$ must be an eigenvalue?

Let $T$ be linear transformation, defined for some space $V$, where $T: V \to V$. Let vector $v \in V$ be called generalized eigenvector of rank $m$ with correspondence to eigenvalue $\lambda$, if $(T ...
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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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Stability of generalized eigenspaces

Let $A$ be an associative $\mathbb{C}$-algebra and let $\mathfrak{r}\subset A$ be a finite dimensional Lie subalgebra of $A$ (for the commutator bracket) and assume that $\mathfrak{r}=\mathfrak{n}\...
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Algebraic multiplicity of a zero eigenvalue, in a generalized eigenvalue problem

Consider the generalized eigenvalue problem (GEVP) $\begin{equation} \boldsymbol{A} \boldsymbol{v} = \lambda \boldsymbol{B} \boldsymbol{v} \tag{1}\label{1} \end{equation} $ where $\boldsymbol{A}, \...
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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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Vector space$V$ is direct sum of eigenspaces of $\phi$ when $\phi $ is diagonalisable

This seems a simple statement but if $\phi \in \text{End}_K (V)$ is diagonalisable then I am trying to show that $$V=V_{\lambda_1} \oplus \cdots \oplus V_{\lambda_n} $$ where $\lambda_1 ,\ldots, \...
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Complex conjugate generalized-eigenvectors of a non-diagonalizable matrix

Let $v_i$ ($i = 1,\ldots m$) be the generalized eigenvectors of matrix $A$ associated with the Jordan block $J(\lambda)$, where $\lambda$ is a complex eigenvalue of $A$. Can we show that the complex ...
Ryan's user avatar
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Generalized eigenvalue problem with real symmetric matrices and Hadamard product

I have a generalized eigenvalue problem of the form $$ (S \circ A) v = \lambda S v $$ where $\circ$ denotes the elementwise or Hadamard product and $S$ and $A$ are real symmetric matrices As far as I ...
b3m2a1's user avatar
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Jordan Normal Form using generalized eigenvector

How do I find the Jordan Normal Form for this matrix $A=\begin{bmatrix} -2 & -3 & 6 \\ 1 & 2 & -2\\ -1 & -1 &3 \end{bmatrix}$ ? This is my method but I am not able to get to ...
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Generalized eigenspace formula

I'm trying to figure out the very last sentence of the proof of this theorem, and I don't understand why what they're saying is true. Theorem: Let $V$ be an $F$ vector space with $\dim_{F} V=n$ and $T:...
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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 ...
esang's user avatar
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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 ...
breakingmath's user avatar
1 vote
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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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Possible or not? Jordan chain does not span the full generalised eigenspace

Given a matrix $A$ with eigenvalue $\lambda$ and a corresponding eigenvector $v_1$: Is it possible that the Jordan chain $v_1$ generates does not span the full generalised eigenspace of $A$ w.r.t $\...
Eric Chan's user avatar
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Is $N_T \subseteq Im(T^l)$?

If $T\in \mathcal L(V)$ is a nilpotent linear map (ie. $T^k=0_V$ for some $k \in \mathbb{N}_{\ne 0}$), let $N_T$ be the nullspace of $T$, $Im(T^l)$ be the image of $V$ under $T^l$. Question: Is $N_T \...
Eric Chan's user avatar
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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 ...
mikasa's user avatar
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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 ...
Luca's user avatar
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2 answers
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Minimal polynomial from Jordan canonical form.

Suppose $T:V\to V$ is a linear operator on $V$ which is finite dimensional and suppose the JCF of $T$ is, $\begin{pmatrix}c & \\1 &c\\ & &c\\ & & &d\\ & & & 1&...
Kishalay Sarkar's user avatar
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Name for this eigenvector/eigenvalue problem

Let $A \in \mathbb{C}^{n \times m}$ and $B \in \mathbb{C}^{m \times n}$ be matrices. Consider the following problem: Find non-zero vectors $u \in \mathbb{C}^m$ and $v \in \mathbb{C}^n$ such that $Au =...
Frederic Chopin's user avatar
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1 answer
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Solve 2nd-order ODE with $n$ variables: Mass-spring-damper overdamped

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\{\...
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2 answers
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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{...
Terence M. Highsmith's user avatar
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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 ...
Carlos Adir's user avatar
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3 votes
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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 ...
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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 $\...
Carlos Adir's user avatar
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2 votes
2 answers
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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} ...
Fr2021's user avatar
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Why can't columns of a generalized modal matrix for the same Jordan block be interchanged?

Suppose we have the matrix $$M=\begin{pmatrix} 1 & 0 & 1\\ 1 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}$$ Then we can find a Jordan normal form $J$ and generalized modal matrix $P$ such ...
Stew Guffin's user avatar
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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 ...
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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. ...
Daniel Wiczew's user avatar
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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 $\...
MathGuy's user avatar
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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 $\lambda$ ...
Bob S's user avatar
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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 $(\...
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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 ...
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