# 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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### 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{...
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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 ...
• 1,136
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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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### 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 ...
1 vote
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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. ...
1 vote
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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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### Perron-Frobenius theorem for reducible non-negative matrices

Let $M$ be a non-negative matrix ($M_{ij}\geq 0$ for all $i,j$). If $M$ is irreducible, then we know that there exists an eigenvalue $\lambda$ of $M$ that equals the spectral radius $\rho(M)$ and has ...
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### Why do GLS and ML estimators coincide for the estimation of a VAR(p) model?

When estimating the coefficients in a VAR(p) model (assuming normality), the coefficient estimators using GLS and MLE coincide. Could anyone explain why this is the case?
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### A basis that makes a matrix triangular.

Find a basis for $\mathbb C^3$ so that the following matrix is in triangular form: \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix} What are the eigenvalues? I ...
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### Check if adjacency tensor of hypergraph is irreducible

Consider an m-uniform, n-dimensional hypergraph. It's adjacency tensor is an $\overbrace{(n \times n \times ... \times n)}^{m}$ -dimensional tensor $T$. As shown in , to calculate the $(H)$-...
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### How can I guarantee that all the number of generalized eigenvectors are equal to the algebraic multiplicity?

I have found many questions in the Stack but, I couldn't find what I want... Since my major is physics, I'm not good at the terms in mathematics. So it was hard to reach understanding Jordan normal ...
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### Number of distinct eigenvectors in generalized eigenvalue problem $A v = \lambda B v$ (with structure on $A, B$)

Consider the generalized eigenvalue problem $$A v = \lambda B v$$ where $v \in \mathbb{R}^n, A, B \in \mathbb{R}^{n \times n}$. Suppose that $A, B$ are symmetric and positive semi-definite. Suppose ...
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### Generalized characteristic polynomial coefficients

Given two square matrices $A,\,B$ of order $n$, of which therefore all the terms are known, let us define the following polynomial: $$p(x) := \det(A - x\,B)\,.$$ I was wondering if in the literature ...
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### How does this problem specifically correspond to a generalized eigenvalue problem?

Given two symmetrical square matrices $A,B \in \mathbb R^{n\times n}$ and a rectangular matrix $W\in \mathbb R^{n,k}$. I want to maximize $$\max_{W} \frac{\det (W^TAW)}{\det(W^TBW)}.$$ I read that the ...
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• 167
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I'm considering the following model. Given $x_i\in\mathbb{R}^n$, $i\in \{1,2,\ldots,N\}$, where $n\leq N$. $X=[x_1,x_2,\ldots,x_N]$ and a full-rank $W\in \mathbb{R}^{N\times N}$ $P=[p_1,p_2,\ldots,... • 45 1 vote 0 answers 33 views ### For what$\lambda$do non-negative solution to$(A-\lambda I)x = b$exist? I am interested in understanding for what values of$\lambda$is$(A - \lambda I)^{-1} b$component-wise non-negative. Here$A$is a$n\times n$matrix and$b$is a$n\times 1$vector. I know there is ... • 1,330 7 votes 2 answers 135 views ### Generalized eigenvector for product of commuting matrices Suppose$A,B$are commuting invertible matrices with a common generalized eigenvector$v$with eigenvalues$a,b$respectively. That is, suppose there exist positive integers$K,L$such that$(A-aI)^K ...
I've been trying to solve the following exercise, In the space of bivariate polynomials of the form $f(x,y)=\sum_{n,m=0}^2a_{n,m}x^ny^m$, the lineal operator $T$ is defined by $Tf(x,y)=f(x+1,y+1)$. ...