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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Help finding Eigenvectors

The matrix is \begin{equation*} A = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & -2 \\ 3 & 2 & 1 \end{pmatrix} \end{equation*} I got the eigenvalues $\lambda_1 = 1, \lambda_2 = 1 + 2i$...
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10 views

EIGS: Why is 'smallestabs' faster than 'largestabs'?

Let $L$ be the Laplacian matrix of a grid graph of $n$ nodes, $D$ a $n\times n$ diagonal matrix of positive values and $s$ a vector of dimension $n$. I want to find $x$, such that $L x = \lambda(D + ...
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29 views

Describing a Matrix of Generalized Eigenvectors

Suppose $A(z)$ and $B(z)$ are $n\times n$ matrices with elements that are functions of $z\in \mathbb C$. Further, suppose that $A(z)$ is diagonal and that $$\det\left(A(z)B(z)-I_n\right)=0$$ has ...
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2answers
51 views

Can there be a different Eigen vector for a particular Eigen value?

Please see the photo. Here, my answer came $k$ $\begin{bmatrix} 1\\ 1\\ -1 \end{bmatrix}$ But their answer is given : $k$ $\begin{bmatrix} ...
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1answer
72 views

Which of the statements is not necessarily true?

Let $A$ be a $3\times3$ matrix and $u, v, w$ be linearly independent vectors in $\mathbb{R}^3$ such that: $Au = 2u, Av = 2v, Aw = 0$. Which of the statements are NOT necessarily true? Option 1: $w$ is ...
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1answer
35 views

Generalized spectral radius and matrix norm

I am reading a paper that discusses the design of an approximate matrix in the context of numerical methods for PDEs. However, I do not understand the following step, which I believe should be ...
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15 views

generalized eigenvalue problem with tri-diagonal matrix admits only real eigenvalues

I am considering this generalized eigenvalue problem: $( A-\lambda B ) x = \mathbf{0}$, here $A$ and $B$ are real tri-diagonal matrix and $A$ is symmetric but $B$ is not . I find this problem usually (...
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29 views

As for generalized eigenspaces, do these results still hold on infinite-dimensional spaces?

Suppose V is a vector space and $T\in L(V)$. Let $G(\lambda,T) $ denote the generalized eigenspace of $T$ corresponding to $\lambda $. In other words, $G(\lambda,T)=\{\;v\in V\;\vert\;{(T-\lambda I)}^{...
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1answer
20 views

Need help finding an eigenvector

I'm trying to compute the eigenvector of the following matrix: $$ A = \begin{bmatrix} 0.3889 & 0.3456 \\ 0.3456 & 0.4044 \\ \end{bmatrix} $$ where one of the ...
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1answer
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“Generalized eigenvectors corresponding to distinct eigenvalues are linearly independent” for infinite-dimensional spaces?

This is form Axler's Linear Algebra Done Right. Please allow me to borrow the screenshot from this Question : 8.13 Linearly independent generalized eigenvectors From the above picture, we can see that ...
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1answer
34 views

definition of generalized eigenspace

If $G_\lambda$ is the generalized eigenspace corresponding to the eigenvalue $\lambda$, and $E_\lambda$, the eigenspace, then why is both of the following true? $G_\lambda = \ker(T-\lambda I)^{\dim V}...
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17 views

Must generalized eigenvectors of distinct eigenvalues be orthogonal?

Let $v_1, v_2$ be generalized eigenvectors of ranks $l_1, l_2 > 0$, which respectively belong to eigenvalues $\lambda_1 \neq \lambda_2$ of the linear operator $A$: $(A - \lambda_i I)^{l_i} v_i = 0$ ...
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59 views

A matrix system of equations involved with a generalized eigenvalue problem

I want to solve the following system w.r.t $A$, $B$, $\lambda_j$ and $\mu_j$: $$H(L\otimes B)(L\otimes B)^TH^TA =HH^TA\Lambda$$ $$V^T(L\otimes A)(L\otimes A)^TVB =V^TVBM$$ $$ \|H^TAe_j\|_2^2=1 , \ j = ...
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1answer
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Finding all cyclic subspaces of $\mathbf{C}^2$ and $\mathbf{R}^2$ w.r.t to rotational linear transformation

I have been studying linear algebra and came across this question that I am not quite sure how to solve. The question begins by letting $T:\mathbf{R}^2 \rightarrow \mathbf{R}^2 $ be a rotation counter-...
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27 views

How to determine the eigenvector in the case of $\lambda=\infty$?

Let $A=\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$, $B=\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$ we want to determine the eigenpairs of the general eigenvalue problem. From the ...
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1answer
56 views

Gerschgorin theorem

One tricky question: (a) Let $A\in\mathbb R^{n\times n}$ be a diagonal matrix and let $\tilde{A}=A+E$, where $E\in\mathbb R^{n\times n}$ is such that $e_{ii}=0$ for $i=1,2,\ldots,n$, and $2||E||_{\...
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Eigen vectos and Calculation of orientation angles from symmetric 2nd order orientation tensor

I have a list of the evolution of symmetric 2nd order orientation tensor $A_{ij}$ with time. ...
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1answer
32 views

Determining the Jordan decomposition of a given matrix

$ \newcommand{\m}[1]{\left( \begin{matrix} #1 \end{matrix} \right)} \newcommand{\l}{\lambda} $ I have the matrix given as follows: $$A := \m{ -2 & -1 & 1 & 2 \\ 1 & -4 & 1 & 2 \...
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2answers
18 views

Invariance of generalized eigenspace

I have a lemma saying that each of the generalized eigenspaces of a linear operator $T$ is invariant under $T$. This means that if $E_j$ is a generalized eigenspace then $T:E_j \rightarrow E_j.$ The ...
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35 views

Looking for an intuition of the definition of Generalized Eigenspaces

The eigenspace of (a square matrix) $A$ corresponding to $\lambda$ is the collection of all vectors $\mathbf{x}$ that satisfy $A\mathbf{x}=\lambda\mathbf{x}$, or equivalently, $(A-\lambda I)\mathbf{x}=...
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27 views

Compute eigenvalues and eigenvectors using block matrices

I want to solve numericaly (using Matlab) the generalized eigenvalues-eigenvector equation: $(K-w^2M)q=0$. Where M and K have the following block structures: $$ M = \begin{bmatrix} M_1 & 0 \\ 0 &...
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1answer
76 views

Generalized eigenspace - Construction algorithm proof

Suppose that $A$ is an $n\times n$ nilpotent matrix. Let $x_1,\ldots,x_m$ be eigenvectors of $A$ forming a basis for the eigenspace of $A$. For $j=1,\ldots,m$, form a Jordan chain $$C\left(x_j\right)=\...
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80 views

linear independence of generalized vectors

I have doubts for the second part of this proof by @daw, would you help explain? Does $A$ have fixed sets of eigenvalues? If so, why can this proof add additional $\lambda_i$ arbitrarily? Why $(\...
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24 views

To find generalised eigenvector of a matrix

The given matrix is A= $\begin{pmatrix} 2 & -1 \\ 1 & 4 \end{pmatrix}$. I needed to find generalised eigenvector. First I found that 3 is it's eigenvalue with algebraic multiplicity 2 but ...
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2answers
55 views

Prove that the intersection of 2 generalised eigenspaces is the zero space

I have searched for my above question and came across the "Trivial intersection of generalised eigenspaces" post on math stack exchange but I do not understand the proof using coprime ...
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50 views

Determine the generalized eigenspace of A

I have this exercise, which I am having some problems with. It says, let $\alpha,\beta$ be scalars in a field characteristic $\neq 2$, and let $A\in$ End(V) be the linear transformation of $V=\mathcal{...
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28 views

Positive eigenvalues of generalized eigenvalue problem

It is known that when $\mathbf{A}$ is real, symmetric and positive definite, all its eigenvalues are real positive (tell me if I'm wrong, though). Looking for a similar condition for generalized ...
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1answer
22 views

A problem on general eigenvector

Suppose T$\in$$\mathcal{L}$(V),dim(V) =n.$λ_{i}$ and $λ_{k}$ are two different eigenvalues of T, $v_{i}$$\not=0$ is an general eigenvetor of T corresponding to $λ_{i}$,prove whether $(T-λ_{i}I)^{n}$$(...
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1answer
26 views

Prove that $A^N v_2 = \lambda^N v_2 + N \lambda^{N-1} v_1$ for any natural number $N$

$A$ is an $n \times n$ matrix with an eigenvalue $\lambda$ and a corresponding eigenvector, $v_1$. $v_2$ is a generalized eigenvector such that $(A- \lambda I)v_2=v_1$ I'm trying to prove $A^N v_2 = \...
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1answer
33 views

Analogue of eigenvalues for matrix of polynomials?

Let $A(x)$ be a matrix with entries that are polynomials in say $\mathbb{Z}$. Suppose furthermore that $A$ is invertible. For any fixed $x$ we can find an eigenvalue of $A$ (in $\mathbb{C}$) and this ...
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1answer
85 views

direct sum of generalized eigenspaces

I'm studying linear algebra and learning about generalized eigenspaces, and i have 3 questions regarding a specific proof which i think i have to write down before i can ask the questions (I'm ...
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1answer
80 views

Solving a repeated eigenvalue ODE

I am trying to solve the system: $$ \bar{x}' =\left(\begin{array}{rr}-8 & 4 \\ 0 & -8\end{array}\right)\bar{x}+\left(\begin{array}{rr}3e^{-8t} \\ e^{-8t} \end{array}\right),\space \bar{x}(0)=\...
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0answers
46 views

A misunderstanding or a bug in LAPACK's solver for generalized eigenvalue problems?

I have two general real matrices $A$,$B$ defined as follows, $$ A=\begin{bmatrix} -s I_3 & A_0 & 0 & 0 \\ A_0^T & -s I_3 & 0 & 0 \\ 0 & A_1 & -s ...
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0answers
33 views

Generalised Eigen Vector may not correspond to an eigen value?

Which of the following is/are true? $(1)$ It is possible for a generalised eigen vector of a linear operator $T$ to correspond to a scalar that is not an eigen value of $T$. $(2)$ Any linear operator ...
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1answer
61 views

Confusion about the description of generalized eigenspace

When learning about generalized eigenspace, there are two statements from two different textbooks which I was learning, seems contradict to the other one. In Chapter 8 of the book Linear Algebra Done ...
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26 views

Finding the nullspace.

I am still struggling to understand the relationship between the row echelon form and the basis of the nullspace. I have a questions that asks me to find the diagonalized normal form of a matrix A. $$...
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16 views

How to write the eigen spaces

I have a questions where I have to to eignevalue deocmposition to get a matrix in normal jordan form. I've found the eigenvalues, the multiplicities and calculated the eigenvectors from the ...
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2answers
61 views

Generalised Eigenvectors | Correct way to Approach

I have a matrix $$A = \begin{bmatrix}1 & 1 \\ -1 & 3\end{bmatrix}$$ I want to find out the generalised Eigenvectors. The Eigen values corresponding to the characteristic equation is $\lambda = ...
2
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2answers
72 views

Diagonalizable operator implies every generalized eigenvector is an eigenvector

I don't understand the proof below from Artin's Algebra. Can someone please explain? My specific questions are below. Suppose $T$ is diagonalizable, so the matrix $\Lambda = [T]_\mathbf{B}$ with ...
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49 views

Transforming semidefinite program into generalized eigenvalue problem (GEVP)

Suppose we have the following optimization problem on $r\in \mathbb{R}$ with constraint on positive definite matrices $A,B$ \begin{eqnarray} \text{minimize } &\; r\\ \text{subject to } &\; A \...
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Solve matrix equation $\|I-2x A + x^2 B\|=1$

Suppose $A$ and $B$ are symmetric positive definite matrices, and $x$ is a positive real number, how do I solve the following equation for $x$ and operator norm $\|\cdot\|$? $$\|I-2x A + x^2 B\|=1$$
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Generalised eigenvectors and composition of endomorphisms

Take a finite dimensional vector space $V$ over $\mathbb{C}$, say of dimension $n$. Consider an endomorphism $\phi$ of $V$ with eigenvalues $a_1,\ldots, a_r$. For each $j \in 1, \ldots, r$, let $v_j \...
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24 views

What transformation is required to find a unique solution of this problem instead of multiple solution?

\begin{align} \text{(P1):} &\ \ \ \ \ \ \ \max\limits_{\mathbf f} \log_2\left(\prod^K_{i=1} \ \ \frac{ \mathbf f^H \mathbf {\mathbf E} ( \mathbf W_i, \Theta, \tau_i) \mathbf f} { \mathbf f^H \...
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1answer
26 views

Support of a generalized eigenvector

Let $A$ be a square matrix with eigenvalue $\lambda$. Let $\mathbf{c}$ be such that $\mathbf{v} := (A - \lambda I) \mathbf{c}$ is an eigenvector, but $\mathbf{c}$ is not an eigenvector. That is, $\...
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1answer
74 views

An invertible linear map over C has a square root (Linear Algebra Done Right 8.33)

$T \in \mathcal{L}(V)$ is a complex finite-dim linear operator. 8.33 proves that if $T$ is invertible, it must have a square root. It shows that $T |_{G(\lambda_i, T)} = \lambda_i (I + \frac{N_i}{\...
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1answer
85 views

Meaning of infinite eigenvalues in generalized eigendecomposition

For a numerical problem I need the generalized eigendecomposition $$\boldsymbol{A} = \boldsymbol{B P D P}^{-1}.$$ In some cases however, the matrix $\boldsymbol{B}$ becomes singular and I get infinite ...
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1answer
22 views

Chain of kernels, generalised eigenvector

Given $A\in M_{n,n}(\mathbb{R})$ and $\lambda$ an eigenvalue, a generalized eigenvector of rank $i$ is defined as $v \in ker(A-\lambda E)^i\setminus ker(A-\lambda E)^{i-1}$. Why does such vector ...
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1answer
26 views

From generalized eigenvector to Jordan form

I can't figure out the following part of Chen's Linear Systems book. How does he "readily obtain" $Av_2=v_1+\lambda v_2$?
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1answer
81 views

Proof of $ A^k=(\lambda_A)^k V_A(V_A)^T +O(k^{m_2}|\lambda_2|^k) $, for A primitive matrix

A primitive matrix is an irreducible matrix such that it has a unique dominant eigenvalue( positive and real). Another definition which does not use irreducibility is: $ A \text{ primitive} \iff \...
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25 views

Relation between the weights of x vectors and corresponding eigenvalue of the covariance matrix of vectors?

There are ${{x}_{i}},\:i=1,\ldots ,N$ vectors with K dimensions. The covariance matrix of weighted samples is defined as below: $M=\frac{1}{N}\sum\limits_{i=1}^{N}{{{\alpha }_{i}}{{x}_{i}}x_{i}^{H}}$ ...