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Questions tagged [generalized-eigenvector]

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Eigenvectors and eigenvalues relational proofs

Let A ∈ R^ n×n How do I prove that "If A has a finite number of distinct eigenvectors then each eigenvector must have a distinct eigenvalue." If A a symmetric matrix in R n×n . A is called positive ...
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Steps to finding a generalised eigenspace, given a linear transformation.

I'm currently taking a linear algebra course and we are reading Sheldon Axler's "Linear Algebra Done Right". On page 254, he gives an example: Suppose $T\in {\cal L}(V)$ is defined by $T(z_1, z_2, ...
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How do I compute the eigenvectors and general eigenvector of this matrix for $\lambda = 3$?

Let $$A =\begin{pmatrix} 177& 548& 271& -548& -356\\ 19& 63& 14& -79& -23\\ 8& 24& 17& -20& -20\\ 42& 132& 55& -141& -76\\ 56& 176&...
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Can there be just one eigenvalue on the invariant subspace (generalized eigenspace) associated with an eigenvalue?

In the proof of Jordan decomposition here, once I know that an indecomposable subspace $V$ is of the form $V=Ker((f-\lambda Id)^n)$, can there be an other eigenvalue $\mu$ for $f\vert_V$?
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Intuition of generalized eigenvector.

I was trying to get an intuitive grasp about what the the generalized eigenvector intuitively is. I read this nice answer, so I understand that in the basis given by the generalized eigenvectors, a ...
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1answer
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Exponential of a Matrix- Repeated AND Complex Eigenvalues

I am seeking a general solution to the initial value problem x' = Ax, x(0) = x_0 that can be written out to include both the eigenvalues and eigenvectors. To cover the case of repeated eigenvalues, ...
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If $u$ is a generalized eigenvector, then $e^{tA}u = e^{t\lambda}e^{t(A-\lambda I})u$

I have come across this statement. If $\mathbf{u}$ is a generalized eigenvector, so that $(\mathbf{A}-\lambda \mathbf{I})^m \mathbf{u} = 0,$ then $$e^{t\mathbf{A}}\mathbf{u} = e^{t\lambda} e^{t(\...
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Can I find eigenvalues for $A$ if I know the eigenvalues from $AA^T$

Let's say we have a real matrix $A$ and I find the eigenvalues for $\sqrt{AA^T}$. Is it possible for me then to find the eigenvalues for $A$ without using $$det(A-\lambda I)=0$$ ?
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Terminology for different types of eigenvalue degeneracy?

Let $T$ be a linear operator on a finite-dimensional complex vector space $V$, and let $\lambda$ be an eigenvalue of $T$ with multiplicity $m$ (defined as the dimension of the subspace of $V$ spanned ...
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Proving that the set of generalized eigenvectors spans the space

The question is formatted as follows. Firstly I am given the following theorem: If T is a linear operator, and A is its matrix representation on $\mathbb{C}^n$, and the minimum polynomial is $m_T(...
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Trace in a generalized eigenspace

I am reading a paper and there is a steep in a proof that I cannot understand. The proof says: Suppose $v_\lambda$ is a generalized eigenvector corresponding to an eigenvalue $\lambda$ of the map $T$...
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2answers
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Finding the Jordan Form of a matrix…

I know that this type of question has been asked on here before but I am still having a hard understanding what is going on. The text that I am learning from is "Linear Algebra Done Right by Sheldon ...
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Decomposition of invariant subspaces

Consider a set of $d$ linearly independent generalized eigenvectors of some matrix $A \in \mathbb{C}^{d \times d}$. Suppose this set is decomposed into $M$ distinct Jordan chains and the $m$-th Jordan ...
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1answer
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What is the simplified form of the generalized eigen space when the characteristic polynomial does not split in the given field.

Let $V$ be a finite dimensional vector space over a field $\mathbb{F}.$ Let $T$ be a inear operator on $V$ and $\lambda \in \mathbb{F}$ be an eigenvalue of $T$ of algebraic multiplicity $m.$ Now ...
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Are they similar matrix

Do $\begin{bmatrix} 0&i&0\\0&0&1\\0&0&0 \end{bmatrix} $ and $\begin{bmatrix} 0&0&0\\-i&0&0\\0&1&0 \end{bmatrix} $ are similar.Is this True/false ...
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Show that a is the length of the biggest Jordan chains corresponding to an eigenvalue.

Let V be a fi nite-dimensional vector space, T belong to L(V) and T has an eigenvalue x with the corresponding space of generalized eigenvectors Ux. Let a = min{k : (T-x)^k|Ux = 0}. Show that a is the ...
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Finding the general solution of an eigenspace corresponding to a system of equations

I'm having trouble understanding how to obtain the set of all vectors of an eigenspace represented in parametric form. For example if we have the system $$\begin{bmatrix}0 & 0 \\ -8 & 4\end{...
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1answer
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Find the general solution $\left[\begin{smallmatrix}x \\ y \end{smallmatrix}\right]$

Let $\begin{bmatrix} \dot x \\ \dot y \end{bmatrix}$$=$$\begin{bmatrix}-5 & -3 \\ 3 & 1 \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}$. a) Find the general solution $\begin{bmatrix}x \\ y \...
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1answer
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How to find Jordan Basis and Jordan Form

I need to find the Jordan Normal form $J$ and a matrix $S$ such that $J = S^{-1} AS$. The matrix is $$ M = \left( \begin{matrix} 1 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 ...
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1answer
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Let $f$ and $g$ be two linear operators, show that eigenvalues of $g$ are of the form $a, a-1, a-2, \ldots, a - (n - 1)$?

Let $f$ and $g$ be two linear operators with $\dim(L) = n$, where $L$ is a vector space over a field $F$ with characteristic zero. Assume that $f^{n} = 0$, $\dim \ker(f) = 1$, and $gf-fg = f$. How it ...
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1answer
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Let $T:V\to V$ be a diagonalizable linear map on a vector space $V$. Does it hold that $T^{**}:V^{**}\to V^{**}$ is diagonalizable?

Edit: I found my own answer. But I will be happy to see other answers as well. Question: Let $T:V\to V$ be a linear operator on a vector space $V$. (a) Suppose that $T$ is diagonalizable. Does ...
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Generalized Eigenspaces Associated to Different Values

Let $V$ be some vector space, and $T \in \mathcal{L}(V)$. If $a \neq b$, then $G(a,T) \cap G(b,T) = \{0\}$, where $G(b,T)$ denotes the generalized eigenspace. I am having a lot of trouble with this ...
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Dual formulations of Generalized Eigenvectors

I've seen two definitions of generalized eigenvectors. One is $(A−λ⋅I)^kv=0$ and one where it's $Av = \lambda B v$. I get that they are the same thing but I'm curious what is the point of both ...
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1answer
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Finding a Jordan Basis after finding the Jordan Canonical Form

The question asked to find the Jordan Canonical Form and Jordan Basis of $\begin{bmatrix}1 & 1 & 0 & -1\\0 & 1 & 0 & 1\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\...
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1answer
101 views

Does maximizing the Rayleigh quotient using the method of Lagrange multipliers require the matrix to be positive semidefinite?

I’m faced with the problem of maximizing a Rayleigh quotient: $$\max_h \,\, \frac{h^t H h}{||h||^2}$$ Which is equivalent to solving $$\max_h \,\, h^t H h $$ $$ s.t. ||h||^2=k >0, k \in \...
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1answer
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A Lie algebra element in a generalized eigenspace is ad-nilpotent

I was reading this book and on page 76 the author defines $\mathcal{N}(\mathbf{g})$ to be the set of all elements of $\mathbf{g}$ that are in the generalized eigenspace of some other element's adjoint,...
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1answer
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Proving there are as many generalized eigenvectors as algebraic multiplicity eigenvalue without the Jordan canonical form

I'm reading some treatment of generalized eigenvectors in a differential equations book. They want to derive that there are as many generalized eigenvectors to a certain eigenvalue $\lambda_i$, as the ...
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Generalized eigenvector in a differential equation system

This is the system: $$\begin{cases} \dot{x}=x+2y+e^{-t}\\ \dot{y}=2x+y+1 \end{cases}$$ Now I first solve the homogeneous one, without the vector $(e^{-t},1)$, so I have to find the eigenvalues of ...
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121 views

Finding generalized eigenvectors from a Jordan form

I am trying to understand the relation between Jordan form, characteristic polynomial and minimal characteristic polynomial. From: Problem5 Consider a matrix A, assume that A has characteristic ...
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1answer
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Can one describe the algebraic multiplicity in terms of generalized eigenspaces and the minimal polynomial?

We defined the algebraic multiplicity of a matrix $A$ with eigenvalue $\lambda$ to be the largest integer $r$ such that $(x-\lambda)^r$ divides the characteristic polynomial of $A$. I would like to ...
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1answer
88 views

Optimizing singular Rayleigh quotient subject to linear constraint

I want to numerically solve $$\min_x \frac{x^TAx}{x^TBx} \quad \mathrm{s.t.}\quad Cx=0,$$ where $A$ and $B$ are large sparse matrices, $A$ is positive semi-definite, $B$ is positive-definite, and $C$ ...
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2answers
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Killing form formula

I am having some trouble with the following proof of a formula for the killing form. Let $\mathfrak{g}$ be a semisimple Lie algebra and $\mathfrak{h}$ a Cartan subalgebra. I know that if $\mathfrak{...
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How to use random projections to find matrices A,B s.t. AX=BY

(This is for my graduate research -- not exactly homework) Given an arbitrary $X \in \Re^{M_X \times N}, Y \in \Re^{M_Y \times N}$, I want to find $A \in \Re^{K \times M_X}, B \in \Re^{K \times M_Y}$...
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1answer
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Constructing matrices with the given eigenvalue and eigenspace

There are several similar questions: (a) If possible, write down a 5 × 5 real matrix with −1 as its only real eigenvalue and where the eigenspace with eigenvalue −1 has dimension 3. (b) If possible, ...
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Generalized eigenvectors from left and right Schur vectors

I am reading up on the generalized Schur decomposition as a means to solve the generalized eigenvalue problem $A\nu = \lambda B \nu $, With $A$ and $B$ matrices, $\lambda$ the eigenvalues and $\nu$ ...
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Second-order matrix equations

Let $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times p}$, $C\in\mathbb{R}^{q\times n}$ and $D\in\mathbb{R}^{q\times p}$ be known matrices. Denote by $I_n$ the identity matrix of size $n\times n$ ...
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Find a constant to bound laplacian norm by gradient norm in finite dimension

I need to prove (P). For prove that, the hint is to use (D). My question is, how to use (D) to prove (P)? Let $T$ a triangle or tetrahedron and $\mathbb{P}_k(T)$ the set of polynomials of degree less ...
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1answer
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Generalized Eigenvectors when algebraic multiplicity greater than 1

Find the Generalized Eigenvectors of $$ \begin{bmatrix} 1 & 0 & 0 & 0 & 0\\ 1 & -1 & 0 & 0 & -1\\ 1 & -1 & 0 & 0 & -1\\ 0 & 0 & 0 & 0 & ...
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1answer
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Eigenspace equals domain

In "Linear algebra done right", the author proves by induction that given a vector space $V$, and a linear operator $T$ on that vector space, $V$ is equal to the direct sum of the generalized ...
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1answer
119 views

Making matrix upper triangular by finding Jordan Normal Form

I must make the following matrix upper triangular: \begin{bmatrix} 2 & 1 & 0 & 0 \\ 0 & 2 & 0 & 1 \\ 1 & 0 & 2 & 0 \\ 0 & 0 & 0 & ...
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Multiplying an eigenvalue equation by a non-invertible matrix: what eigenvalue characteristics are retained?

Suppose I have an eigenvalue equation $$M v=\lambda v$$ and I have characterized the eigenvalues. Maybe $M$ is Hermitian and $\lambda$ is real, for example. Given a non-invertible matrix $P$ (I'm ...
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1answer
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question regarding generalized eigenvectors for Differential equation

I also have a question about the generalized eigenvector, I learned how to find the generalized vector in class, but for this practice problem, I am not sure how to find the generalized eignvectors ...
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Generalized Eigenvectors for Systems of ODEs

If I'm solving a linear system $\vec{x}^{\,'}=A\vec{x}$ with constant square matrix $A$ and if $A$ has an eigenvalue $\lambda$ with multiplicity $3$ but there's only 2 eigenvectors $\vec{v_1},\vec{v_2}...
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When will there exist a basis of eigenvectors?

From what I have found, "there exists a basis of eigenvectors if and only if the algebraic and multiplicity of each eigenvalue is the same." However, I cannot relate the multiplicity of eigenvalues to ...
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Is $\text{dim}\, G(T_\mathbb C ,\lambda)=\text{dim}\, G(T,\lambda)$ when $\lambda$ is real?

Is $\text{dim}\, G(T_\mathbb C ,\lambda)=\text{dim}\, G(T,\lambda)$ when $\lambda$ is real? I am studying the theorem 9.23(c) of the S.Axler's Linear Algebra Done Right (3rd Edition) on page 284 of ...
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333 views

Generalized eigenvectors for Jordan canonical form (and theory)

I am learnnig how to find a Jordan normal form of a matrix, and I got stuck on four items. Can you please explain to me and check if I am right with my understanding? (I do not have a real example on ...
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1answer
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Can a polynomial with degree (r+s-1) have (r+s) distinct roots?

While reading about generalised eigenvectors, i came to a strange proof. It goes like If p is a polynomial which has r distinct roots {λ1,λ2...,λr}, and q is a polynomial with s distinct roots {μ1,μ2....
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How to find generalized Eigen vectors of a matrix with Eigen vectors already on diagonal?

I have a matrix A= \begin{bmatrix} 1 & -1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & -1\\ 0 & 0 & 1 & 1 \end{bmatrix} This matrix has ...