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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1answer
18 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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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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0answers
17 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: $M=\frac{1}{N}\sum\limits_{i=1}^{N}{{{\alpha }_{i}}{{x}_{i}}x_{i}^{H}}$ ...
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19 views

Question involving eigenspaces and generalized eigenspaces

I need to solve the question given above; however, I am unsure of how to proceed exactly; am I required to use the invertibility of $U$ to somehow show the two equalities? Any help would be ...
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1answer
41 views

How to determine the Basis transformation with first finding eigenvector?

How to determine the Basis transformation and all the explicit lambdas with the matrix (let say $A$) below? \begin{pmatrix}1&a&0\\ a&1&0\\ 0&0&b\end{pmatrix} I know that we ...
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1answer
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Prove that $T$ has at most two distinct eigenvalues [duplicate]

I need help with a problem in Axler's Linear Algebra text. Any hint would be great. Let $V$ be a vector space over $\mathbb{C}$ of dimension $n$, and $T: V\to V.$ If $$\dim \ker(T^{n-2}) \neq \dim \...
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1answer
26 views

Are the subspaces corresponding to Jordan blocks unique?

Let $T$ be a linear operator on a complex vector space $V$, where $n<\infty$, and let $A_1,\dots,A_m$ be the Jordan blocks of the matrix of $T$ with respect to some Jordan basis. For each $A_i$ (of ...
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7 views

Random symmetrical matrices for generalized eigenvalue problem - How to solve

Assume that we have $A$ and $B$ and they both are real symmetrical random matrices, e.g they have both negative and positive real values. Now I want to solve $$AV = BVD$$ where $V$ are real ...
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1answer
29 views

Distinct generalized eigenspaces have trivial intersection

Let $V$ be a finite dimensional vector space and $T \in \mathrm{End}(V)$. Let $\alpha,\beta$ also be two distint eigenvalues for $T$ and $V_\alpha, V_\beta$ are the generalized eigenspaces relative to ...
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1answer
38 views

Caclulating generalized eigenvectors of matrices

I have the linear transformation $T \in \mathcal{L}(\mathbb{C}^{5})$ defined by $T(x_1,x_2,x_3,x_4,x_5)=(2x_1,x_2+x_4,2x_3,x_2+x_4,-x_1+x_3+2x_5)$ The matrix with respect to the standard basis is $...
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26 views

How do I find complex eigenvectors if I know the complex eigenvalues?

Assume that we know the complex eigenvalues $\lambda_i$. Is there a way for us to find the complex eigenvectors by finding the nullspace? $$(A-\lambda_i I)W_i = 0$$ Where $W_i$ is the complex ...
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1answer
33 views

Complicated Defective Eigenvalues

The usual, simple defective eigenvalue problem has algebraic multiplicity $2$ or $3$, and geometric multiplicity $1$. Then a complete set of generalized eigenvectors is obtained by using the lone ...
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15 views

Finding the Jordan Canonical form using known Ranks for a generalized eigenvector to certain powers.

Can someone please expand on this idea from wikipedia https://en.wikipedia.org/wiki/Jordan_normal_form, under Generalized Eigenvector - Uniqueness. Particularly, how you can use the rank of $(A - \...
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2answers
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EigenVector, Eigenvalues, System of ODEs, and Solution Verifaction

$\newcommand{\align}[1]{\begin{align} #1 \end{align}} \newcommand{\diffx}{\frac{dx}{dt}} \newcommand{\diffy}{\frac{dy}{dt}}\newcommand{\equation}[1]{\begin{equation} #1 \end{equation}}$ I am given ...
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0answers
15 views

Generalized eigenvectors product

Let's consider a real square matrix $A$ with eigenvalues $\lambda_n$ and eigenvectors $\mathbb x_n$, i.e. $A \mathbb x_n = \lambda_n \mathbb x_n$. Suppose there are some generalized eigenvectors $\...
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1answer
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what to do with a system with a repeated zero eigenvalue, where you cannot impose a condition on one of the components of the generalised eigenvector?

I have the non-linear system $$ \frac{dx}{dt}=ax-bxy \\ \frac{dy}{dt}=cx $$ This has infinitely many steady states of form $x^*=(0,y)$, but my problem is with steady state $x*=(0.\frac{a}{b})$ I ...
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85 views

The problem of Generalized Eigenvalues, How to solve $AV=V\Lambda$?

The problem is how to solve $$AV=V\Lambda, \quad\quad (1)$$ where $A$ is an $n\times n$ matrix, $V$, the "generalized eigen-vector-matrix", is $n\times q$, and, $\Lambda$, the "generalized eigen-value-...
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1answer
37 views

Product of an invertible diagonal matrix and a diagonalizable matrix is diagonalizable?

I encountered one problem. Suppose $\textbf{A}$ is a diagonal invertible matrix and $\textbf{B}$ is a diagonalizable matrix of same size. Is the product matrix $\textbf{AB}$ is diagonalizable? Here ...
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10 views

Symmetric polynomial eigenproblem and algebraic/geometric multiplicity

In a numerical application I encounter the polynomial eigenproblem $$\Phi(s)x=0$$ where $$\Phi(s)=\sum_{i=0}^{k} A_i s^i$$ and $A_i$ are real symmetric $n\times n$ matrices, $x$ is an $n$-vector ...
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3answers
42 views

Find the jordan canonical form of the following matrix

A = $$ \begin{matrix} 2 & 1 & 1 & 1 \\ 0 & 2 & 0 & 1 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & 0 & 2 \\ \end{matrix} $$ I calculated the ...
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0answers
28 views

Confusion on commutativity of matrices/operators in a given proof: $(T-\mu I)(T-\lambda I)^{p-1}(x)$ = $(T-\lambda I)^{p-1}(T-\mu I)(x)$ = $0$

The proof is given in a linear algebra textbook for the proposition: Let T be a linear operator on a vector space $V$, and let $λ$ be an eigenvalue of $T$. Then for any scalar $μ \ne λ$, the ...
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1answer
24 views

orthonormal vectors in generalized eigenvalue problem

Consider the generalized eigenvalue problem $$A\textbf{v}=\lambda B \textbf{v},$$ where $A$ is assumed to be symmetric, nonsingular with distinct eigenvalues and $B$ symmetric and positive definite ...
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46 views

Trying to solve generalized eigenvalue problem Av = λBv for multiple linear regression

I'm trying to understand some mathematics for solving a multiple linear regression problem to predict multiple target values, $Y \in \mathbb R^{n\times p}$, where $n$ is the number of observations and ...
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0answers
21 views

Neumann-Laplacian Eigenvalue Problem Reformulation

Let $\Omega\in R^2$ with boundary $\tau$, and $n$ be the normal to the boundary $\{-\Delta v(x,y)=\lambda v(x,y) \quad \mbox{if }(x,y)\in\Omega, \quad n\cdot\nabla v(x,y)=0 \quad \mbox{if }(x,y)\in\...
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0answers
12 views

Common eigenvalues for two Sturm-Liouville problems

Does exist in literature any results concerning the common eigenvalues for the two eigenvalue problems of the form $$y''(x)=\lambda^2 y(x)+\lambda a(x) y(x), \ x\in(0,1), $$$$z''(x)=\lambda^2 z(x)-\...
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0answers
34 views

Generalized Eigenvalue Problem - How to solve? Fisherfaces

I'm trying to do picture recognition. There are 3 types of methods from OpenCV library. Eigenfaces, Fisherfaces and Local Binary Pattern Histogram. These are good, but in practice, Fisherfaces is the ...
3
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1answer
47 views

Transition matrix for Jordan canonical form

I have this matrix: $$A = \begin{bmatrix} 3 & 1 & 2 \\ 0 & 3 & -1 \\ 0 & 0 & 3 \end{bmatrix}$$ The characteristic polynomial is $(3-\lambda)^3,$ so the eigenvalue is $\...
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1answer
57 views

Solve ODE system with generalized eigenvectors

So I’m trying to solve the system $x’=Ax$ with the initial conditon $x(0)=v_4$ with $A$ a 4x4 matrix with constant coeficients. And I am given the following properties; $$Av_1=2v_1\,;Av_2=-3v_2\,; ...
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27 views

Solving biased eigendecomposition problem, $\mathbf{AV_B}=\mathbf{V_B\Lambda_B} + \mathbf{Y_B}$

Given a matrix $\mathbf{A}\in\mathbb{R}^{N\times N}$ with $\mathbf{\Lambda_B}\in\mathbb{R}^{B\times B}$ as the diagonal matrix containing $B\ (B\ll N)$ eigenvalues of matrix $\mathbf{A}$. Is there any ...
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1answer
80 views

Algebraic multiplicty of an eigenvalue greater than its depth

In the book, Linear Algebra Done Wrong , Sergey Treil says that one can notice that the algebraic multiplicity of an eigenvalue $\lambda$ is greater than its depth (minimum $k$ such that $(A-\lambda ...
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45 views

SVD for Diagonalizing Differential Equation Systems

It is said that only a linear system of differential equations with all real and complete eigenvalues can be decoupled by applying a spectral decomposition to its coefficient matrix. Since the ...
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1answer
28 views

Proving $\overline{\ker(A-\lambda {\rm Id})^{r}} = \ker(A-\overline{\lambda}{\rm Id})^{r}.$

Is there a simple proof of $\overline{\ker(A-\lambda {\rm Id})^{r}} = \ker(A-\overline{\lambda}{\rm Id})^{r}$, with $A \in M(n,\mathbb{R})$? I think induction might work, but if there are other ...
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1answer
34 views

Confused between two methods of calculating the generalized eigenvectors.

For some reason, this question, in particular, has been causing me some problems. I'm revising some notions about eigenvalues, and I'm confused between two methods of calculating the generalized ...
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0answers
25 views

Finding the Jordan basis of a $4 \times 4$ matrix [duplicate]

I have to find the Jordan basis of the following matrix: $$A=\begin{pmatrix} 2&0&-1&1\\ 1&1&-1&1\\ 1&-3&2&-1\\ 0&-3&2&-1\\ \end{pmatrix} $$ I ...
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0answers
38 views

The sequence $\{\ker(f-\lambda Id)^{k} - \ker(f-\lambda Id)^{k-1} \}_{k}$ is decreasing

Given $V$ a vector space over a field $\mathbb{K}$ and a string $[d_{1} < d_{2} < \cdots < d_{k} = n]$, where $d_{i} = \dim \ker(f-\lambda Id)^{i}$. I know that $$\{0\} \subset \ker(f-\...
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1answer
21 views

If exists an Eigenspace $V_{\lambda}$ with dim($V_{\lambda}) \geq 2$, the $f$-invariant subspaces are infinite

Let $V$ a $\mathbb{K}$-vectorial space,with char($\mathbb{K}$) = 0. Is it true that if exists an Eigenspace $V_{\lambda} \subset V$, with dim($V_{\lambda})\geq 2$, the $f-$invariant subspaces are ...
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1answer
130 views

Is every invariant subspace equal to some $\text{null}(T-\lambda I)^n$?

This is a natural follow-up to this question. Let $V$ be a finite-dimensional complex vector space and let $U$ be a subspace of $V$ invariant under the linear operator $T$: $$\forall u\in U: Tu \in ...
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1answer
117 views

Is every invariant subspace also a generalized eigenspace?

Let $V$ be a finite-dimensional complex vector space and let $U$ be a subspace of $V$ invariant under the linear operator $T$: $$\forall u\in U: Tu \in U.$$ Must $U$ also be the subspace associated ...
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0answers
20 views

Optimize sum of the square of hermitian forms

For M different hermitian forms $h_i(\vec{c}_0,\vec{c}_1)$, where $\vec{c}_j$ are complex vectors in $\mathbb{C}^N$, I want to calculate $$\min_{\vec{c} \in \mathbb{C}^N\backslash\{\vec{0}\}} \frac{\...
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Is this notion of an eigenvector for an $r$-tuple of matrices known?

Given $r$ complex matrices $A_1,\ldots,A_r$, each of size $m$-by-$n$, we say that a nonzero $x \in \mathbb{C}^n$ is a generalized eigenvector of $(A_1,\ldots,A_r)$ if $$A_1x \wedge \cdots \wedge A_rx ...
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0answers
25 views

Reducing trace minimization to generalized eigenvalue problem

I'm looking for insight in solving the following optimization over symmetric matrices A and positive-definite H. $$R=\max_{A}\frac{\text{tr}(HA)^2+2\text{tr}(HAHA)}{\text{tr}(AHA)}$$ The paper ...
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1answer
42 views

How did we find the generalized eigenvectors corresponding to the eigenvalue 2 here pg.305 in Golan.

The example is given below: How did we find the generalized eigenvectors corresponding to the eigenvalue 2 here? I watched this video https://www.youtube.com/watch?v=msFp3vOYIoA and I followed ...
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0answers
28 views

Derivative of Schur (QZ) decomposition

I am looking for a closed form solution of the derivative of the Schur (QZ) decomposition: $A=QSZ^*$ and $B=QTZ^*$ w.r.t. elements $A_{ij}$ and $B_{ij}$ of matrices $A$ and $B$ (complex, non-...
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0answers
59 views

Self-consistent non-linear eigenproblem

Given two Hermitian matrices $\mathbf{A}$ and $\mathbf{B}$ of dimension $M \times M$, search for a set of $N \leq M$ orthonormal vectors $\mathbf{v}_n$ for which $$\left( a_n \mathbf{A}+\mathbf{B} \...
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1answer
29 views

Generalized eigenvector question (ODE system)

This is an example in Boyce-Diprima. I have the following system$$x'=\begin{bmatrix} 1 & -1 \\ 1 & 3 \end{bmatrix}x$$ I solve for the eigenvalues, which is just $\lambda=2$ in this case, and ...
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1answer
44 views

Generalized eigenspace calculation

Let $\alpha, \beta$ be scalars in a field $F$ of characteristic $\neq 2$, and let $A \in End(V)$ be the linear transformation of $V=F^3$ which is represented by \begin{equation} [A] = \begin{...
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0answers
34 views

Generalized Schur Decompostion - Reordering Unitary Matrices

My problem concens two matrices of the following form$$V=\begin{pmatrix}\,0&I\,\\\,A&B\,\end{pmatrix}\quad\text{and}\quad W=\begin{pmatrix}\,I&0\,\\\,0&C\,\end{pmatrix}$$ Here, $A,B$ ...
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1answer
66 views

Generalized eigenvectors of AB and BA proof

Let $A \in R^{m\times k}, B\in R^{k \times m}$ be non-square matrices. A vector $v \in R^m$ is said to be a generalized eigenvector of $AB$ corresponding to $\lambda \neq 0$ if $(AB- \lambda I)^m v =...
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0answers
28 views

Eigenvalue-like problem for block matrices — numerical solutions

Consider the following matrix equation $$ \left[\begin{array}{cc}{A_{11}} & {A_{12}} \\ {A_{21}} & {A_{22}}\end{array}\right]\left[\begin{array}{c}{v_1} \\ {v_2} \end{array}\right] = \left[\...
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0answers
44 views

How to get “true” (orthonormal) eigenvectors when I got generalized eigenvectors by solution Hv=eSv

I solved equation $H[v] = eS[v]$ using some numerical package (e,V = scipy.lingalg.eig(H,S)) There $[v]$ is matrix of all generalized eigenvectors $v_i$, $H$ is ...