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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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 [1], 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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Solving a systemof eqns by given initial conditions

I have the system $x'(t)=2x+8y$ $y'(t)=-x-2y$ Which has the I.C. $X(0)=(6,-2)$. So I take that it means this: $6=2x+8y$ $-2=-x-2y$ and then it is solved as any other system? We get eigenvector $(1,1/2)...
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Phase diagram and criterion for linear nullclines

I have two equations $\frac{dq}{dt}$= $Rq + K - 1$ and $\frac{dK}{dt}$=$ Nq + N$. Here, N and R are just constants so I was ignoring them and just assigning a certain arbitrary value. Solving ...
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What can we say about this eigenvalue problem?

Consider the generalised eigenvalue problem \begin{equation} \lambda \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} v= \begin{bmatrix} 1 & 2 \\ 3 & 1 \end{bmatrix}v. \tag{1} \end{...
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How to identify which eigenvectors correspond to generalized eigenvectors?

I have a specific example that I am hoping to use to learn something more general. The matrix $$A = \begin{bmatrix} 1 & 0 & 0 & 1 \\ 0 & 3 & -1 & 0 \\ -1 & 0 & 2 & ...
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Are some eigenvectors more special than others?

Question in Short. Are some eigenvectors more special than others in that they correspond to generalized eigenvectors while others do not? Specifically, if I have a matrix with an eigenvalue $\lambda =...
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Confusion in proof of theorem 8.33 in Axler's Linear Algebra done right

Suppose $V$ is a complex vector space and $T\in L(V)$ is invertible then $T$ has a square root. Let dim $V=n$. The proof goes along these lines: Let $\lambda_i'$s, ($1\le i\le m)$ be the distinct ...
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Relationship between eigenvalues of two SPD matrices and their generalized eigenvalues

I have two real, symmetric, positive definite matrices A and B. These matrices can be diagonalized by U and V respectively, such that U$^\top$AU=$\Lambda_1$ and V$^\top$AV= D$_1$. We know that these ...
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Sequence of subspaces $Ker(\lambda I-A)^k$ stationary at algebraic multiplicity?

Let $A$ be a square $n\times n$ matrix, and $\lambda$ be an eigenvalue of $A$. Then the nullspace or subspace $Ker(\lambda I-A)$ is different from $\{0\}$. And we can obtain an increasing sequence of ...
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Why is generalized eigenvalue problem the solution to multiclass Linear Discriminant Analysis?

If you have two classes, A and B. The best way to have a distance between then is to find the ...
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Obtain a eigenvector for a matrix

Recall that a matrix $A$ is diagonalizable if there is a diagonal matrix $D$ and an invertible matrix $P$ such that \begin{equation*} A = PDP^{-1}. \end{equation*} Next we study the possibility ...
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If $u$ is a generalized eigenvector of $A$, what can we say about $Au$

Not much, right? I think that if $u$ is in $N((\mu_i I - A)^j)$, so is $Au$. But beyond that, I don't see much. Are there other helpful properties?
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Solve the generalized eigen value problem by Wang's algorithm

I found a paper about how to solve the generalized eigen value problem in the easy way. But when I tried it out in GNU Octave, then did not gave me same results as ...
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A special generalized eigenvalue problem

I have to solve a generalized eigenvalue problem, $A x = \lambda B x $. Here $A,B$ are both $n\times n $ symmetric and in particular, $B$ is positive definite. I just need the smallest generalized ...
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Do time-invariant self-adjoint operators have locally orthogonal eigenfunctions?

Let $T$ be some self-adjoint, time-invariant (in that it commutes with any shift) operator on $L^2(\mathbb{R})$. Let $u$, $v$ be generalized eigenfunctions of $T$. Is it true that the product $uv$ ...
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Increase number of eigenvectors

Let $x$ be an n-dimensional vector, and $M$ be an $ n \times n$ matrix. Then, $M$ has at most $n$ unitary eigenvectors, i.e., vectors such that $Mx = x$. Now, let $\phi: \mathbb{R} \rightarrow \mathbb{...
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Linearly Independent Generalized Eigenvectors (Question about Axler's proof in his textbook)

I would like to ask a question about the proof of the Thm in the title. The proof is from "Linear Algebra Done Right" by Axler. The statement is "Let $T \in L(V)$. Suppose $\lambda _1, ....
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Spectrum of sum of two commuting self-adjoint operators with different spectrum classification

I got curious while studying quantum mechanics. Suppose we have two self-adjoint operators $A$ and $B$ and let's say they do commute as well. Let $\sigma(A)$ be a pure point spectrum of $A$ and $\...
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2 votes
1 answer
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A proof about generalized eigenvalue problem

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,...
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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 ...
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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 ...
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The Jordan Canonical Form of linear operator in two variables polynomial

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 $Tf(x,y)=f(x+1,y+1)$. ...
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What is a pseudo-eigenvector where AV=VD?

Reading through the C++ documentation of Eigen::EigenSolver, I came across "pseudo-eigenvector". Based on the description below, how is this different than the regular eigen vector? Is ...
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Non-linearly Independent Result in Calculating Generalized Eigenvectors

I am trying to find a basis for the generalized eigenvectors of the operator whose matrix is $T=\begin{bmatrix}2&1&0&4\\0&2&-1&0\\0&0&1&1\\0&0&0&3\end{...
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Finding generalized eigenvectors of a matrix

I would like to know how to find the generalized eigenvectors to the following matrix $A$, so that I can express $A$ as $PJP^{-1}$. $$ A = \begin{bmatrix} 1 & -3 & 1\\ 1 & 5 & -1\\2 &...
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Question on what maximum means in the phrase "maximum number of independent generalized $\lambda$-eigenvectors"

I was studying generalized eigenvalues, and I read the following property of the algebraic multiplicity of an eigenvalue $\lambda$: the algebraic multiplicity of $\lambda$ is the maximum number of ...
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Why eigenvector of Symmetric Matrix to be orhogonal for repeated eigenvalues. [duplicate]

I know that for a symmetric matrix, its eigenvector are orthogonal. But for a repeated eigenvalue for a symmetric matrix, why still its eigenvalue must be still orthogonal. I read somewhere that ...
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Given a canonical Jordan form respond each question

Sea $T$ a linear operator on a subspace finite dimensional $V$ such that the Jordan Canonical Form is $$\begin{pmatrix} 2 & 1 & 0 & 0 & 0 &0 & 0 &0 &0 & 0 \\ 0 &...
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Eigenvectors of the derivative operator are the exponential function

The eigenvectors of the derivative operator are the exponential function. The derivative operator is not Hermitian, so we get complex eigenvalues. $\frac{d}{dx} f(x) = \lambda f(x) \Rightarrow f(x)=Ce^...
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Eigenspace problem

I am trying to solve the following problem: Let $(\lambda,v)$ be an eigenpar of $A \in \mathbb{R}^{n \times n}$ such that the algebraic multiplicity of $\lambda$ is $2$, but with geometrical ...
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1 answer
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How do you determine generalized eigenvectors using Jordan form?

I have the following matrix $$ A= \begin{bmatrix} 0 & 0 & 0\\ 0 & i & 1\\ 0 & 0 & i \end{bmatrix} $$ I can see that it is in Jordan normal form. I read that you can find the ...
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Proving that Non-Defective Matrices with Repeated Eigenvalues Have Complete Eigenspaces

Is it always the case that for non-defective matrices the geometric multiplicities (dimensions of the eigenspaces) of the eigenvalues will equal the algebraic multiplicities ? What theorems are used ...
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To find the eigen-vectors for an infinitely differentiable function

Let $$V = \{f : \Bbb{R} \to \Bbb{R} | f \text{ infinitely many times differentiable functions} \}$$ be the vector space over $\Bbb{R}$. Find the eigenvectors of the differential operator $d/dt : V \...
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Prove or give a counterexample: $T$ is nilpotent $\iff$ $G(0,T)=V$ .

Suppose $V$ is a vector space and $T$ is a linear map on $V$. $\ G(\lambda,T) $ denotes the generalized eigenspace of $T$ corresponding to $\lambda $. In other words, $G(\lambda,T)=\{v\in V\mid {(T-\...
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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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1 vote
0 answers
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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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-1 votes
2 answers
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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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1 vote
1 answer
163 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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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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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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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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2 votes
1 answer
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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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1 vote
1 answer
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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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