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.
196
questions
0
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0
answers
31
views
non linear eigen value [closed]
I want to find eigen values of a non linear eigen problem
$$ k1 =
\begin{matrix}
4 & 5\\
5 & 8
\end{matrix}
$$
$$ k2 =
\begin{matrix}
.004 & .005\\
.005 & .008
\end{matrix}
$$
$$ ...
0
votes
0
answers
21
views
Why are cycles of generalized eigenvectors disjoint
Let $γ_1,...,γ_p$be cycles of generalized eigenvectors of a linear operator $T$ with respect to an eigenvalue $λ$, such that the initial vectors of the cycles are linear independent. Prove that if the ...
- 2,981
3
votes
1
answer
121
views
What are the eigenvalues of a squared matrix?
Suppose matrix $A$ has eigenvalues $\lambda_1$ and $\lambda_2$.
Are the eigenvalues of $A^2$: $\lambda_1^2$ and $\lambda_2^2$?
If so, can I prove this by simple diagnolization, where $T$ is the ...
- 41
0
votes
0
answers
26
views
A nilpotent operator $T:V\to V$ has a basis which is a union of disjoint chains.
Suppose $V$ is a finite dimensional vector space and $T:V\to V$ be nilpotent.We define a linearly independent set $\mathcal S=\{x_1,x_2,...,x_k\}$ is a $T$-chain if $x_2=Tx_1,x_3=Tx_2,...,Tx_{k-1}=x_k,...
- 2,957
1
vote
0
answers
31
views
Prove existance of orthonormal basis of $\mathbb{R}^3$ consisting of eigenvectors of generalized eigenvalue equation
Context
I am studying normal modes oscillations and normal modes [1,2]. In an earlier post [3], I asked for a proof that the generalized eigenvalue equation in normal-mode analysis has positive ...
- 913
0
votes
0
answers
23
views
How to recover matrix from generalized eigenvectors?
I am working on a graph problem and I am interested in whether it is possible to recover the the graph Laplacian or the adjacency matrix from its generalized eigenvectors.
More specifically, consider $...
- 372
0
votes
0
answers
12
views
Linear System O.D.E.: 1<dim(Eigenspace)<eigenvalue multiplicity
Let’s be:
the linear system $\boldsymbol{x}’=A\boldsymbol{x}$ where $A\in M_{n\times n}(\mathbb{R})$ is constant and
an eigenvalue ($\lambda$) with multiplicity $m>1$.
What happens if the ...
- 211
0
votes
0
answers
20
views
Possible or not? Jordan chain does not span the full generalised eigenspace
Given a matrix $A$ with eigenvalue $\lambda$ and a corresponding eigenvector $v_1$:
Is it possible that the Jordan chain $v_1$ generates does not span the full generalised eigenspace of $A$ w.r.t $\...
- 141
1
vote
1
answer
28
views
Is $N_T \subseteq Im(T^l)$?
If $T\in \mathcal L(V)$ is a nilpotent linear map (ie. $T^k=0_V$ for some $k \in \mathbb{N}_{\ne 0}$), let $N_T$ be the nullspace of $T$, $Im(T^l)$ be the image of $V$ under $T^l$.
Question: Is $N_T \...
- 141
2
votes
1
answer
62
views
An element both in $V_\mu$ and in $V_\overline{\mu}$ is zero?
I was doing an exercise with following notations:
$V$ is a $\mathbb{R}$-vectorspace
$\alpha \in$Hom$_\mathbb{R} (V,V)$ and $\alpha_\mathbb{C}\in$Hom$_\mathbb{C} (V_\mathbb{C},V_\mathbb{C})$ its ...
- 291
0
votes
1
answer
40
views
How to solve a non-linear eigenvalue problem based on experimental data
I have a matrix $A(\operatorname{Im}(\lambda))$ whos terms $a_{ij}$ depends on the imaginary part of $\lambda$ with a relationship that is experimentally assessed (to give a bit more context, I'm ...
- 183
1
vote
2
answers
72
views
Minimal polynomial from Jordan canonical form.
Suppose $T:V\to V$ is a linear operator on $V$ which is finite dimensional and suppose the JCF of $T$ is,
$\begin{pmatrix}c & \\1 &c\\ & &c\\ & & &d\\ & & & 1&...
- 2,957
0
votes
1
answer
45
views
Name for this eigenvector/eigenvalue problem
Let $A \in \mathbb{C}^{n \times m}$ and $B \in \mathbb{C}^{m \times n}$ be matrices. Consider the following problem:
Find non-zero vectors $u \in \mathbb{C}^m$ and $v \in \mathbb{C}^n$ such that $Au =...
- 1,001
1
vote
1
answer
58
views
Solve 2nd-order ODE with $n$ variables: Mass-spring-damper overdamped
Introduction:
I have a mass-spring-damper system with $q$ degrees of freedom. We model it by using a 2nd-order vectorial ODE \eqref{1} on variable $\left\{X\right\}$:
$$
\left[M\right] \cdot \left\{\...
- 1,136
2
votes
2
answers
52
views
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{...
0
votes
1
answer
29
views
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
3
votes
1
answer
74
views
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 ...
- 2,437
0
votes
1
answer
112
views
Rewrite generalized eigenvalue problem as standard eigenvalue problem
I have two matrix $\mathbf{A}$ and $\mathbf{B}$ and I want to find the values of $\lambda$ such that
$$
\mathbf{A} \cdot \mathbf{v} = \lambda \cdot \mathbf{B} \cdot \mathbf{v}
$$
$\mathbf{A}$ and $\...
- 1,136
2
votes
2
answers
95
views
Maximization of $tr[U^HAU(U^HBU)^{-1}]$
I am looking for the solution of
\begin{align*}
\max_{{U}\in\mathbb{C}^{M\times P},\ {U}^H{U}={I}_P } &tr\left[ {U}^H{A}{U}\left({U}^H{B}{U}\right)^{-1}\right],
\end{align*}
where $P\leq M$, ${A} ...
- 101
1
vote
1
answer
53
views
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 ...
- 11
1
vote
0
answers
35
views
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 ...
- 624
0
votes
0
answers
18
views
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. ...
- 101
1
vote
0
answers
27
views
What gurantees the existence of generalized eigenvectors of rank m?
Let $A$ be a $n\times n$ matrix. If $\lambda$ is an eigenvalue of A with multiplicity $m>1$. Then there could be two cases:
There are $m$ linearly independent eigenvectors of A corresponding to $\...
- 11
0
votes
0
answers
27
views
What is the connection between the adjoint and left/right generalized eigenproblems?
I would like to understand the connection between the adjoint eigenproblem and the left/right eigenproblems
A generalized eigenproblem $Ax = \lambda Bx$ can be re-arranged to be $(A-\lambda B)x = 0$, ...
- 43
1
vote
0
answers
48
views
Proving the relationship between left and right eigenvalues for a complex generalized eigenproblem
A generalized right eigenproblem is defined as:
$A q = \lambda B q$
where $A$ and $B$ are complex matrices in general and $q$ represents the right eigenvector, which is a column vector, and $\lambda$ ...
- 43
0
votes
0
answers
30
views
Generalised Eigenfunctions
Given $\mathbf{J}$, the Jacobian matrix of a function $\mathbf{F}:\mathbb{R}^n\to\mathbb{R}^n$, the system
$$
\mathbf{J}\mathbf{v}=\lambda \mathbf{v}
$$
has solutions corresponding to eigen-pairs $(\...
- 2,998
5
votes
0
answers
57
views
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 ...
0
votes
1
answer
146
views
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 ...
- 1,123
0
votes
0
answers
32
views
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?
2
votes
1
answer
71
views
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 ...
- 1,327
1
vote
0
answers
33
views
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 $(H)$-...
- 279
3
votes
2
answers
71
views
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 ...
0
votes
0
answers
44
views
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 ...
- 31
2
votes
0
answers
97
views
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 ...
- 249
2
votes
1
answer
107
views
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 ...
- 1,300
1
vote
2
answers
56
views
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)...
- 2,145
0
votes
0
answers
37
views
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 ...
3
votes
1
answer
79
views
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{...
- 505
1
vote
1
answer
44
views
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 & ...
- 2,297
2
votes
1
answer
128
views
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 ...
- 10.1k
1
vote
2
answers
47
views
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 ...
- 805
1
vote
0
answers
51
views
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 ...
- 869
0
votes
0
answers
50
views
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 ...
- 1,072
1
vote
1
answer
36
views
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{...
- 167
1
vote
1
answer
58
views
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, ....
- 1,240
0
votes
0
answers
139
views
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 $\...
- 17
2
votes
1
answer
98
views
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,...
- 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 ...
- 71
6
votes
2
answers
126
views
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 $Tf(x,y)=f(x+1,y+1)$. ...
- 65