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Eigenvalues are numbers associated to a linear operator from a vector space $V$ to itself: $\lambda$ is an eigenvalue of $T\colon V\to V$ if the map $x\mapsto \lambda x-Tx$ is not injective. An eigenvector corresponding to $\lambda$ is a non-trivial solution to $\lambda x - Tx = 0$.

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3
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2answers
17 views

Divergence of fixed-point iteration for real starting values

Consider the linear system of equations $Ax = b$ with invertible $A\in \mathrm{GL}(n,\mathbb R)$ and $b\in\mathbb R^n$. For $A = M - N$ with invertible $M$ the solution $x_* = A^{-1}b$ is a fixed ...
0
votes
1answer
39 views

Find Eigenvalues and Eigenvectors of A

Let $\mathbf{A}\mathbf{x}=\mathbf{a} \times \mathbf{x}$, where $\mathbf{x} $ and $\mathbf{a}$ are in R$^3$ and $\mathbf{a}$ is a fixed or constant vector. Find the eigenvalues and eigenvectors of A.
0
votes
0answers
18 views

When is “$\Re(\lambda) \gt 0$ for $\lambda \in \sigma(A),A \in \mathbb{R}^n $” true?

Let $A \in \mathbb{R}^{n \times n}$ and $\sigma(A)$ the spectrum of $A$. I am searching for a fast way to check whether $\Re(\lambda) \gt 0$ for all $\lambda \in A$. If $A = A^t$, one only has to ...
0
votes
1answer
26 views

Help to in finding the Eigenvectors for the following $2\times2$ Matrix

Please help in finding the eigenvectors for the following $2\times2$ matrix. This is very urgent, required for my examination. Your help will be greatly appreciated. Thank you. Matrix $$ A = ...
1
vote
1answer
21 views

Equation for minimum/maximum eigenvalue

It is well known that for a hermitian matrix $A$ we have $\lambda_{min}(A)=min{x\ne 0} <x,Ax>/<x,x>$, which we can see be diagonalizing $A$. Now here is my question about the following I ...
0
votes
0answers
27 views

Find the eigenvalues and eigenvectors of T in V

Let $\mathbf{V}$ be the linear span of the functions 1, cos x, sin x. Let the operator T on V be given by the rule $T y(x)= y(x+\pi/4)$. Find the eigenvalues and eigenvectors of T in V. I'm not sure ...
1
vote
4answers
41 views

Find eigenvalues and eigenvectors of the operator $A$

The question is: Find the eigenvalues and eigenvectors of the operator $A$ on $\Bbb{R}^3$ given by $A\mathbf{x}=|\mathbf{a}|^2 \mathbf{x}- (\mathbf{a} \cdot \mathbf{x}) \mathbf{a}$, where $\mathbf{a}$ ...
1
vote
2answers
30 views

$rank(T^n) = rank(T^m)$ for any positive integer $m \geq n$

Let $T$ be a linear operator on a finite dimensional space $V$. Prove that if $rank(T^n) = rank(T^{n+1})$ for some positive integer $n$, then $rank(T^n) = rank(T^m)$ for all positive integer $m \geq ...
0
votes
1answer
11 views

Eigenvalues of Sub-Matrix Formed from subset of Columns

I have an n-by-p matrix $X$ and I consider the eigenvalues of the p-by-p matrix $X^{'}X$. Let's denote the largest and smallest eigenvalues of $X^{'}X$ with the usual notation $\lambda_{1}(X^{'}X)$ ...
0
votes
1answer
22 views

Need help regarding Subspace of matrix and its basis

I need some kind of hint to get me going for this question as I'm so lost at it. Any sort of help would be appreciated. Let E be the set of all 2x2 matrices that have $v={(1,-1)}$ as an eigenvector. ...
0
votes
0answers
8 views

Auto-Thresholding PCA Eigenvalues

I'm applying a PCA on a dataset consisting of about 70k histograms with 153 bins each. So far everything is working fine except that I'm stuck on the decision which eigenvalues/vectors to throw away ...
0
votes
0answers
26 views

An analytic characterization of eigenvalues of a Hermitan matrix.

If $A$ is a Hermitian matrix with eigenvalues, eigenvectors $\{ \lambda_i , e_i \}_{i=1}^n$. (where we have $\lambda_i > \lambda_{i+1}$) for a vector $v$ let its component along the corresponding ...
0
votes
0answers
14 views

What conditions must an operator meet, to have only real eigenvalues?

Given the problem $Lu = \lambda u$, what properties must $L$ have, for all its eigenvalues to be real? An answer in the context of (partial) differential equations would be appreciated.
0
votes
3answers
41 views

Evaluating a determinant for eigenvalues

I need to evaluate $$\left| {\matrix{ {3 - \lambda } & 1 & 1 \cr 2 & {4 - \lambda } & 2 \cr 1 & 1 & {3 - \lambda } \cr } } \right|$$ A direct computation ...
0
votes
2answers
12 views

Same Eigenvector to Transformation raised by nth power

Why does $ T^nv=\lambda ^nv$ for an eigenvector $v\in V, \lambda\in \mathbb{F}$ and $T:V \to V$? would appreciate an explanation how from $ Tv=\lambda v$ we get $ T^nv=\lambda ^nv$
0
votes
0answers
24 views

Eigenvalue and Eigenvector for the linear transformation in $ \mathbb{Z}_2^4$

I'm trying to find the Eigenvalue and Eigenvector for the Linear transformation: $T:\mathbb{Z}_2^4 \to \mathbb{Z}_2^4: (x_1,x_2,x_3,x_4)=(x_1+x_3,-2x_1-x_3,x_2+x_4,x_2-x_4)$ My problem is with ...
0
votes
1answer
31 views

How to know if a linear map matrix is diagonalizable knowing about the kernel

Let $T: \mathbb{R}^4 \to \mathbb{R}^4$ be a linear map with characteristic polynomial $pt(x)$. Is $T$ diagonalizable in the following cases? $pt(x) = x^4-1$ $pt(x) = x^3(x+1)$ and $\dim \ker(T) = ...
0
votes
0answers
8 views

relation between eigen vectors of laplacian matrix and eigen vectors of the weight matrix

Let's denote laplacian matrix of a graph as $L = D-W$ where $W \in R^{n \times n}$ is the weight matrix, $D \in R^{n \times n}$ is the degree matrix such that $D_{ii} = \sum_{j=1}^n W_{ij}$ ($D$ is ...
2
votes
0answers
29 views

Eigenvalues of 5x5 matrix given equation involving matrix

I have been given the matrix $A$ and we are told it is a $5\times 5$ matrix s.t. $A^4=A^2\neq A$. I want to find the eigenvalues so I tried $A^2(A-I)(A+I)=0$ so the eigenvalues are $0, 1, -1$ but I ...
0
votes
1answer
18 views

prove characterstic polynomial of $2\times 2$ matrix is $C_{A}(x)=x^2-(\lambda_{1}+ \lambda_{2})x+\lambda_{1} \lambda_{2}$

Let $$ A=\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{bmatrix}$$ Let $\lambda_{1}, \lambda_{2}$ not necessarily distinct, be the eigenvalues of A. Show that $$ ...
0
votes
2answers
30 views

Eigenvector multiplication

I don't understand how multiplying eigenvetors by an expression like $e^{-2t}$ works, and results in this graph. Can someone explain this to me?
-1
votes
0answers
23 views

Matrix determinant, eigenvalues [on hold]

"...has the determinant $x*y*(1-ab)$. Since $L<bK<b(aL)$, $1-ab<0$ and the fixed point is a saddle. So I know that for the fixed point to be a saddle, one eigenvalue must be positive and one ...
0
votes
0answers
11 views

Alpha and Omega Limit Sets for Linear Systems [duplicate]

What are all the possible $α$- and $ω$-limit sets of points for the linear system of equations $x'=Ax$ given by the following matrices $A$: I know how to calculate the eigenvalues and the ...
1
vote
1answer
20 views

Eigenvalues of linear operator TS and ST for infinite dimensional space

Here is the original problem: Let $S$ and $T$ be linear operators on a finite-dimensional vector space $V$. Show that $TS$ and $ST$ have the same eigenvalues. I can prove it. However, my question is: ...
1
vote
2answers
35 views

Show that if $A$ is an $n \times n$ matrix that commutes with $B$

Suppose that $A$ is an $n\times n$ matrix with distinct eigenvalues. And suppose $B$ commutes with $A$. Show that $B$ is diagonable; i.e., show that $B$ is similar to a diagonal matrix. I get that ...
1
vote
1answer
26 views

What happens to other eigenvalue? - steady state heat equation

A circular plate is bounded by circles of radii $r=2$ and $r=4$, its surface is insulated and temperatures along boundaries are given by $u(2,\theta)=10\cos\theta + 6\sin\theta$ and ...
0
votes
1answer
21 views

What is my change of basis matrix?

When we are diagonal a matrix we do the following: $$P^{-1}MP$$ Where $P$ is the matrix with columns as the eigenvectors of $M$. Let us say that $M$ is representing some linear map in the basis $E$ ...
1
vote
3answers
196 views

Sum of eigenvalues of a symmetric matrix

Problem to calculate the sum of eigenvalues of a matrix: $$ \begin{pmatrix} 1 & 1 & 3 \\ 1 & 5 & 1 \\ 3 & 1 & 1 \\ \end{pmatrix}$$ I can ...
1
vote
1answer
31 views

Jordan Cell as Jordan Form implies Commuting Matrices are in Polynomial

If $X$ is a matrix such that its Jordan form is a single Jordan cell, then show that all matrices $Y$ that commute with $X$ are polynomials in $X$ (there is a polynomial $f$ such that $Y=f(X)$).
0
votes
0answers
63 views

Eigenvalues of this 3x3 matrix

I am trying to find the eigenvalues of A where A = [1 1 2][1 2 1][2 1 1]. I'm stuck after writing out the equation (1-λ)(2-λ)(1-λ)-6(1-λ) = 0. I have tried solving this in two different ways (using ...
0
votes
3answers
53 views

Unit Eigenvalue if Determinant of an Orthogonal matrix is 1 [closed]

For a (2n+1)x(2n+1) orthogonal matrix M, det(M)=1. Show M has a unit eigenvalue.
1
vote
2answers
25 views

Help to determine a basis for eigenspace

Please find a basis for the eigenspace corresponding to eigenvalue=3 for the following matrix: $$ \pmatrix{3&1&0\\0&3&1\\0&0&3} $$ [3 1 0] [0 3 1] [0 0 3] I have already ...
0
votes
0answers
16 views

Physical Significance of EigenValues and EigenFunctions? [closed]

Do please explain me the physical significance (Practical Applications) in the context of Signal Processing or any other fields about EigenValues and EigenFunctions? ThankU All..... Regards, Sanjay ...
1
vote
1answer
30 views

Eigenvalues, polynomials and minimal polynomials

I have proved (a) by: Let $\lambda$ be an eigenvalue of $AB$ $ABv=\lambda*v$ Then $BABv=\lambda*B*v$ so Bv is an eigenvector of BA with eigenvalue $\lambda$. For B, I have found the formula in ...
2
votes
2answers
34 views

Eigenvalues and eigenvectors of similar matrices.

Suppose there is a transformation $T$ and let $A$ be a matrix representation of $T$ with chosen basis. If I find out the eigenvalues of matrix $A$, these eigenvalues will be the eigenvalues of the ...
0
votes
0answers
30 views

What exactly do the eigenvalues of a Jacobian matrix mean intuitively in a dynamical system? [closed]

I read that if they are negative the system is stable but I do not understand why.
0
votes
1answer
39 views
0
votes
1answer
33 views

Sum of eigenspaces is direct sum

I know, thanks to a kind user of this forum, that the sum of the eigenspaces of an endomorphism $A:V\to V$, with $\dim(V)=n$, is a direct sum. A clear complete proof for the case where the ...
1
vote
2answers
108 views

$AB=BA$ with same eigenvector matrix

I read in G. Strang's Linear Algebra and its Applications that, if $A$ and $B$ are diagonalisable matrices of the form such that $AB=BA$, then their eigenvector matrices $S_1$ and $S_2$ (such that ...
0
votes
1answer
42 views

Hopf bifurcation and limit cycles

$$dV/dt=10(V-\frac{V^3}{3}-R+I_{input})$$ $$dR/dt=0.8(-R+1.25V+1.5)$$ Use $I_{input}$ as the relative parameter to prove that there these equations undergo 2 hopf bifurcations and indicate whether ...
-1
votes
1answer
48 views

Fixed points and stability of them

Find the fixed points and classify them for the system of equations: $$x'=v$$ $$v'=-x+wx^3$$ $$w'=-w$$ $$0=v$$ $$0=-x+wx^3$$ $$1=wx^2$$ $$0=w$$ is the only fixed point (0,0,0)?? jacobian: ...
-1
votes
0answers
38 views

Are Eigen vectors unique?

$x = [x_0 x_1 … x_N]$ and $y=[y_0 y_1 … y_N]=Hx+n$ where $n$ is a zero mean random vector and independent of $x$. $A=E(xy^{T})=E(xx^T)H^T$ and $B=E(yy^{T})=HE(xx^T)H^{T}+E(nn^{T})$ are $ N \times N$ ...
1
vote
4answers
39 views

similar matrices have the same eigenvalues [duplicate]

how do I show similar matrices have the same eigenvalues? I really have no idea, any detailed explanation would be thoroughly helpful. thank you
0
votes
0answers
19 views

Understand singular vectors and unit-phase factor

Wikipedia says "Non-degenerate singular values always have unique left- and right-singular vectors, up to multiplication by a unit-phase factor $e^{i\theta}$". I don't understad it. Can you explain it ...
2
votes
1answer
47 views

similar matrices, real eigenvalues, matrix rank,

I'm not quite sure how to tackle this problem: Consider a real nxn matrix A, where all elements are zero except those on the diagonal and those in the first row and first column. Also, assume that ...
0
votes
1answer
34 views

How to find eigenvalues and eigenfunctions of this boundary value problem?

I want to find eigenvalue and eigenfunction of this problem: $$ y''+ \lambda y=0, 0<x<l \\ y(0)=0, ly'(l)+ky(l)=0 $$ And $y'$ stands for $\frac{dy}{dx}$ and similar for $y''$. I get the ...
2
votes
1answer
28 views

Finding eigenfunctions and eigenvalues to Sturm-Liouville operator

I'm struggling to understand how to find the associated eigenfunctions and eigenvalues of a differential operator in Sturm-Liouville form. For instance, one question that I am trying to solve is the ...
0
votes
0answers
37 views

Generalized Eigenvalue Problem $(B-\lambda A)\xi = 0$

I have two matrices, $$ A = \begin{pmatrix} 2 & 0 & 3 \\ 0 & 3 & 0 \\ 3 & 0 & 6 \end{pmatrix} B = \begin{pmatrix} 2 & 0 & 3 \\ 0 & 2 & 1 \\ 3 & 1 & 5 ...
0
votes
1answer
24 views

Decomposition of a rectangular matrix

I am looking to decompose a rectangular matrix $X$ into the product of an orthogonal matrix $U$ and a diagonal matrix $S$ i.e. X=$US$. Any possible solution?
0
votes
1answer
21 views

Find matrix that implements Jordan normal form

I have a matrix $$B=\begin{pmatrix}1&2&3\\0&4&5\\0&0&6\end{pmatrix}$$ I have calculated the eigenvectors: ...