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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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10 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 ...
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20 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 ...
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4answers
33 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 $\mathbf{a}$ ...
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2answers
27 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 ...
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1answer
10 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 $\lambda_{1}(X^{'}X)$ ...
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1answer
18 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. ...
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7 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 ...
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0answers
21 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 ...
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0answers
13 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.
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3answers
40 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 ...
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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$
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23 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 ...
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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) = ...
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0answers
6 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 ...
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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 ...
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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 $$ ...
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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?
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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 ...
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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 ...
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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: ...
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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 ...
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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 ...
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1answer
20 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 $E$ ...
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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
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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 $Y=f(X)$).
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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 ...
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3answers
51 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.
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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 ...
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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 ...
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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 ...
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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 ...
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0answers
29 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.
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1answer
38 views
0
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1answer
32 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 ...
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2answers
107 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 ...
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1answer
41 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 ...
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1answer
47 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: ...
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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$ ...
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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
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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
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1answer
46 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 ...
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1answer
29 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
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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 ...
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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 ...
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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
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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: ...
2
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1answer
36 views

Eigenvalues of an integral operator on $L^2[-1, 1]$

Find the eigenvalues of the integral operator $K: L^2[-1, 1] \to L^2[-1, 1]$ defined by $(Kx)(t) = \int_{-1}^1 (1 - 3t \tau)x(\tau) d\tau$. I began with the fact that eigenvalues must be values ...
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2answers
38 views

Are two matrices similar iff they have the same Jordan Canonical form?

Are two matrices similar if and only if they have the same Jordan Canonical form? Does the Jordan form have to have ordered eigenvalues? For example, if $\lambda_1$ and $\lambda_2$ are eigenvalues ...
1
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1answer
26 views

Linear Algebra Characteristic Polynomials

Let $p(t) = t^n+a_{n-1}t^{n-1}+a_{n-2}t^{n-2} + \cdots + a_1t+a_0$. Show that the characteristic polynomial of the matrix A below \begin{bmatrix} 0 & 0 & \cdots & & & -a_0\\ 1 ...
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3answers
43 views

Multiplicity of an Eigenvalue and the Minimal Polynomial

Let $V$ be a finite dimensional vector space over $\mathbf C$ and $T:V\to V$ be a linear transformation. Let $p(x)=(x-\lambda_1)^{k_1}\cdots(x-\lambda_m)^{k_m}$ be the minimal polynomial of $T$. ...