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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Linear independence of Eigenvectors - repeated eigenvalue

Suppose an n$\times$n matrix has n-1 distinct eigenvalues, $\{\lambda_1,\lambda_2,..,\lambda_{n-1}\}$. The eigenvalue $\lambda_{n-1}$ has algebraic multiplicity 2 and geometric multiplicity 1.The ...
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51 views

Prove/Disprove: $v$ is an eigenvector of $T^n$ implies $v$ is an eigenvector of $T$

Prove/Disprove: $v$ is an eigenvector of $T^n$ implies $v$ is an eigenvector of $T$ I'm pretty sure it's not necessarily true, but can't think of a counter example. Can you help me think of ...
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1answer
11 views

principal eigenvectors of an unknown matrix

Do you have any idea about how we can find the principle eigenvectors of an unknown matrix ${H}$. The only information that we have is that $H$ has only a few (up to 3) dominant eigen modes regardless ...
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19 views

Eigenvalues and Eigenvectors of symmetric matrix (using Householder)

I have used the Householder method to reduce the symmetric matrix, A, to a tridiagonal form, T, but I'm not sure of the next step to take to calculate the eigenvalues and eigenvectors of the matrix T ...
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1answer
39 views

Proof that an involutory matrix has eigenvalues 1,-1

I'm trying to prove that an involutory matrix (a matrix where $A=A^{-1}$) has only eigenvalues $\pm 1$. I've been able to prove that $det(A) = \pm 1$, but that only shows that the product of the ...
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1answer
40 views

Can -3 and 2 be eigenvalues of the following matrix?

Can $-3$ and $2$ be eigenvalues of and nxn matrix B such that $A = B^{2}+B-6I$ and A's determinant is $0$? So this is what I concluded: At first glance, it can be seen that the matrix $A$ can be ...
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35 views

Eigenvalues of a matrix with special form

Let $p,a_1,...,a_n\in(0,1)$ and $\sum_{i=1}^na_i=1$. Now consider the following matrix: $$ \left(\begin{array}{ccccc} (1-p) & \sqrt{p(1-p)}a_1 & \sqrt{p(1-p)}a_2 & ... & ...
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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 ...
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1answer
41 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.
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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 ...
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1answer
31 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 = ...
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1answer
23 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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30 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
44 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
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 ...
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1answer
12 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
23 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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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 ...
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31 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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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.
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3answers
43 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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25 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
32 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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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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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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32 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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Matrix determinant, eigenvalues [closed]

"...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
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 $E$ ...
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3answers
197 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 ...
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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
59 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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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
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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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1answer
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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 ...
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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 ...
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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 ...
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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: ...
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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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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 ...