Tagged Questions

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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1answer
25 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
4 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
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0answers
27 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
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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
27 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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0answers
22 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
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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 ...
1
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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 ...
1
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1answer
25 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
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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
195 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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0answers
62 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
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3answers
43 views

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

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? [on hold]

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
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1answer
29 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
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0answers
29 views

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

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 ...
1
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2answers
106 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
40 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
46 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
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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
38 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
votes
1answer
45 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
28 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 ...
0
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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 ...
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?
-4
votes
1answer
31 views

eigenvalues of square matrix [closed]

suppose A is a 2*2 square matrix so that A≠±I and A=A−1. show that eigenvalues of A are given by λ=±1? prove or disprove:if − λ1 and λ2 are eigenvalues of A and B respectively,λ1λ2 is an eigenvalue of ...
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: ...
2
votes
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 ...
1
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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
vote
1answer
25 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 ...
0
votes
3answers
42 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 $T$. ...
2
votes
1answer
49 views

Is there any graph property which is equivalent to that the spectral radius of its adjacency matrix is less then $1$?

Let $G$ be a directed graph and $A$ the corresponding adjacency matrix. I'll denote with $\rho$ the spectral radius, and with $I$ the identity matrix. What can we say about $G$ when the spectral ...
1
vote
1answer
44 views

Determining matrix of similarity

Two matrices $A$ and $B$ are similar if there is an invertible matrix $P$ such that $A=PBP^{-1}$. If the two matrices are similar, how do you find $P$?
0
votes
1answer
12 views

Confused about the dimension of a span of a set of vectors ls

The question is: What is the dimension of the following subspace of $\mathbb{R^5}$? $$span\left( \begin{pmatrix} 1 \\ 0 \\ -1 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ -1 \\ 1 \\ 0 ...
2
votes
1answer
37 views

Is the product eigenvalues less than or equal to the product of singular values?

On a numerical math course I recently saw the following statement without a proof. How would one prove it? Let $A$ be an $n$ x $n$ real matrix with singular values $s_1 \geq s_2 \geq \ldots \geq ...
1
vote
2answers
26 views

complex eigenvalues and invariant spaces

I am currently reading Guillemin and Pollack's Differential Topology, and the following claim is made without proof: Given a linear isomorphism $E: \mathbb{R}^k \to \mathbb{R}^k$, with $k>2$ and ...
1
vote
1answer
19 views

Eigenvalue of the Power of a Matrix

Let $A$ be an n×n matrix with eigenvalues $\lambda_1,\dots,\lambda_n$. Show that $\lambda_1^k,\dots,\lambda_n^k$ are the eigenvalues of $A^k$. I don't know where to start.
0
votes
1answer
48 views

Alpha and Omega limit sets (dynamical systems)

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 guess I don't really know how to get started, Could I ...
0
votes
1answer
40 views

$2X2$ matrix $A$ such that $A$ has one independent eigenvector while $A^{2}$ has two independent eigenvectors

Give an example of $2X2$ matrix $A$ such that $A$ has one independent eigenvector while $A^{2}$ has two independent eigenvectors. I would like to know a systematic answer of how to get this. My guess ...
1
vote
1answer
16 views

Maximum of a generalized Rayleigh quotient

Given two symmetric positive definite matrices $A,B\in\mathbb{R}^{n\times n}$ and $x\in\mathbb{R}^n$. How do I prove that the generalized Rayleigh quotient $R(A,B,x):=\dfrac{x\cdot A\cdot x}{x\cdot ...
1
vote
2answers
46 views

Multiplicity of an Eigenvalue of the Exponential of an Operator

I am trying to prove the following: Let $T:\mathbf R^n\to\mathbf R^n$ be a linear operator. Then $$\det e^T=e^{\text{Trace}(T)}$$ To do this I took the following apporach. Let ...
5
votes
2answers
58 views

Why $\rho(AABABB)=\rho(ABAABB)$?

Let $A,B$ be two matrices, $\rho$ be spectral radius, which is the top eigenvalue of a matrix. I discovered that $$\rho(AABABB)=\rho(ABAABB).$$ But I could not find the reason. By the way, all I had ...