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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Example of matrices with some interesting properties like same characteristic and minimal polynomial etc.

Looking for two matrices $A$ and $B$ with entries in the field $F_2$ with the following properties: $A$ and $B$ both are invertible,have same minimal polynomial,Characteristic polynomial,same ...
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Looking for proof of “ two congruent symmetric matrices with real entries have the same numbers of positive, negative, and zero eigenvalues ”

For any real square matrix $X$ let $P(X)$ denote the no. of its positive eigenvalues counting multiplicity . Let $A$ be a real symmetric $n \times n$ matrix and $B$ be a real invertible $n \times ...
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How do I find the characteristic polynomial and eigenvalues?

For the following matrix, compute its characteristic polynomial its eigenvalues $$A = \begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 2 & -5 & 4\end{bmatrix}$$ So I think I ...
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Interperate Jacobian Determinant - Stability of Equilibriums

In my SIR model, I have the following Jacobian Matrix \begin{align*} J =\begin{bmatrix} -\alpha I & -\alpha S & \zeta & 0 \\ \alpha I & \alpha S - \beta - \rho & 0 & 0 \\ 0 ...
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22 views

Zero eigenvalue of an Operator [on hold]

In my functional analysis notes, there is a claim with proof(which I seem to understand, but don't get the point) is the following. Consider the bounded linear operator L: H$ \rightarrow $ H, where H ...
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Does Orthogonal matrix have complex eigenvectors with the same absolute value? If it is true, how can I prove it?

Does Orthogonal matrix have complex eigenvectors with the same absolute value (or modulus or magnitude)? If it is true, how can I prove it?
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24 views

Matrices, Transition matrix

I have a matrix $B:= \begin{bmatrix}0 & 1\\-1 & -\lambda\end{bmatrix} $ I need to diagonalise it and work out the transition matrix. I have worked out that the eigenvalues are $ \mu_± = ...
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If $\vec{v}$ is any non-zero vector perpendicular to $\vec{u}$, show that $\vec{v}$ is an eigenvector of $S$ [on hold]

I have the following problem.. I solved the first one however i can't find out how to solve the second (b). Suppose $\vec{u}$ is a unit row-vector in $\mathbb R^n$ , and $A=uu'$ matrix. (a) ...
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18 views

Eigenvalue value problem [on hold]

Let $\Omega$ be a bounded, connected open subset of $\mathbb R^n$ $ \Delta u =\lambda u \, \text{at} \, \Omega \\ \frac{\partial u}{\partial \vec{\nu}}=0 \;\, \text{at} \;\, \partial\Omega \, ...
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Bound spectral radius of a certain matrix.

Consider a stochastic matrix $P$, i.e. real, non-negative, square, rows sum to one. Consider $\Xi = \text{diag}(\xi)$ to be a diagonal matrix with a principal left eigenvector $\xi$ of $P$ (i.e. ...
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23 views

Ways to prove that a matrix is nihilpotent/invertible

What are the ways to prove a matrix to be nilpotent/invertible? Showing that det(A) =! 0 is not possible and I can't find a way to have the polynomial in a recursive way. The dimension of A is ...
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The inverse of $(I-A)$ and the spectral radius of a nonnegative $A$ matrix

Suppost that $A$ is a nonnegative matrix, and let denote the identitiy matrix with $I$ and the spectral radius of $A$ with $\rho(A)$. Note that because $A$ is nonnegative according to the ...
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Inverse Power Iteration converges to largest eigenvalue instead of smallest?

I am trying to write a Matlab function that takes a matrix and an iteration count and performs inverse power iteration to output the smallest eigenvalue. The problem is, as k increases, the function ...
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56 views

New proof about normal matrix is diagonalizable.

I try to prove normal matrix is diagonalizable. I found that $A^*A$ is hermitian matrix. I know that hermitian matrix is diagonalizable. I can not go more. I want to prove statement use only this ...
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48 views

Find the eigenvalues of the following matrix

Consider $A =\left( \begin{array}{ccc} -1 & 2 & 2\\ 2 & 2 & -1\\ 2 & -1 & 2\\ \end{array} \right)$. Find the eigenvalues of $A$. So I know the characteristic polynomial is: ...
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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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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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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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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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63 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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41 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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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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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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35 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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24 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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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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$rank(T^n) = rank(T^m)$ 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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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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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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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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32 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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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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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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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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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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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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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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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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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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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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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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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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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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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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198 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 ...