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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Significance of an eigenvector being equal to a unit vector?

I was reading ahead in my math book when I came across a matrix denoted as A = $\begin{bmatrix} 1 & -1 & 0\\ 2 & -2 & 0\\ 6 & 0 & -2\\ \end{bmatrix}$. I then found the ...
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eigenvalues of homogeneous integral equation of second kind, with singular kernel

There is a homogeneous integral equation of second kind with a singular kernel(non-symmetric). The equation has the form: $\int_{a}^{b} k(x,t)Γ(t)dt =λΓ(x).$ It's 2-norm is infinity, $||k(x,t)||_2 ...
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How to find the rules by which eig(A) = diagonal entries

Suppose A = [a b c;d e f;g h i] how to find a series of rules such that eig(A) = a, e, i such as h = 0 or f = 0? Is there any rule for this? (I know we can find this by principal submatrix, but I ...
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17 views

Eigenvalue of Adjacency matrix

If I have a 2D-Graph which is invariant under traslations in the x direction but not over the y one. And I want to calculate its eignevalue and its eigenvector. What can I do? If it was an ...
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30 views

What does my teacher mean by 'choosing' from a vector?

I'm revising some lecture notes from a class I missed, I'm just struggling to figure out what she means at this point. What is choosing x1=0, x2=1... etc mean? Could someone explain?
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2 views

Can centrality of given network decreased on removal of higest central node

Actually I am calculating the Eigenvector Centrality of whole graph with each node centrality also.Then I remove the highest central node(high eigenvector centrality) in graph by making respective ...
6
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2answers
57 views

A quick way to estimate eigenvector/eigenvalue of a matrix

Is there a quick way to give a raw estimation of an eigenvector/eigenvalue of a matrix? By "quick" I mean some method which can be computed without a computer or paper and pencil...something you could ...
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3answers
22 views

How to get An eigenvalue and eigenvectors of a matrix that contain both zero column and zero row?

Could anyone help in how to get the eigenvalue and eigenvectors of a matrix that contain both zero column and zero row like : \begin{pmatrix} -1 & 1 & 0\\ 1 & -1 & 0\\ 0 ...
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0answers
14 views

The lower bound of Cheeger Inequality as the degree goes to infinity

Consider an undirected graph $G(V,E)$ with adjacency matrix $A$ and define the graph Laplacian as \begin{equation} L = D - A \end{equation} where $D$ is a diagonal matrix such that $D(i,i) = d_i$. ...
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3answers
30 views

For two p.d. matrices $A$ and $B$, prove that $\lambda_1(AB)\leqslant \lambda_1(A) \cdot\lambda_1(B)$

If $A$ and $B$ are two nxn positive definite matrices, then show that $$\lambda_1(AB) \leqslant \lambda_1(A) \cdot \lambda_1(B),$$ where $\lambda_1(\cdot)$ denotes the largest eigenvalue.
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Let A : $R^2$ to $ R^2$ be a linear transformation with eigenvalues 2/3 and 9/5

Let A : $R^2$ to $ R^2$ be a linear transformation with eigenvalues 2/3 and 9/5 . Then, there exists a non-zero vector $v$ in $R^2$ such that (a) $||Av||$ > 2$||v||$; (b) $||Av||$ < 1/2$||v||$; ...
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1answer
15 views

Are eigenvalues of A all with positive real parts if and only if $x^TAx>0$?

Are eigenvalues of $A$ all with positive real parts if and only if $x^TAx>0$ for any $x$? $A$ is non symmetric. If this is true, if $B=-B^T$, then if the eigenvalue of $A$ are with positive real ...
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1answer
53 views

eigenvectors of a matrix

Good Day, I have a matrix of \begin{bmatrix} 28 & 10\\ 10 & 19 \end{bmatrix} I have found the eigenvalues... first eigenvalue (v1) : 24 + 5sqrt5 = rd off to 35.18 second eigenvalue (v2) ...
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1answer
25 views

On the eigenvalues / properties of a specific matrix.

I'm not sure how to better phrase the title of the question, because I don't know the specific name of the matrix I am after, but I want to consider matrices of the form $$ \begin{align*} ...
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0answers
17 views

A question about eigenvalues of a special block matrix

Thanks for anyone who views or answers this question! $N\times N$ matrix $L$ can be partitioned into $p\times p$ blocks, ...
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2answers
53 views

Showing that matrix admits an eigenvector?

Let A= a b c d be a 2 x 2 matrix, where a,b,c and d are real numbers. We say that A admits an eigenvector if there exists a unit vector u and a real ...
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21 views

Why do we want that the determinant of the coefficients is $0$?

Eigenvalue problem with periodic boundary conditions-complete Fourier series $$y''+\lambda y=0, 0 \leq x \leq L$$ $$(*): \begin{cases} y(0)=y(L)\\[4pt] y'(0)=y'(L) \end{cases}$$ $$$$ It's a ...
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2answers
52 views

When do eigenvectors converge?

Let $A_n$ be a sequence of self-adjoint $N\times N$ matrices that converge in the operator norm to $A$. The sequence of eigenvalues of $A_n$, denoted $\lambda_n$, converges to an eigenvalue of $A$, ...
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10 views

Hilbert-Schmidt theorem

In the Hilbert-Schmidt theorem what it means : $A e_n=\lambda_n e_n$ ? Thank you .
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1answer
25 views

General eigenspace & general eigenvector

Consider the following: 1.Suppose $k_i$ is an eigenvalue of $A$ with algebraic multiplicity $n_i$ 2.dim $V_i = m_i$ (geometric multiplicity), $V_i$ is the eigenspace corresponding to $k_i$. 3. $m_i ...
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1answer
29 views

How to get an eigenvector of a $3\times 3$ matrix that has first column and a row of zeros

I have the following matrix $$ \begin{bmatrix} 1& 0& 0\\ 0& 1& 1\\ 0& 1& 1 \end{bmatrix} $$ First I got the eigenvalues which are $0$, $1$, $2$. I tried to get the ...
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0answers
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Stieltjes transformation of e.d.f. sample eigenvalues

If the eigen-decomposition of sample covariance matrix is $S=PDP'$ where $D$ is a diagonal matrix with eigen value of $S$ and $P$ are eigenvectors. If we define the empirical distribution function of ...
2
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1answer
46 views

How does negating a matrix affect its eigenvalues?

I'm working on the following problem: "If $Ax = \lambda x$, find an eigenvalue and an eigenvector of $e^{At}$ and also of $-e^{-At}$." So far, I have figured that $e^{\lambda t}$ will be an ...
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Assume that the square matrix A has an eigenvalue of 0. Is A invertible? Why or why not?

Just wanted some input to see if my proof is satisfactory or if it needs some cleaning up. Here is what I have. Proof:Suppose $A$ is square and invertible and for the sake of contradiction let $0$ ...
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0answers
14 views

Finding characteristic roots and characteristic vectors

V is a two-dimensional vector space over the field of real numbers, with a basis $v_1, v_2$. Find the characteristic roots and corresponding characteristic vectors for T defined by $v_1(T) = v_1 + ...
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1answer
25 views

Any transformation in $A(V)$ satisfies a polynomial of degree $2$ over $\mathbb F$. [on hold]

$V$ is a two-dimensional vector space over a field $\mathbb F$; prove that every element in $A(V)$, the ring of linear transformations on $V$, satisfies a polynomial of degree $2$ over $\mathbb F$.
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1answer
15 views

How to prove that the spectral radius of a linear operator is the infimum over all subordinate norms of the corresponding norm of the operator.

I am trying to understand a proof I have seen of the following theorem: $$\rho(A)=\inf_{\|\cdot\|}\|A\|.$$ I understand that to do this, the idea is to show that 1) $\rho(A)\leq\|A\|$ for any norm, ...
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1answer
26 views

Eigenvalue of (1-0) matrix

Assume I have 2 matrices, each of size nxn with only 1 and 0 as entries in both. (n>10) The first matrix (call it A) has each row summing up to 2 (ie: on each row, it has two "1" and n-2 "0"). It is ...
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0answers
44 views

Question about an eigenvalue problem

I have a question... How can I show that the eigenvalue problem $$y''+λy=0$$ $$y(0)=0,$$ $$ y'(0)=\frac{y'(1)}{2}$$ is NOT a Sturm-Liouville problem?
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What are eigenvalues of higher order finite differences matrices?

I know (at first empirically, then read somewhere) that for for second order finite differences matrices like $$\begin{pmatrix} -2&1&0&0&0\\ 1&-2&1&0&0\\ ...
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1answer
87 views

Second Derivative of Eigenvalue

Consider the potential $V_t=tx$ and the corresponding Schrödinger operator $H_t=- \frac{\partial ^2}{\partial x^2}+V_t$ on $L^2([0,R])$ with Neumann or Dirichlet boundary conditions. Let ...
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eigen values and vectors in this question

I have the following matrix to be solved for landa. here is the matrix with my answer (sorry for the wrong equation writing format): but the correct equation from this matrix should be (when ...
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2answers
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Optimization of Product of Different Objective functions (Ex.: Maximize The Product of projections of a complex vector)

Suppose We have this optimization problem which is convex $\mathbf{x}={\arg}\: \underset{\mathbf{x}}\max f_{i}\left (\mathbf{x} \right )$ But the product of different objective function is ...
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1answer
44 views

Find an $n\times n$ integer matrix with determinant $1$ and $n$ distinct positive eigenvalues

I feel pretty stupid for doing this, but here goes anyway. Earlier today I asked: Find an $n\times n$ integer matrix with determinant 1 and $n$ distinct eigenvalues. As it turns out, for my problem I ...
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0answers
23 views

3-Species Population Model

I am trying to solve a 3-species predator-prey system in matlab. Here is the equation: $$\frac{d}{dt} \begin{bmatrix} N_1 \\ N_2 \\ N_3 \\ \end{bmatrix} = \begin{bmatrix} N_1 & 0 & 0 \\ 0 ...
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Find an $n\times n$ integer matrix with determinant 1 and $n$ distinct eigenvalues

Pretty much what the title suggests: for any positive integer $n$, I'm looking for an $n$-by-$n$ matrix with integer entries, determinant $1$ and $n$ eigenvalues. In case it is absolutely useless to ...
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1answer
12 views

Calculating eigenvectors where there is only 1 non zero number in matrix

So I am attempting to find the equilibrium points of a nonlinear system and I am getting the following jacobian matrix: $$ \begin{pmatrix} 1 & 0 \\ 0 & 1.5 \\ ...
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1answer
44 views

Is there a significance to a matrix having an eigenvector equal to a column vector within a matrix?

Consider matrix $A$: $\begin{bmatrix} 2 & -2\\ 3 & -3\\ \end{bmatrix}$ After little computing, we find the eigenvectors (and their corresponding eigenvalues) to be equal to $E_{\lambda=0}= ...
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1answer
16 views

How would you compute eigenvectors from this linear system?

I am stuck on a problem and I do not know how to obtain the eigenvectors: $\frac{dY}{dt}=\bigl(\begin{smallmatrix} -2&0\\ -3&1 \end{smallmatrix} \bigr)Y$ Work: I obtained the eigenvalues ...
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4answers
186 views

If all eigenvalues are 1 or -1, is then $A^{12}=I$?

True or false: If all the eigenvalues of A are either $\lambda=1$ or $\lambda = -1$ then $A^{12}$= I If we have a matrix $$\mathbf A = \begin{pmatrix}1&0\\0&-1\end{pmatrix}$$ this has ...
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4answers
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Every n × n-matrix A with real entries has at least one real eigenvalue. [duplicate]

I have a true/false question: Every n × n-matrix A with real entries has at least one real eigenvalue. I am thinking that this is true but I would like to hear ...
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Apply MDS when all eigenvalues are negative

I want to map the vectors from n-dimensional to two-dimensional space. I am using MDS ( Multidimensional scaling) method as described in the paper: An Introduction to MDS. On 10 page it is written ...
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1answer
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Complex Eigenvalue Periodic

This is probably a simple question, but I'm having trouble finding a clear answer. Let's say we have a system, with the complex eigenvalue: $$\lambda = \alpha + i\beta$$ I know that $\alpha < 0$ ...
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Low rank matrix square root

I'm trying to perform canonical correlation analysis (CCA) between matrices $X$ ($n \times p$) and $Y$ ($n \times k$), with covariance matrices $S_{X}=XX^T/(n-1)$ and $S_{Y}=YY^T/(n-1)$ respectively, ...
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Characteristic vector components expressed as polynomial in $A$ and $\lambda$

If $\lambda$ is a simple root of $A$, a characteristic vector $x$ associated with $\lambda$ can always be taken to be a vector whose components are polynomials in $\lambda$ and the elements of $A$. ...
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2answers
34 views

Finding real, distinct eigenvalues for arbitrary constants

Let $A= \begin{bmatrix} 1 & 1 & 0 \\ -4 & -3 & 1 \\ k & 0 & 0 \end{bmatrix}$. Find all values of $k$ such that $A$ has three real distinct eigenvalues. I have obtained the ...
4
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1answer
25 views

Eigenvectors of sums of matrices

Suppose we know all eigenvalues and eigenvectors of the hermitian matrices $A$ and $B$, what does this say about the eigenvectors of $A+\varepsilon B$ for small $\varepsilon$?
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1answer
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Proof of a theorem connecting Gerschgorin circles and eigenvalues

How do I prove that, given $A\in\mathbb C^{n\times n}$, if $A$ is irreducible and $\lambda$ is an eigenvalue of $A$ such that $\lambda\in\partial\left(\displaystyle\bigcup_{i=1}^nK_i\right)$, where ...
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1answer
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Common eigenvector of a sequence of compact operators

Let $H$ be a separable, infinite-dimensional Hilbert space and suppose we have a sequence of norm-one compact operators $(A_n)$ on $H$ which all have 1 as an eigenvalue. Can we pass to a subsequence ...
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1answer
37 views

Problem determining eigenvalues of a Hermitian matrix

Suppose that you've got an $n \times n$ irreducible matrix $A$ with strictly positive real entries and eigenvalues $\lambda_i$, $i=1,...,m$, arranged so that $|\lambda_1| > \cdots > ...