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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11 views

Equating eigenvalues of Hermitian matrix and correlating symmetric/antisymmetric matricies

I have a matrix $AH$ which is created by adding $AS$ and $i*AA$, which are the symmetric and antisymmetric components of the matrix $A$ So $AS=(A+A')/2$ $AA=(A-A')/2$ $AH=AS+i*AA$ AH has ...
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1answer
24 views

Finding eigenvalue and eigenvectors of a matrix containing an imaginary number

How do you solve for the eigenvalues given the matrix? \begin{matrix} i & -2 \\ 1 & 0 \\ \end{matrix} I know how to get the characteristic polynomial Ca(X); X^2 - ...
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1answer
17 views

linear algebra characteristic values [on hold]

Let $T$ be the linear operator on $\mathbb{R}^4$ which is represented in the standard ordered basis by the matrix $$ \left( \begin{matrix} 0 & 0 & 0 & 0 \\ a & 0 ...
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13 views

Limit of the projection of a matrix when the projection is not continuous

Consider two real matrices: the $n\times n$ matrix $A$ the $n\times m$ matrix $B$ of rank $m$, with $m<n$. Let, for $a\in\mathbb{R}$, $$S_a=A-aI_n,$$ and denote by $P_a$ the orthogonal ...
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0answers
34 views

Argument for the zero vector not being defined as an eigenvector

Two days ago, my lecturer of Advanced Numerical Methods gave a review on the topic about eigenvalues and eigenvectors. Just as the lecturer presented the definition of eigenvalues and eigenvectors, a ...
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2answers
41 views

Process of finding the eigenvalues of a 3x3 matrix

I'm trying to find the eigenvalues of a 3x3 matrix in order to eventually find an orthogonal matrix $Q$ and diagonal matrix $D$ such that $Q^TAQ = D$, where $A$ is a symmetric matrix, however I'm not ...
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1answer
18 views

How to find a “flag base” to an endomorphism?

I found several exercises that ask me to find a flag base for a given matrix, for example: $$ A=\left( \begin{array}{ccc} -1 & 1 & 0 \\ 2 & 2 & 4 \\ -1 & -2 & -3 \end{array} ...
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0answers
25 views

Spectral Radius of a Block Matrix

I have real matrix $P$ obtained from numerical solution (FEM) of a physical problem, as \begin{equation} P=P_1+P_2= \begin{bmatrix} A_{2n \times 2n}&B_{2n \times n}\\C_{n \times 2n}&D_{n ...
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0answers
29 views

Is it possible to have a matrix with eigenvalues that cannot be constructed from a finite number of basic arithmetic operations, and nth roots?

For example, a characteristic polynomial $ p(\lambda) = \lambda^5 - \lambda -1 $ has the root 1.167304..., but this number cannot be written as a finite number of arithmetic operations (addition, ...
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1answer
35 views

How to find a eigenvector with a repeated eigenvalue?

The eigenvalues of my matrix are $x_1= 1$ and $x_2=3$ I get an eigenvector $V = t~[ 4~~~~~~ 3 ~~~~~1 ]^T $ but how can I diagonalize the matrix if I have the same column repeated twice. Should I ...
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1answer
55 views

Largest eigenvalues of AA' and A'A [on hold]

Prove that for every real matrix $A$, the largest eigenvalue of $A'A$ equals the largest eigenvalue of $AA'$ (where ' means transpose). Thanks!
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1answer
24 views

Eigenvectors and geometrical transformation

$$A= \begin{pmatrix} 2/3 & 2/3 & -1/3 \\ 2/3 & -1/3 & 2/3 \\ -1/3 & 2/3 & 2/3 \\ \end{pmatrix}$$, I need to understand that kind of ...
2
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2answers
38 views

Linearly independent eigenvectors

The matrix A=$\begin{pmatrix}0 & 1 &1 \\1& 0&1 \\ 1 & 1 & 0\end{pmatrix}$ has eigenvalues $\lambda=2$ with algebriac multiplicity $1$ and $\lambda=-1$ with multiplicity $2$ ...
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1answer
30 views

Eigenvector and Eigenvalues of a square matrix.

Define what is meant by saying that v is an eigenvector with associated eigenvalue λ for the square matrix A. Just a definition question that I was hoping to get help with. It's from a past exam ...
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1answer
11 views

diagonalizing a matrix with random elements

Consider the matrix $A = \begin{pmatrix} cY & 0 \\ 2 & 1\end{pmatrix}$, where $c \in \mathbb{R}$ and $Y$ is a random variable that is uniformly distributed over $[0,1]$ (That is, $Y \sim ...
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0answers
31 views

Eigenspace calculation?

Based on $\lambda$ of 4 I got the below matrix in reduced row echelon form $$\left(\begin{array}{ccc} 1 & \frac{1}{3} & -\frac{1}{3} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ ...
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3answers
33 views

Eigen values of a transpose operator

Let $T$ be linear operator on $M_{nxn}(R)$ defined by $T(A)=A^t.$ Then $\pm 1$ is the only eigen value. My try : Let $n=2,$ then $[T]_{\beta}$ = $ \begin{pmatrix} a & 0 & 0 & 0 \\ ...
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0answers
17 views

Scaling of the Dirichlet Laplacian eigenvalue [on hold]

Let Ω be a smooth domain and let $λ_1(\Omega)$ be the first (or lowest) eigenvalue of the Dirichlet Laplacian in $\Omega$. How can I show that for $\beta>0$, $λ_1(\beta\Omega)=\frac{1}{\beta^2} ...
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0answers
29 views

Relation between Eigenvectors and the commutivity between square matrices?

I'm stuck on this in a proof, any help is greatly appreciated! Thanks so much in advance! [EDIT] Okay, so basically I'm stuck on a proof, I need to proof that all eigenvectors of B are unique, ...
2
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2answers
29 views

Relation Between Eigenvalues of Block Matrices

Is there any relation between eigenvalues, or spectral radii, of $M$, $M_1$, and $M_2$ block matrices? \begin{equation} M= \begin{bmatrix} A&B\\B^T&C \end{bmatrix} \end{equation} ...
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0answers
35 views

Eigenvector and eigenvalue of an infinte, symmetrical matrix

How to get eigenvectors and eigenvalue of an infinite matrix like $$ A= \begin{pmatrix} 1&0&1&0&\dots\\ 0&1&0&1&\dots\\ 1&0&1&0&\dots\\ ...
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25 views

Eigenvalues of transition matrix of a random walk on a line

Consider the following $n\times n$ stochastic matrix describing a simple random walk on a line: $$ P=\begin{bmatrix}\frac{1}{2} & \frac{1}{2} & 0 & \cdots & 0\\ \frac{1}{3} & ...
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80 views

Proving that the algebraic multiplicity of an eigenvalue $\lambda$ is equal to $\dim(\text{null}(T-\lambda I)^{\dim V})$

Without using induction, prove that the the algebraic multiplicity of an eigenvalue $\lambda$ is $$\dim (\text{null} (T-\lambda I)^{\dim V});$$ here, the algebraic multiplicity of an eigenvalue ...
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1answer
28 views

If $A $ is a square matrix of size $n$ with complex entries such that $Tr(A^k)=0 , \forall k \ge 1$ , then is it true that $A$ is nilpotent ? [duplicate]

If $A$ is a square matrix of size $n$ with complex entries and is nilpotent , then I can show that all the eigenvalues of $A^k$ , for any $k$ , is $0$ , so $Tr(A^k)=0 , \forall k \ge 1$ . Now ...
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76 views
+50

Maximize the largest eigenvalue of a Hermitian matrix constrained by quadratic polynomials

I am looking for a method to maximize under $\mathbf{y}$ the largest eigenvalue of the following Hermitian matrix \begin{equation} S = \left [ \begin{array}{ccc} \mathbf{y}^{H}S_{11}\mathbf{y} ...
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1answer
29 views

Non Distinct Eigenvalues

If the Eigenvalues are not distinct like in this problem I have attached i.e the eigenvectors are not linearly dependent because of that? and does that change the answer? please clear my doubt
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1answer
33 views

Exist a eigenvector $v=\begin{bmatrix} x \\ y \end{bmatrix}$ such that $x,y >0$

Problem: Let $\begin{bmatrix} a &b \\ c&d \end{bmatrix}$ is a real $2 \times 2$ matrix such that $a,b,c,d>0$. Prove that exist a eigenvector $v=\begin{bmatrix} x \\ y \end{bmatrix}$ ...
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1answer
43 views

Determining diagonalizability of a matrix containing complex enteries

$$A=\left[\begin{matrix}3-8i&-11+7i\\-1-4i&-2+6i\end{matrix}\right]$$ I've determined the $tr(A) = 1-2i$, and the $det(A)=3-3i$. From here I should be able to use the characteristic equation ...
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21 views

Prove a matrix is Hermitian, if its eigenvalues are real and satisfy an orthogonality relation

Prove a matrix is Hermitian, if: (a) Its eigenvalues are real, and (b) the eigenvectors satisfy $ r_{i}^\dagger r_{j} = \delta_{ij} = \left<r_{i}|r_{j}\right> $ I can see this is the reverse ...
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18 views

Prove that minimum of the matrix norm is achieved at certain parametres

Given matrix $A\in R^{n\times m}$ prove that minimum of the $||A-xy^T||$, $||B||=tr(B^TB)$, is achieved when $x$ is an eigenvector of $AA^T$, corresponding to its greatest eigenvalue, and $y$ is an ...
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2answers
36 views

Prove a matrix is not diagonalizable

To show that a matrix is not diagonalizable, I would just have to show that there are no eigenvalues present in the matrix. So, for example, if I want to prove that $$A=\begin{bmatrix} 0 & -1 ...
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1answer
69 views

How to prove that the matrix $A^k$ approaches $0$ as $k$ approaches infinity

First of all, what does it mean to say an eigenvalue is "less than unity"? I'm not exactly sure what this means. Secondly, how do I show that $\lim_{k\to\infty} A^k=0$ given that all eigenvalues of ...
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1answer
21 views

Eigenvalue Bound of Block Matrices

I have the following eigenvalue problem for block matrices A and B \begin{equation} \left[ \begin{array}{ccc} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & ...
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2answers
48 views

Determine the coefficient of polynomial det(I + xA)

Given matrix an n-by-n matrix $A$ and its $n$ eigenvalues. How do I determine the coefficient of the term $x^2$ of the polynomial given by $q(x) = \det(I_n + xA)$
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1answer
74 views

smallest eigenvalue of rank one matrix minus diagonal

Let $x$ be a $d$-dimensional real vector with $\| x\| = 1$. Define $X := xx^T - \mathrm{diag}(xx^T)$. Is it possible to show that $\lambda_{\mathrm{min}}( X ) \geq - 1/2$? Running a bunch of random ...
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27 views

Question about representation of the eigenvalues of second order elliptic operator

let $Lu:=-\text{div}(A\cdot\nabla u)$, where $A$ is symmetric. Eigenvalues of $L$ is $\lambda_1<\lambda_2<\cdots$. By definition. (If exists a nontrivial solution $w$ such that $Lw=\lambda w$, ...
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2answers
31 views

Finding Eigenvalues and Eigenvectors for Leslie Matrix

A Leslie Matrix is given by $$L =\begin{pmatrix}0 & (3/2)a^2 & (3/2)a^3\\1/2 & 0 & 0\\ 0 & 1/3 & 0\end{pmatrix}\cdot$$ Find the Eigenvalues and determine the dominant ...
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2answers
41 views

Same roots, same polynomial? How to prove characteristic polynomial of $AB = BA$?

I'm giving a (simple) proof that the characteristic polynomial of $AB$ = characteristic polynomial of $BA$ (without using the fact that $AB$ and $BA$ are similar). $det(AB) = det(A)det(B) = ...
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1answer
18 views

Not understanding why equation 2 and equation 3 has to be multiplied with $v(x)$

The discrete differential operator $L_h$ is defined as: $(L_h v)(x_j)=-\frac{v(x_{j+1})-2v(x_j)+v(x_{j-1})}{\Delta x^2}$ (Equation 1). The contineous problem had solution $v(x_j)=sin(\beta x_j)$. It ...
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4answers
44 views

Proving properties about matrix $A$ s.t. $A^2 = -I$

$A$ is $nxn$ matrix, and $A^2=-I$. Show $A$ is non-singular and show that $n$ is even. (Further questions, show that $A$ has no real eigenvalues. Show det(A) = 1. But I'm not there yet.) So if $A$ ...
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1answer
79 views

Proving the continuation of the Cayley-Hamilton theorem from Schur's triangularization theorem

The Cayley-Hamilton theorem says that every square matrix can satisfy its own characteristic equation, $p(\lambda) = 0$, or $p(\mathbf{A}) = \mathbf{0}$. The question is to show how the ...
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0answers
14 views

The smallest value of the Laplacian of a subdivided graph

Given a graph $G$, the normalized Laplacian $\Delta$ acts on the real-valued space of functions on the vertices of $G$ by $$\Delta f(v) = f(v) - \frac{1}{m(v)}\sum_{u - v} f(u)$$ where $m(v)$ is the ...
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2answers
47 views

Finding eigenvalues of given matrix

$A = \left(\begin{array}{ccc}0 & -1 & 0 \\0 & 0 & 1 \\-1 & -3 & 3\end{array}\right)$ I want to find eigenvalues of this matrix. The answer given to me is: 1,1,1. I cannot ...
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1answer
23 views

How page rank relates to the power iteration method

I do not see how pageRank relates to the power method. Since for the pageRank we are looking for the steady stable state (vector) for a Markov (transition) matrix and the matrix has already an ...
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3answers
36 views

Why is this theorem (on eigenvalues and invertibility) important?

Theorem 5.1.5 in this book says A square matrix $A$ is invertible if and only if $\lambda=0$ is not an eigenvalue of $A$. I'd like to know if this theorem is important for practical purposes. ...
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2answers
25 views

Finding a non-singular matrix $C$ such that $C^{-1}AC$ is diagonal

$A = \left(\begin{array}{cc}1 & 0 \\1 & 3\end{array}\right)$. I find the eigenvalues = 1,3. The eigenvector corresponding to 1 = $t(1,-2)^t$. The eigenvector corresponding to 3 = $r(1,0)$. ...
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If all eigenvalues of $A$ (matrix representing linear transform) are distinct, then dimension eigenspace $E_A(\lambda) = 1$ [closed]

If all eigenvalues of $A$ (matrix representing linear transform) are distinct, then the dimension eigenspace $E_A(\lambda) = 1$. Why is the above statement true? I saw it in a proof that matrices ...
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0answers
20 views

Eigenvectors and Eigenvalues of Linear Transformation 1 [closed]

I am still having a hard time approaching this type of problem, can someone show me a step by step guide on how to do it. Much help is needed! Thank you!
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15 views

What is the glb and the lub for the spectral radius of a nonnegative matrix with row sum either 1 or 0?

What is the greatest lower bound and the least upper bound for the spectral radius of a nonnegative matrix with row sum either 1 or 0 (cannot be all zeros)? Also diagonal entries of that matrix are ...
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3answers
82 views

Prove that $\det(AA^T+I)\ge 1$

If $A$ is a matrix with real entries, prove that $$\det(AA^T+I)\ge 1.$$ I tried using the eigenvalues. One thing came into my mind: maybe $AA^T$ is positive definite (I don't know whether this is ...