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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Simple Eigenvalue finding question (by gauss elimination)

I saw a method for finding eigenvalues by using Gauss elimination to find an upper triangular matrix, then just taking the diagonal elements as the eigenvalues. It seems to work except for this case: ...
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27 views

Finding similar matricies

I'm trying to find a matrix N similar to the scalar matrix M = $ \begin{pmatrix} a & 0 \\ 0 & a \\ \end{pmatrix} $ Such that $M = ANA^{-1}$. I have no idea ...
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185 views

Eigenvalues and roots of unity

Let $A \in \mathcal{M}_{n}(\mathbb{C})$ such that $A^{n} = \mathrm{I}_{n}$ and the family $(\mathrm{I}_{n},\ldots,A^{n-1})$ is linearly independent. I would like to prove that $\mathrm{Tr}(A) = 0$. ...
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Self-adjoint operator and eigenbasis

Let us assume that we have a self-adjoint operator $A: D(A) \subset L^2 \rightarrow L^2$ and we know that $A$ has a purely discrete spectrum and the eigenvalues of $A$ are simple. Does that mean that ...
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1answer
50 views

Prove that $\det(A) > 0$

Let $A \in \mathcal{M}_{n}(\mathbb{R})$ be a real $n \times n$ matrix such that : $A^{3} = A + \mathrm{I}_{n}$. Prove that $\det(A) > 0$. Here is what I tried : $X^{3}-X-1$ is a null polynomial ...
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25 views

Know eigenvalues, get $Q$ of $A=QLQ'$

$A=\begin{bmatrix} 1 & -2 & 2\\ -2 & -2 & 4\\ 2 & 4 & -2 \end{bmatrix}$ I have calculated that the eigenvalues $\lambda=2,2,-7$. When $\lambda=2$, the eigenvector is ...
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10 views

Find the eigenvector for an operator on a linear span

Let $V$ be the linear span of the functions $1,cos(x),sin(x)$. Let the operator $T$ on $V$ be given by the rule $Ty(x)=y(x+ \pi/4)$. Find the eigenvalues and eigenvectors of T in V. I know how to ...
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1answer
17 views

Drawing toral automorphisms

How do we set about drawing a toral automorphism as in figure 5.1 in the picture above. How do we know where the points highlighted in yellow are? What happens if the eigenvalue (Im guessing some ...
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25 views

Why does Givens rotation avoid iteration and Jacobi rotation doesn't in case of reducing a symmetric matrix to tridiagonal?

I am currently implementing symmetric matrix reduction to tridiagonal. I read that Givens rotation avoids iteration when it is used for reducing a matrix to tridiagonal whereas Jacobi rotation is ...
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24 views

Why is QR algorithm using plane rotation followed by givens rotation better than just plane rotations?

To find eigenvectors from a tridiagonal matrix, it[ref:Numerical Recipes] says that QR algorithm using plane rotation followed by givens rotation(QR algorithm with implicit shifts) better than just ...
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21 views

What do they mean by corresponds to the same eigenvector x in this question?

I'm getting confused by the wording of this question. What do they mean by corresponds to the same eigenvector x? Question: Suppose $\lambda$ and $\ell$ correspond to the same eigenvector x? Show ...
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Weighted undirected graphs, complex Laplacian, complex eigenvalues & spectral clusering

I am rather puzzled and confused, I have been trying to get a clear understanding of how would spectral clustering work for an undirected weighted graph, I have used the normalized Laplacian, but I ...
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7 views

Parametric dependence of the maximum eigenvalue of a positive matrix

One of my friends asked me this question: If $A$ is a positive matrix i.e. $a_{ij}>0\forall\ i,j$, and if $a_{i,j}$ are all $C^1$ functions of a parameter $\theta\in [0,\infty)$, then how ...
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12 views

Efficient way to compute the strong convexity modulus of a function?

I have a strongly convex function $f:X\to\mathbb{R}$, where $X\subseteq \mathbb{R}^n$, with strong convexity parameter $\sigma>0$. By definition $f$ satisfies, for all $x,y\in X$ and $t\in[0,1]$, ...
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1answer
22 views

eigen value is a 'continuous function' of matrices

I have a doubt in linear algebra basically about polynomials. If a sequence of real matrices $A_n$ converges to a matrix $A$, does it imply that in $\mathbb{C}^n$, the spectrum vectors $\sigma_n$ ...
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29 views

Eigenvalues with constraints?

Note: This is a short version of About diagonalizing a matrix for a quadratic expression (with the goal of uncoupling mixed terms) For a $n$-dimensional symmetric matrix A, orthogonal matrix C exists ...
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1answer
23 views

Linear Algebra - Give an example for $3x3$ matrix for these eigenvalues

I'm having trouble with this problem : Give an example for matrix $A$ with these eigenvalues $\lambda_1-1,\lambda_2=1,\lambda_3=0$ while : $$v_1=(0,1,1)$$ $$v_2=(1,-1,1)$$ $$v_3=(0,1,-1)$$ ...
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17 views

About diagonalizing a matrix for a quadratic expression (with the goal of uncoupling mixed terms)

my question is originated from a physical problem. I will try to present the problem as simple as possible, but I fear it will still be long since I'm bad at expressing myself briefly. It starts with ...
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1answer
27 views

Get normalised eigenvectors

I am given the matrix: $\begin{pmatrix} a & b \\ b & -a \end{pmatrix}$ and I already calculated the eigenvalues $\lambda = \pm \sqrt{a^2+b^2}$. Now, I want to get the normalised ...
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3answers
62 views

Roots of a cubic equation with coefficients based on unknown values $a$, $b$ and $c$.

I want to find the eigenvalues of the following matrix: $$ \left( \begin{array}{ccc} 0 & a & b \\ a & 0 & c \\ b & c & 0 \end{array} \right) $$ So, I found the characteristic ...
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Why must every vector in V belongs to one of the generalised eigenspaces of $T: V \to V?$

Why must every vector in V belongs to one of the generalised eigenspaces of $T: V \to V?$ Is there a simple proof for this? Can someone provide me with an intuition behind it? Note that V is an ...
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47 views

Prove that a matrix with a given characteristic polynomial is diagonalizable

Matrix $A$ is defined over real number. Characteristic polynomial : $p(x)=(x+3)^2(x-1)(x-5)$ It also known that : $$\text{rank}(A+2I)+\text{rank}(A+3I)+\text{rank}(A-5I)=9$$ prove $A$ ...
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Linear Algebra - Prove trival solution eigenvalue

A is an $2\times2$ matrix with $\operatorname{trace}=1$, and $\det A=-6$. Prove that $(2A+5I)x=0$ has only trival solution. I need to show that $(-A-\frac{5}{2}I)x=0$ Therefore I need to show that ...
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11 views

Multigrid eigensolver: properties of the Laplacian at different levels of the hierarchy

I'm not entirely sure this is the right place, but I could really use some help. I'm attempting to implement a hierarchical eigensolver specific to graph Laplacians $L_0$, but after one iteration, the ...
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17 views

median eigenvalue

When I look at the spectral density plots of my (usual) laplacian graphs, they spike at the median eigenvalue. But what significance for the graph/matrix (which originates from a network) does the ...
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58 views

What is the quickest way to find the characteristic polynomial of this matrix?

Let $e_k$ be the $k$-th vector of the canonical base of $\mathbb R^n$ and let $$B = [e_2 \mid e_3 \mid \dots \mid e_n \mid e_1]$$ What it the quickest way to show that the charachteristic polynomial ...
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1answer
47 views

Eigenspace conceptual question

Is it true that a vector space V is a direct sum of all its eigenspace? What happens if T is not diagonalisable? Does this only apply to a vector space over an algebraically closed field? Similar to ...
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15 views

Eigenvalues of a Self-Adjoint Operator

It is easy to see that eigenvectors corresponding to distinct eigenvalues of a self-adjoint operator $T$ are mutually orthogonal. However, from this, it is supposed to be easy to see that for a given ...
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1answer
47 views

Eigenvalues of tridiagonal matrix

on page 13 of the paper here there is a proof in theorem 4 that all eigenvalues of this tridiagonal matrix, which has strictly positive entries down the subdiagonals, are simple. Unfortunately, I ...
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Why can we solve eigenvalue problems which are non-convex by Lagrange multiplier methods and get global minima?

while reading the paper "Some Modified Matrix Eigenvalue Problem" by Golub this doubt occurred to me. there he writes that we can minimize $x^TAx$ subject to $x^TBx=1, Cx=0$ As far as I understand ...
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Diagonalization of Hermitian matrix

I would like to perform diagonalization of a Hermitian matrix $A$ and I know the steps but at the end I am not getting diagonal matrix with eigenvalues on the main diagonal, can anyone help me why? ...
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Two Matrices with Negative Eigenvalues of Each Other?

I have two matrices, $A$ and $B$. I was (perhaps naively) expecting them to be more-or-less similar ("more-or-less" because this is in a numerical setting), but instead of having exactly the same ...
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Why is the number of Jordan blocks with eigenvalue $\lambda$ and size $\geq l =$ dim (Ker $f_\lambda$ $\cap$ Im $f^{l-1}_\lambda$)?

Why is the number of Jordan blocks with eigenvalue $\lambda$ and size $\geq l =$ dim (Ker $f_\lambda$ $\cap$ Im $f^{l-1}_\lambda$)? Note that $f_\lambda$ = $f - \lambda I$. This is stated in my ...
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How to prove inverse of a linear operator is diagonalizable using concept of eigenspaces?

Let T be an invertible linear operator on a finite dimensional vector space V. Given for any eigenvalue $\alpha$ of T, $\alpha$^(-1) is an eigenvalue of T^(-1). I first proved that the eigenspace of ...
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11 views

Principal Components vs Principal Directions

I'm trying to do statistical downscaling of some climate data and there is a module of principal component analysis by regression method required. I am confused with the different terms here. What is ...
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40 views

List of eigenvalues for the Schrödinger equation

I'm writing an algorithm which computes the eigenvalues $E$ of the Schrödinger equation with potential $V(x) = x^2$, ie the harmonic oscillator. The equation is defined as follows $$ y''(x) = ...
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1answer
27 views

Eigenvalues of linear transformations

Determine the eigenvalues of each linear transformation. Give brief explanations. (Hint: you do not need to find a matrix representing the linear transformation.) (a) $\mathcal P : \Bbb ...
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12 views

Eigenvalues of Hankel matrices

Let $\mathbf{A}$ be a $4-$ dimensional symmetric matrix with real entries, whose elements are given as \begin{equation} \mathbf{A} = \left( \begin{array}{cccc} a & b & c & d \\ b & c ...
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2answers
32 views

Eigenvalues of 3D rotation matrix

I'm having some trouble calculating the eigenvalues for this rotation matrix, I know that you subtract a $\lambda$ from each diagonal term and take the determinant and solve the equation for $\lambda$ ...
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38 views

Eigenvectors of $\left( \begin{array}{ccc} 0 & -b \\ a & 0 \end{array} \right)$

This is similar to my previous question in that I when I form a system of simultaneous equations and solve them all the terms cancel and I don't get any information on the eigenvectors. The matrix in ...
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3answers
65 views

Eigenvectors of $\left( \begin{array}{ccc} a & 0 \\ 0 & -b \end{array} \right)$

I calculated the eigenvalues of the following matrix to be $a$ and $-b$. $J = \left( \begin{array}{ccc} a & 0 \\ 0 & -b \end{array} \right)$ But when I use the formula $(J - \lambda I)v = 0$ ...
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4answers
49 views

Linear Algebra - Prove Isomorphism.

Let $T : \Bbb R^n \rightarrow \Bbb R^n$ Linear transformation. Prove that there is a real number $\alpha$ that the transformation $\alpha I-T$ is isomorphism. isomorphism is only if $\ker T={0}$ or ...
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30 views

Generalizing the idea of Eigenvectors

I was thinking about the idea of an eigenvector as an element of a set that is closed under a transformation. So if $A$ represents the transformation, and if $\vec{u}$ is an element for which there ...
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Show that the inverse of a strictly diagonally dominant matrix is monotone

I have been struggling with this problem for awhile. Given that $A$ is a strictly diagonally dominant matrix with positive diagonal entries and non-positive off-diagonal entries, show that $A$ is ...
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Linear Alegbra - Is this linear transformation isomorphism?

Let $\mathbb R^4 \rightarrow \mathbb R^4$ linear transformation. That : $$\dim\operatorname{Im}(T+I)=\dim\ker(3I-T)=2$$ Is $T-I$ isomorphism? The only thing I come up with is that 3 and -1 are ...
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Eigenfunctions of the exponential of the derivative

In the space of smooth functions of one variable is there a way to tell what are the eigenfunctions of the operator $\exp(\partial_x)$, i.e. what is the solution of the eigenvalue problem ...
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Solution of differential system

Consider the following differential system in $\mathbb{R}^{n}$: $$u'=Au+(x^{2}+1)^{-1}u---(1)$$ where $A$ is an $n\times n$ matrix with real entries such that all eigenvalues of $A$ have ...
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98 views

Give conditions on a,b,c, and d such that A has two, one, and no eigenvalues?

I am given that matrix $$A= \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} $$ and I need to find conditions on a,b,c, and d such that A has Two distinct ...
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Find the eigenvalues of the matrix and give the bases for each of the corresponding eigenspaces

I'm having issues with this problem. I have solved for the eigenvalues but am having trouble finding the bases for both eigenvalues. The pictures below contain my work for solving for the eigenvalues ...
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48 views

Geometric Interpretation of Eigenvectors

I just want to make sure I'm thinking about this correctly. I've been given a matrix A and I need to find the eigenvalues and eigenvectors geometrically. I have found the eigenvalues. It wasn't too ...