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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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How are eigenvalues relevant to the invariants of a system?

For a matrix $\mathbf{A} \in \mathbb{R}^{2\times2}$, what can one say about its eigenvalues $\gamma_1, \gamma_2 \in \mathbb{C}$, if: $$\mathbf{S}_0 \in \mathbb{R}^{2}$$ $$\mathbf{S}_{n+1} = ...
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1answer
18 views

Questions about eigenvalues and eigenvectors

I've started studied eigenvalues and eigenvectors. If there is a transformation T: V->V I can find out a matrix of T with fixed basis and characteristic polynomial of T. With this characteristic ...
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2answers
32 views

Can anyone help me with this proof of matrix?

If there is $n \times n$ matrix $A$, and $A^2= - I$? Can anyone explain me why there exists no real eigenvalues? and $\det A=1$?
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finding clusters in a network from eigengaps

I have a usual Laplacian matrix, which describes a network. From the matrix I get the eigenvalues and from these I can compute a metric of modularity in my network based on the largest eigengap. Let's ...
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24 views

Range of vectors that turn into eigenvectors after recursive multiplication by a matrix

Suppose $\mathbf{x}$ is a vector, and $\mathbf{A}$ is a square matrix. Which $\mathbf{x}$'s will satisfy the equation $\mathbf{A}^n\mathbf{x} = \lambda\mathbf{A}^{n-1}\mathbf{x}$, where $\lambda$ is ...
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14 views

Can the Lanczos algorithm converge very fast by choosing initial guess smartly?

Suppose I have the two lowest eigenvectors $v_1$, $v_2$ of a matrix $M$. If slightly change $M$ to $M'$. Can I use $v_1$ or $v_2$ as an initial guess for $M'$? If so, which one should be used, $v_1$ ...
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1answer
11 views

Why left eigenvector complex conjugate transpose of right eigenvector?

My teacher today stated the following: For a matrix $A\in \Bbb R^{n \times n}$, any left eigenvalue $e^*$ is simply the transpose of the conjugate of a right eigenvector $e$ of $A$, so $e^* = ...
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1answer
25 views

Need help to understand a line of a proof of diagonalizability of real symmetric matrices

I was reading a proof of diagonalizability of real symmetric matrices using the concept of generalized eigenvalues and understood all except the very starting (and fundamental) line of the proof " if ...
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2answers
29 views

Definiteness of a symmetric matrix of order $3\times 3$

Let $a,b,c$ be three positive real numbers such that $b^{2}+c^{2}\lt a\lt 1$. Consider $3\times 3$ matrix$$A = \left[ \begin{matrix} 1 & b & c \\ b & a & 0 \\ c & 0 & 1 ...
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2answers
38 views

Prove statement about determinants.

$A$ is a $3\times 3$ matrix over $\mathbb{R}$, I want to show that if $$\det(A + I_3)=\det(A+2I_3),$$ then $$2\det(A+I_3) + \det(A-I_3) + 6 = 3\det A.$$ Can you help me?
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1answer
46 views

Does all Eigenvectors of $A$ lie on the vector space of $Ax$?

The problem is with the last part of the following question: I will write my results to the first parts which are correct here : Three Eigenvalues: $$\lambda_1=1 , \lambda_2=2 , ...
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1answer
30 views

Properties of annihilators and eigenspaces

I'm stuck on the following practice question, and I'm uncertain of how to proceed with the proof. We've only very briefly touched on annihilators in class, and I really don't understand how to ...
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0answers
6 views

Projection matrix I_V*V', can it change null space of orignial matrix?

I have a matrix, M. suppose i have null(M)=10 and rank(M)=50. the eigenvectors of M are v1-v10 represents the null space and v11-v50 represent the other eigenvalues. now i have a projection matrix ...
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1answer
23 views

Converting between SVD and Eigenvector-based expressions

For the well known ordinary least squares there are the well known solution $$\beta=(X'X)^{-1}X'Y$$ This can be expressed in "canonical form" using eigenvector decomposition. ...
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1answer
16 views

$T$ on $\mathbb R^n$ is diagonalizable over $\mathbb R$ with $T^k = I$ , for some $k$ , then is $T^2=I$ [on hold]

If $T$ is a diagonalizable (over $\mathbb R$ ) linear transformation on a finite dimensional real vector space such that $T^k$ is the identity operator for some positive integer $k$ , then is it true ...
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1answer
19 views

$T$ is diagonalizable on finite dimensional v.s. $\implies$ $(T^2+T+I)(\vec v) \ne \vec 0 , \forall \vec v \ne \vec0$?

Let $T$ be a diagonalizable (over $\mathbb R$) operator on a finite dimensional real vector space ; then is it true that there is no non-zero vector $\vec v$ such that $(T^2+T+I)(\vec v)=\vec 0$ ? ...
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1answer
30 views

eigen spaces of similar matrices

Suppose $A,B $ are similar matrices.Suppose $\alpha $ is an eigen value of $A$.Will the eigen space of $A$ with respect to $\alpha$ be the same as that of the eigen space of $B$ with respect to ...
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2answers
146 views

Why an eigenspace is a linear subspace, if the zero vector is not an eigenvector?

I've started studying Eigenvector and Eigenvalue. It says in my book that 0 is excluded from being an eigenvector because it breaks the uniqueness of eigenvalue associated with each eigenvector. ...
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2answers
37 views

Let A and B be nxn matrices, each with n distinct eigenvalues. Prove that A and B have the same eigenvectors if and only if AB=BA

I have been working on this problem for an hour and I need some help. I'm terrible at proving things and I was just hoping someone could help be refine what I have so far. A= PD$P^{-1}$ for some ...
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4answers
84 views

Proving that $A+A^2+A^3$ has eigenvalue $\lambda+\lambda^2+\lambda^3$ where $\lambda$ is an eigenvalue of $A$

I know that $$P= \begin{pmatrix} 4 & 5 & 1 \\ -9 & -6 & 0 \\ 8 & 4 & 2 \end{pmatrix}$$ I also know that $$D= \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ ...
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2answers
59 views

Powered matrices and its eigenvalues

Let $A$ be a n by n matrix, with eigenvalues $\lambda_1$, ...$\lambda_n$. I want to prove that the eigenvalues of $A^k$ are eigenvalues of $A$, raised to power $k$. I managed to prove that if ...
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26 views

Quantum Mechanics in Electric Field

I asked this problem in Physics SE but I did not get any useful answers except one. I believe asking this question here would be more beneficial owing to the Mathematical nature of the problem. I am ...
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1answer
29 views

Number of positive, negative eigenvalues and the number of sign changes in the determinants of the upper left submatrices of a symmetric matrix.

How do we prove that the number of sign changes in the sequence of the determinants of the upper-left matrices of a symmetric matrix $A$ corresponds to the number of positive and negative eigenvalues ...
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26 views

Can the number of sign changes in a sequence of determinants tell us how many negative eigenvalues a symmetric matrix has?

From notes, I've gathered that given a symmetric matrix, the number of sign changes in its characteristic polynomial is equal to the number of positive eigenvalues of $A$. Proof: Let $p(x)$ be a ...
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26 views

Roots of characteristic polynomial have negative real parts implies positive coefficients of the polynomial

Can you help me prove that if all the eigenvalues $\lambda_i$ of a square n-dimensional matrix $A$, have a strictly negative real part then prove that all the coefficients $a_j$ of the characteristic ...
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0answers
16 views

Eigensystem of a real symmetric Toeplitz matrix of large order

My question is related to this one. I am looking for the eigenvalues and eigenvectors of a square, symmetric, real Toeplitz matrix of order N where N is large. There are some references in the above ...
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3answers
292 views

A method of finding the eigenvector that I don't fully understand

Let $$A=\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & t \\ \end{pmatrix}$$ Which has a known eigenvalue : $\lambda$ Find the corresponding eigenvector Over the ...
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0answers
26 views

Linear systems, eigenvectors

For each of the following linear systems of differential equations, (i) find the general real solution (ii) show that the solutions are linearly independent (iii) draw the phase portrait a. $$\dot ...
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1answer
19 views

eigenvectors, linear systems

For each of the following linear systems of differential equations, (i) find the general real solution (ii) show that the solutions are linearly independent (iii) draw the phase portrait a. $$\dot ...
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1answer
47 views

Why for random matrix one of eigenvalues is so big?

Here i have the image of eigenvalues for matrixes obtained with $rand(n,n)/sqrt(n)$, why one of the eigenvalues for every matrix is so big on real axis comparatively to the others? EDIT: i have ...
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1answer
31 views

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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1answer
35 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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3answers
193 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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29 views

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 ...
2
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1answer
51 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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1answer
30 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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0answers
24 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
20 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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1answer
28 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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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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0answers
24 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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0answers
8 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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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
23 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
25 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
28 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
68 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 ...