Number associated to a linear operator from a vector space $V$ to itself: $\lambda$ is and eigenvalue of $T\colon V\to V$ if the map $x\mapsto $\lambda x-Tx$ is not injective.
1
vote
2answers
57 views
Adjacency, Laplacian and Maximum Degree
I am relevantly new to http://math.stackexchange.com/ and this will be my first question, although I have been lurking around for some time now! So, pardon me if I am missing any posting etiquette ...
1
vote
0answers
23 views
finding the decomposition of Laplacian matrix with position of zero elements unchanged
I'd like to know whether it's possible to find the decomposition of a Lapalacian matrix $A$
$B^TB = A$
where $B$ has the same dimension with $A$ and the position of zero elements in $B$ is the same ...
3
votes
1answer
50 views
Alternative representation for Perron Frobenius Eigenvalue
While explaining the application of Geometric programming to Minimizing Spectral radius Boyd says that $\lambda_{pf}$ can also be characterized as:
$\operatorname{inf}\{\lambda|\exists{v}>0, ...
3
votes
1answer
46 views
Eigenvalues and Eigenvectors Diagonilization
Let $ A=\begin{bmatrix}
-7 & -1 \\
12 & 0 \\ \end{bmatrix} $ . Find a matrix $ P $ and a diagonal matrix $D$ such that $PDP^{-1} = A$.
Ok so the first thing I need to look ...
0
votes
2answers
79 views
If A is a square matrix, and $A^3$=I, how can I prove that eigenvalues of A are either 1 or -1?
If $A$ is a square matrix, and $A^3$=I, how can I prove that eigenvalues of $A$ are either $1$ or $-1$?
Thanks for all your input, I figured it out! :)
-5
votes
1answer
56 views
Example of $5\times 5$ matrix with exactly $2$ distinct eigenvalues
What is an an example of a $5\times 5$ matrix with exactly $2$ distinct eigenvalues? Also, what would be the eigenvalues in the example?
1
vote
2answers
45 views
Diagonalization, Solving a System of Linear Equations
I have questions regarding the following task, which is to diagonalize the matrix A:
$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 9 & -20 \\ 0 & 4 & -9 \end{pmatrix}$
What I have ...
3
votes
2answers
63 views
Solving for eigenvectors (Introduction to Linear Algebra by Serge Lang, example 2 page 240)
In "Introduction to Linear Algebra" by Serge Lang, example 2 page 240, there is the matrix $\begin{pmatrix}1&4\\2&3\end{pmatrix}$. The characteristic polynomial is $(t-5)(t+1)$. The system to ...
1
vote
1answer
62 views
A quesion in Fulton & Harris book “representation theory a first course”
In Section 11.2 A little plethysm, it discusses the tensor product of two different representations of $sl_2\mathbb{C}$. It says
"If $V=\bigoplus V_{\alpha}$ and $W=\bigoplus W_{\beta}$ then ...
2
votes
1answer
56 views
complex eigenvectors with non zero real parts
I'm wondering about how to deal with complex numbers in eigenvectors that have non zero real parts, as in my eigenvector is $\bigl[\begin{smallmatrix}1-2i\\-1\end{smallmatrix}\bigr]$ that is supposed ...
2
votes
1answer
81 views
Inequality between Neumann and Dirichlet eigenvalues
Let $\Omega$ be a fixed, smooth and bounded domain in $\mathbb{R}^n$. If we denote with $\{\lambda_n\}_{n \ge 1}$ the nondecreasing sequence of eigenvalues of the Dirichlet problem
$$\left\{
...
1
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1answer
32 views
Are the eigenvalues limited?
While studying elliptic operators, I encountered the following problem, which I'm having problems to prove or give a counter-example:
Let $\Omega$ be an open subset of $\mathbb{R}^m$, and suppose ...
0
votes
3answers
94 views
Confused about Eigenvectors
Consider the Matrix $\pmatrix{E&t\\t&-E}$
The eigenvalues are $\lambda_1=\sqrt{E^2+t^2}$ , $\lambda_2=-\sqrt{E^2+t^2}$.
Consider the first (+ve) eigenvalue.
To find the eigenvectors, we ...
2
votes
1answer
180 views
Prove that if the sum of each row of $A$ equals $s$, then $s$ is an eigenvalue of $A$.
Let $A$ be an $n$ x $n$ matrix.
$i)$Prove that if the sum of each row of $A$ equals $s$, then $s$ is an eigenvalue of $A$.
$ii)$Prove that if the sum of each column of $A$ equals $s$, then $s$ is an ...
0
votes
4answers
70 views
If a real square matrix has only real eigenvalues, is it symmetric?
If a real square matrix has only real eigenvalues, is it symmetric?
I know that a real symmetric matrix has only real eigenvalues, but I'm wondering if the counter implication follows.
1
vote
1answer
68 views
Finding the eigenvalues and the eigenvectors
The matrix:
$$
A=\begin{pmatrix}
1 & -1 & 1 \\
-1 & 1 & -1 \\
-1 & 1 & -1 \\
\end{pmatrix}
$$
has two real eigenvalues, one of ...
1
vote
2answers
36 views
Eigenvalue of anti triangular block matrix (skew matrix?)
I have an real anti-triangular matrix
$M=\left[
\begin{array}{cc}
A & B \\
I & 0 \\
\end{array}
\right]$
where I is an identity matrix. $A$, $B$, $I$, $0$ are all square real ...
1
vote
0answers
36 views
Eigenvalues of discretized linear integral operator
Suppose I have the following kernel operator:
$Af(x) = \int_{-1}^1 K(x-y)f(y)dy$
which is also positive and compact. Hence, it has a countable set of positive eigenvalues. Suppose those eigenvalues ...
2
votes
2answers
48 views
Link between the norm $1$ of a matrix and its biggest eigenvalue
I am working on a set of matrices for a project, studying their highest eigenvalue, let's call it $\lambda_{1}$.
I was curious and plotted the norm 1 of the matrix, ie $ \frac{1}{n^{2}}\sum_{i,j} ...
0
votes
2answers
93 views
How do I write this matrix in Jordan-Normal Form
I have the matrix $A=\begin{pmatrix}2&2&1\\-1&0&1\\4&1&-1\end{pmatrix}$, I want to write it in Jordan-Normal Form. I have $x_1=3,x_2=x_3=-1$ and calculated eigenvectors ...
3
votes
1answer
84 views
Diagonalizable Matrix $A^2$
How can I find a matrix $A$ such that $A^2$ is diagonalizable but $A$ is not?
I have tried many different ways, but to no avail. Is there something that I am missing in the question that gives a ...
2
votes
1answer
68 views
Eigen Values of A?
I just got a quick practice question here that I think should be simple but I can't find a definitive answer.
Let $A$ be a square matrix such that $A^3=A$. What can you say about the eigen values of ...
0
votes
1answer
59 views
Inner product on the vector space $p_2$ of polynomials of degree 2 or less
For this question, dene an inner product on the vector space of $P_2$ of polynomials,
through the formula
$$p(x)q(x) = \int p(x)q(x)dx$$
What are the lengths $$\|\ 1\ \|, \|\ x\ \|,\|\ x^2\ \|$$
...
2
votes
1answer
62 views
Finding the eigenvalues of $A^2$ and $A^{-1}$ from the characteristic polynomial of $A$
In my question we are given a characteristic polynomial $x^3 - 4x^2 + x + 6$ for a matrix A and are told to find eigenvalues for both $A^{-1}$ and $A^2$. I just want to confirm that I am approaching ...
1
vote
1answer
77 views
How do you find the (complex) eigenvalue and each eigenspace over C
My book only has eigenvalue and eigenspace and does not say anything about complex eigenvalue and eigenspace.
$$A = \begin{pmatrix} 0&4\\-1 & 0 \end{pmatrix}\hspace{10pt}B =\begin{pmatrix} ...
2
votes
3answers
179 views
how do I prove that two matrices with same determinant and trace have different eigenvalues?
Assuming that they are both Hermetian, positive definite and have the same full rank. (To show the converse that if two matrices have the same eigenvalues, they must have the same determinant is ...
0
votes
1answer
47 views
Bounding the smallest eigenvalue of an ergodic Markov Chain
I am trying to prove that the smallest eigenvalue of an ergodic Markov chain is greater than -1. Can we do that using proof by contradiction, i.e. assuming the smallest eigenvalue being -1, etc.? The ...
1
vote
1answer
101 views
An eigenvector is a non-zero vector such that…
Various sources define eigenvalues and eigenvectors in slightly different ways (context independent). For example, both of the following definitions seem not to exclude the zero-vector as an ...
0
votes
0answers
21 views
Condition number and Chebyshev systems
Suppose I have a square matrix $A$ of size $n$ with elements $a_{mn}=\phi_m(x_n)$ where $\phi_m(x)$ can be thought of as a very friendly function: orthogonal, bandlimited, bounded and analytic. Also, ...
2
votes
4answers
81 views
Eigenvalues of $A$
Let $A$ be a $3*3$ matrix with real entries. If $A$ commutes with all $3*3$ matrices with real entries, then how many number of distinct real eigenvalues exist for $A$?
please give some hint.
...
1
vote
1answer
28 views
Property of sequence of eigenvalues of an operator.
For a positive (self adjoint) operator $A$ with eigenvalues $\lambda_k$, is it possible to have the case when neither $\lambda_k\to \infty$ or $sup_k \lambda_k<\infty$ for example if a subsequence ...
2
votes
0answers
67 views
Is there a generalization of eigenvalues/eigenvectors to $L^p$ for $p\neq2$?
Given symmetric positive definite $m\times m$ matrix $\Sigma$, $1 \le p,$ and constant $c > 0$, consider the solution to the optimization problem:
$$
\min_{v : \left\|v\right\|_p \ge c} ...
5
votes
1answer
127 views
Eigenvalues of the sum of a stochastic matrix and a diagonal matrix
Let $D$ be a real diagonal matrix $D=\operatorname{diag}(a_1,a_2,\ldots,a_n)$ with $a_1\le a_2\le\ldots\le a_n$. Assume that at least one of the $a_i$ is positive. Let $P$ be an irreducible, real, ...
0
votes
1answer
83 views
Show the stable age structure
Considering the population process described by
where $γ$ is the dominant eigenvalue of $L$ $l$ denotes the survival function of the Leslie matrices and $L$ is the Leslie matrix below
We are trying ...
1
vote
1answer
248 views
How to determine if a matrix is positive/negative definite, having complex eigenvalues?
I am trying to deal with an issue: I am trying to determine the nature of some points, that's why I need to check in Matlab if a matrix with complex elements is positive or negative definite. After ...
1
vote
1answer
106 views
Do the non-zero eigenvalues of AB and BA have the same algebraic multiplicity (for AB and BA not square)?
I know that if A and B are square nxn matrices, then AB and BA have the same characteristic polynomial and thus the same eigenvalues (and same algebraïc multiplicity).
I'm wondering though if this ...
1
vote
0answers
62 views
Solving a Sturm-Liouville differential equation variationally
This is a problem from Haim Brezis' functional analysis book (Exercise 8.41). I solved parts of it, but am stuck on some parts/want confirmation on the method. The problem is as follows:
Let $q ...
1
vote
0answers
54 views
The value interpretation of eigenvectors.
My question is may be strange but I wanna lie it any way.
The direction of an eigenvector is the most important as we normalize it. This view is right but what about the value of this eigenvector in ...
1
vote
1answer
46 views
Expressing a matrix as an expansion of its eigenvalues
This shouldn't be too difficult but I can't find a satisfactory proof.
Show that a real, symmetric matrix A satisfying the eigenvector
equation $Au_{i} = \lambda u_{i}$ cam be expressed as an ...
7
votes
3answers
81 views
Let $A$ be a $n× n$ real matrix with $A^2 = A^T$. Show that every real eigen-value of $A$ is either $0$ or $1$.
Let $A$ be a $n×n$ real matrix with $A^2 = A^T$. Show that every real eigen-value of $A$ is either $0$ or $1$.
my thoughts:
$A^2 = A^T$
$\implies$ $A.A=A^T$
$\implies$$(A.A)^T=A$
$\implies$ ...
0
votes
0answers
16 views
Particular generalized eigenvalue problem with parameters
If the generalised eigenvalue problem is given by Ax=uDx where D is a diagonal matrix with two parameters, one, say a for all the diagonal elemens, except the last one, say b. How does the eigensystem ...
2
votes
2answers
79 views
Eigenvalues of $A^{T}A$
Let $\lambda_{i}(M)$ denote the $i$th eigenvalue of the square matrix $M$, and $T$ denote the matrix transpose.
Is it true that $|\lambda_{i}(A^{T}A)|=|\lambda_{i}(A)|^{2}$ for every square matrix ...
1
vote
1answer
47 views
Boundary-value problem in differential equations
Consider the problem:
$$u^{(4)} + \lambda u = 0, \ \ \ 0<x<\pi; \ \ \ u(0) = u(\pi) = u''(0) = u''(\pi) =0$$
Find the eigenvalues.
How should one proceed about this problem? I am complete ...
2
votes
3answers
122 views
Spectrum of eigenvalues and eigenfunctions
Our O.D.Es professor had the "amazing" idea of heavily introducing advanced linear algebra material (which is not an official prerequisite for the course) along with boundary value problems. Not being ...
4
votes
2answers
108 views
Math hack for solving system of equations
Is it a "standard" Math/Numerical-Analysis hack to add a relatively small number e.g. 1*10E-5 to the diagonal of a squared matrix to ensure LU Decomposition (or whichever decomposition algorithm is ...
1
vote
3answers
44 views
Differentiation operator and eigenvalues
Let $V = \{p(x) \in F[x] \ | \ \deg(p(x)) \le n\}$. Let $T : V \to V$ be given by differentiation, in essence $$T(p(x)) = p'(x)$$
It seems to me that the only eigenvalue that can exist is $\lambda ...
3
votes
1answer
173 views
Evaluating eigenvalues of a product of two positive definite matrices
Let $A,B\in M_n(\mathbb{R})$ be two symmetric positive definite matrices, i.e.: $$\forall x\in\mathbb{R}^n, x\neq 0, (Ax,x)>0, (Bx,x)>0,$$ where $(\cdot,\cdot)$ is the usual scalar product in ...
1
vote
2answers
115 views
A basic example of basis for eigenspaces
Hi I'm just learning how to find bases for eigenspaces and I ran into a very basic case that confused me.
So the matrix is $$A = \begin{pmatrix}2 & 1\\0 & 2\end{pmatrix}$$ and $\lambda = 2$.
...
1
vote
1answer
101 views
Finding approximate eigenvalues of perturbed matrix
Assume I have some constant matrix $A$ to which I add a perturbation, resulting in $M(\epsilon )=A+\epsilon B$ the perturbed matrix ($B$ is constant as well), and that I can easily find the ...
3
votes
1answer
63 views
How to compute $\text{trace}((A+D)^{-1}A)$
Give a diagonal perturbation matrix $D$ (which is not an identity matrix), is there a simple way to compute
$$\text{trace}((A+D)^{-1}A)$$
Or is there a good approximation?
