Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [eigenvalues-eigenvectors]

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 $\lambda x - Tx = 0$.

0
votes
0answers
7 views

Exponential of a Matrix- Repeated AND Complex Eigenvalues

I am seeking a general solution to the initial value problem x' = Ax, x(0) = x_0 that can be written out to include both the eigenvalues and eigenvectors. To cover the case of repeated eigenvalues, ...
1
vote
2answers
18 views

Eigenvalues of anti-symmetric real matrix and non-singularity

I am trying to solve the following problem (book: Classical Mechanics by Goldstein, exercise 4.4): By examining the eigenvalues of an anti-symmetric $3\times3$ real matrix $A$, show that $I\pm A$ is ...
1
vote
0answers
25 views

Proving that a matrix is nonnegative if its powers are nonnegative

I am working on a problem involving doubly stochastic matrices where I must prove that $P$ is doubly stochastic if and only if $P^k$ is doubly stochastic for $k = 1, 2, ...$ It is easy to show that if ...
0
votes
0answers
12 views

Plotting vector field lines and eigenvectors on Gnuplot

I have been trying to draw the phase portrait of a system in 3D and I need to use Gnuplot for it but I really could not find anything in the internet about drawing 3D-eigenvectors and the vector field ...
3
votes
2answers
47 views

Eigenvalue of a complex matrix

Wonder if this is correct: If $$ A=\begin{pmatrix}a & b\\c & d \end{pmatrix} $$ is a complex matrix that has a real eigenvalue, then the matrix $$ B=\begin{pmatrix}\overline{a} & \...
0
votes
3answers
71 views

5.6 Axler's Linear Algebra Done Right

Consider the following theorem: Theorem 5.6 Equivalent conditions to be an eigenvalue: Suppose V is finite-dimensional, $T \in \mathcal{L}(V),$ and $\lambda \in \mathbb{F}.$ Then the following ...
0
votes
1answer
51 views

Graphs with rational eigenvalues

Let $A$ be the adjacency matrix of a graph with eigenvalues $\lambda_i$. My questions are: Is there any assumption/conditions for a graph to have all rational eigenvalues ($\lambda_i \in \mathbb{Q} \;...
2
votes
1answer
42 views

Largest solution of a linear system

Given an $n\times m$ matrix $A$ of full-column rank, and a vector $\vec b$ of size $n$. We consider the solution of the linear system: $$ A\vec{x}=\vec{b} $$ Since $A$ is full-column rank, the ...
1
vote
0answers
29 views

left eigenvector as unique solution?

Consider the following equation in linear algebra: $\ \lambda·x·y-(x·M·y)=0$ , with $\ x$ and $\ y$ being respectively 1xn and nx1 vectors, and $\ M$ a nxn real semipositive and irreducible matrix. ...
0
votes
3answers
41 views

Does multiplication operator have eigenvectors? [on hold]

Given the infinite dimensional space of all complex polynomials $p(z)$ does the operator $O$ representing multiplication by $x$ have any eigenvectors? Since the multiplication operator is $O[p(z)]=xp(...
1
vote
1answer
20 views

Bounds on eigenvalues of product of matrices

I have 4 matrices $\mathbf{V}_1$, $\mathbf{V}_2$, $\mathbf{V}_3$, $\mathbf{V}_4$, all with eigenvalues $0 < |\lambda^i_n| <1$, where $\lambda^i_n$ is the $n$th eigenvalue of the $i$th matrix, ...
0
votes
1answer
25 views

Properties of eigenvalues from differential equation

I have the matrix equation $$ M'(t) = A(t) M(t)$$ with the initial condition $M(0) = I$, with $I$ the identity matrix, and where both $M$ and $A$ are $3\times 3$ matrices, $A(t)$ is real and can be ...
1
vote
1answer
13 views

About irreducible polynomial over field & characteristic or minimal polynomial of matrix

Let $F$ be a field and $K$ be a finite extension of $F$, and let $\alpha\in K$. Consider a linear map $T:K\to K$ is defined by $T(\beta)=\alpha\beta$ for all $\beta\in K$, where $K$ viewed as a ...
1
vote
0answers
13 views

Eigenvalue of nonlinear elliptic equation.

For any $u \in H^2(\mathbb R^n)$, consider operator $$ Lu = -\frac{1}{2}\Delta u + u - 3u_0^2u $$ where $u_0$ is a solution of $$ -\frac{1}{2}\Delta u + u - u ^3 =0 $$ how to show $L$ has only one ...
1
vote
2answers
63 views

How to show that $\dot z=(M^{-1}AM)z\implies \dot z=\begin{bmatrix}a &-b\\b & a\end{bmatrix}z$?

Consider the linear system $$\dot x=A x,$$ with complex eigenvalues. Let $z=M^{-1}x$, where $M=[v_R\; v_I]$ with $v_R=1/2(v_1+v_2)$ and $v_I=i/2(v_1-v_2)$. Both $v_1,v_2$ represent the eigenvectors of ...
0
votes
0answers
25 views

Solving system of differential equations with unknown eigenvalues

I have 4 differential equations and a characteristic polynomial like $ λ ^4+ \frac{w^2*λ^2}{ɛ} - \frac{2kw^2}{ɛm}=0 $ where I denoted $ɛ$ as small deviation approximately zero, $m$: mass, $w$: angular ...
0
votes
0answers
17 views

“Eigen images” for categorical data

I have a sequence of categorical images. For a two category image, each image pixel can have one of two values. I would like to analyze these images using a technique like eigen images. The goal is to ...
0
votes
0answers
30 views

Eigenvector normalization, poles and residue

I am trying to understand an eigenvector normalization procedure described in an article [1](appendix B). The problem involves a complex valued matrix $\mathbf{Z}$, function of the complex number $s$,...
2
votes
0answers
48 views

How do I calculate $e^{tA}$

I want to calculate $e^{tA}$, and eigenvalues are $\lambda_1=$ trace A , $\lambda_2 = 0 \DeclareMathOperator{\tr}{tr}$ so $P_0=I$ and $P_1 =(A-\lambda_1I)=A-(\tr A)I$ $r_1=e^{(\lambda_1)t} = e^{(\tr A)...
2
votes
0answers
17 views

NS Condition for all eigenvalues contained in the field F?

Suppose $T: V(F) \rightarrow V(F)$ is a linear transformation, where $V$ is a finite dimensional vector space. What is the necessary and/or sufficient condition that $F$ contains all eigenvalues of ...
0
votes
1answer
32 views

Distinct Eigenvalues of a matrix

In one of the competitive exams, they asked the below question. Q). The number of distinct eigenvalues of the matrix A is equal to ___ and The matrix A is given below, ...
0
votes
2answers
66 views

Eigenvalues eigenvectors

Good evening; can you help me with the problem? Let an $ n\times n$ matrix A have eigenvalues $\alpha_{1},...\alpha_n$ and eigenvectors ${a}_{1},.. {a}_{n}$ Find eigenvalues and eigenvectors for ...
0
votes
0answers
52 views

Negative eigenvalues of a real symmetric matrix? And/or numerical stability of GSL “eigen_symmv” algorithm

I am calculating the linearly independent modes (and their uncertainties) of a large set of correlated variables by diagonalizing their covariance matrix, $C$, using GSL's ...
0
votes
0answers
10 views

Does Eigenvalue decomposition (EVD) of a matrix $\mathbf{A}$ mean $\mathbf{Q}$ in wiki,if $\mathbf{A}$ ignore the rank-1 constraint

In this paper ( https://liu.diva-portal.org/smash/get/diva2:1245887/FULLTEXT01.pdf ) the author told me two things 1.$\mathbf F_k= \mathbf f_k \mathbf f^H_k, \mathbf f_k$ is a $N$ by $1$ matrix,and $\...
0
votes
0answers
37 views

How to analyze eigenvalues of design matrix of categorical features obtained from different encoding schemes

I have a dataset with multiple strings (example shown as below) and would like to apply machine learning methods to mine possible patterns: ABBABCDCCD CABBABCDCC BBABCDCCDB AABABCDCCD ABBABCDCDD ...
0
votes
2answers
37 views

Linear Transformation, eigenvalues ​and eigenvectors

Could someone help me with this linear algebra question: Let $T:\mathbb R^3 → \mathbb R^3$ be a linear transformation, $B = \{v_1, v_2, v_3\}$ a basis of $\mathbb R^3$ and $U = [v_1, v_3].$ Knowing ...
0
votes
0answers
14 views

Approximate graph

Let $L_{G}$ be the Laplacian of a graph $G$ with irrational eigenvalues. I am curious to know: Is there any efficient way to find an approximate graph $\hat{G}$ such that all the eigenvalues of this ...
0
votes
0answers
9 views

Routines for diagonalization of banded hermitian matrices?

I have a problem in which I need to diagonalize matrices with many thousands of complex elements three times. I know that the matrices are hermitian and sparse. Specifically, they consist of 9 bands ...
0
votes
1answer
19 views

How do i generalize theory to arbitrary trace and arbitrary determinant?

Given a matrix $A\in \Bbb R^{2\times 2}$ Assume that trace $A = 0$. Then: a. If $\det A = 0$, then $0$ is the only eigenvalue. b. If $\det A <0$, then eigenvalue is $\pm\sqrt{-\det A}$ c. If $\det ...
0
votes
1answer
44 views

How to calculate the eigenvalues of nonhomogeneous system of LDE?

The system \begin{align} \dfrac{dx}{dt}&=-8 - 4 y + 2 x y, \\ \dfrac{dy}{dt}&=-x^2 + 4 y^2. \end{align} I have linearized it for one of the singularity point(-1,-2): \begin{align} \dfrac{dx}...
2
votes
0answers
14 views

Need to conclude the interlacing property among the eigenvalues of $A$ and $A - \mathbb{E}A$.

Suppose $\mu_1 \leqslant \mu_2 \leqslant \cdots \leqslant \mu_n$ be the eigenvalues of the matrix $A$ and $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$ be the eigenvalues of the ...
0
votes
0answers
14 views

Matrix Numerov Method in three dimensions

Hi can anybody help me to write the following equation in form of matrix by using Numerov's method. $\left(\frac{d^2}{d x_1^2}+\frac{d^2}{d x_2^2}+\frac{d^2}{d x_3^2}+x^2_1+x^2_2+x^2_3+x_1x_2+x_2x_3+...
1
vote
2answers
84 views

Bounding the coefficients of the characteristic polynomial of a matrix

For a given $n \times n$-matrix $A$, the characteristic polynomial of $A$ is $\lambda^n+a_{n-1}\lambda^{n-1}+\cdots+a_1\lambda+a_0$. I am curious to know if we can upper bound the coefficients of this ...
1
vote
0answers
24 views

Proving Lie algebra isomorphism to $\mathfrak{sl}(2, \mathbb{C})$ [duplicate]

The question: Let $L$ be a three dimensional simple Lie algebra over $\mathbb{C}$. Say that $x \in L$, $x \neq 0$, and that $ad_x$ has an eigenvalue $\lambda \neq 0$. Show that $L \cong \mathfrak{sl}(...
-1
votes
2answers
26 views

if we know eigenvector of a matrix, what will be eigenvector for cube of matrix?

How can I verify $(1,- {1 \over 2},0)$ is an eigenvector of M³? I can consider a diagonal matrix with all diagonal entries $⁻3$, or I can consider a general diagonal matrix. But my question is how I ...
5
votes
2answers
394 views

Finding eigenvector only knowing others eigenvectors.

The matrix $A \in M_3(\mathbb{R})$ satisfy $A^t=A$ and $(1,2,1), (-1,1,0)$ are eigenvectors of $A$. Which vector is also an eigenvector of $A$? Alternatives: $(0,0,1)$; $(1,1,-3)$; $(1,1,3)$; There is ...
0
votes
0answers
12 views

Why is the maximum value of an orthonormal dictionary of size $N \times N$ less than that of another having size $M \times M$, where M>N?

I was trying to compare the maximum value in DCT dictionary (representing an orthonormal dictionary) of size $N \times N$ to that of another DCT dictionary of size $M \times M$, where $M>N$. I ...
1
vote
2answers
35 views

Eigenvalue of a matrix $A$

WTS: A scalar $\lambda$ is an eigenvalue of a matrix $A$ $\iff$ $\det(\lambda I-A)=0$ My proof: Assume $\lambda$ is an eigenvalue of A. So $Av=\lambda v$ for a non-zero vector, v.This is equivalent ...
0
votes
2answers
19 views

Diagonalizable Matrix $A$ with Eigenvalues scalar=c, is $A=cI$

Let $A$ be an $n \times n$ diagonalizable matrix such that all of it's eigenvalues are equal to a scalar, $c.$ Then $A=cI$. Is this true? Why or why not? I'm thinking not, because I can come up with ...
0
votes
0answers
8 views

Monotone eigenvector of the normalized Laplacian

Let $u_0 \geq u_1 \geq \cdots \geq u_{n-1}$ be positive numbers and define a matrix $n\times n$ by $M_{i,j} = u_{\left|i-j\right|}$ for all $i,j$. Let $L = I - D^{-1/2}MD^{-1/2}$ be the normalized ...
-1
votes
0answers
27 views

How important is the imaginary part of the eigenvalues and eigenvectors?

I'm building a library in C for embedded systems that can compute linear algebra. Available on my GitHub. I going to code so the library can use eigenvalues and eigenvectors, also Schur ...
0
votes
1answer
29 views

solve coupled second-order differential equation

How can I solve the following set of coupled 2nd order differential equations? \begin{equation} \ddot{x}_{1}= -a^{2}x_{1}+ b^{2}x_{2} \end{equation} \begin{equation} \ddot{x}_{2}= b^{2}x_{1}-c^{2}x_{2}...
1
vote
1answer
37 views

What matrix has only negative or zero real part for all the eigenvalues?

Say $X \in \mathbb{R}^{m\times m}$, Is it possible to have a constraint on $X$, such that all the eigenvalues has negative or zero real part? What I conjecture The following $X$ has only negative ...
1
vote
1answer
30 views

Finding the square of an eigenvalue in a Generalized eigenvalue problem

I'm looking at a generalized eigenvalue problem that is associated with a FEM structure: $$\mathbf{A}\mathbf{v}-\lambda \mathbf{B}\mathbf{v}=\mathbf{0}$$ Where $\mathbf{A}$ is a stiffness matrix, $\...
0
votes
2answers
21 views

Fast way to compute eigenvalue of block identity matrices

Let $M \in \mathbb{R^{2n \times 2n}}$ have the form \begin{equation} M = \begin{pmatrix} \mathbb{I}_{n} & - a \mathbb{I}_{n} \\ b \mathbb{I}_{n} & \mathbb{I}_{n} \end{pmatrix} , \end{equation} ...
1
vote
1answer
19 views

Matrix representation of a linear transformation under change of basis.

For a matrix $A\in M_n(\mathbb{F})$ consider the linear transformation $T_A:\mathbb{F}^n\rightarrow \mathbb{F}^n$ such that $x\mapsto Ax$. Suppose A is diagonalizable and $B=\{v_1,...,v_n\}$ is a ...
2
votes
2answers
49 views

Finding the Eigenvectors given Eigenvalues

My matrix is $$A=\begin{bmatrix}0 & -1\\-1 & 0\end{bmatrix}$$ I used $$det(\lambda I-A)=0$$ And found my two eigenvalues, $1$ and $-1$. If I from here try to use $\lambda = -1$ it ...
1
vote
2answers
28 views

Let $V$ be a $n$-dimensional real vector space and let linear operator $ T \in L(V) $ satisfy the equation$ (T^2+I) *(T^2+4I)=0$.

Let $V$ be a $n$-dimensional real vector space and let linear operator $ T \in L(V) $ satisfy the equation $$ (T^2+I) *(T^2+4I)=0$$. Find the eigenvalues for $T$ and prove that $n$ is even. I'm a bit ...
0
votes
1answer
28 views

Proof - Two entries on main diagonal will be the same for all powers

Prove that in any power of the matrix $$\begin{bmatrix} 2&1&1\\0&2&1\\1&1&1\end{bmatrix}$$ two entries on the main diagonal will be the same. I was able to prove this by ...
2
votes
2answers
14 views

Finding normal operator matrix from characteristic polynomial

Let $ A \in L ( \mathbb{C}^4) $ be a normal operator with characteristic polynomial $ k_{A} = (\lambda - 1)^2 * (\lambda - 2)^2$. Is then the matrix for the operator just a diagonal matrix with ...