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 $\lambda x - Tx = 0$.

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9 views

What can be inferred from the shape and contour plot (2D reconstruction) of Eigenmodes?

Let us say we solve an eigen value problem $k B q=A q$, like this: $$ \begin{array}{c} -i \omega \hat{u}=-i k \bar{U} \hat{u}-\bar{U}_{y} \hat{v}-i k \hat{p} \\ -i \omega \hat{v}=-i k \bar{U} \hat{u}-\...
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eigen value of the $n \times n $ Matrix $x y^T$

Let $x,y$ be two non zero $n \times 1$ vectors. If $y^T$ denotes the transpose of the matrix $y$, then what are the eigen value of the $n \times n$ matrix $x y^T$? Attempt: Denoting $x= \big(x_1,\...
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Proof verification: $A, B$ are real symmetric commuting matrices then they share a common eigenvector.

I'm asked to prove if $A, B$ are real symmetric matrices that commute then they share a common eigenvector. My approach: Since $A$ is real symmetric, it has an orthonormal eigenbasis. Now if all of ...
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22 views

eigenvalues and kernel

Say that A is a nxn matrice with eigenvalue $\lambda$. Let vector $\vec{v}$ ∈ Ker((B − λIn)$^2$) but not an element of Ker(B − λIn). (I have proved the contrary to be true though). How do i show that ...
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Linear Algebra Proof on Eigenvectors

Let $A$ and $B$ be $n\times n$ matrices, with $B$ invertible. Assume that $\lambda$ is an eigenvalue of $A$ with eigenvector $v$. Prove that $\lambda$ is still an eigenvalue of $B−1AB$ with ...
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34 views

Solving a tensor equation from singularity condition

I apologize if this question has already been asked but I'm not sure what the best key-words are. I have a tensor equation of the form: $(A + x\otimes b)c=0$. Here, $A$ is a second-order tensor/matrix,...
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Determinant of the sum of tridiagonal and anti-tridiagonal matrices

$A$ is a tridiagonal matrix and $B$ is an anti-tridiagonal matrix, both of size $n\times n$, such that $B^2$ is a diagonal matrix. Is it possible to express $\det(A+B)$ in terms of $\det(A),\det(B),n$ ...
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Commutability of symmetric positive-definite matrices

Suppose $A, B \in R^{n \times n}$ are symmetric, positive-definite. Can we say that they commute? If not, what additional conditions are required?
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Eigenvalues of square of a matrix

It can easily be seen that if matrix $A$ has eigenvalue $a_1$ and eigenvector $v_1$, then matrix $A^2$ has eigenvalue $a_1^2$ and eigenvector $v_1$. However, I was wondering if the converse is also ...
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Some questions about eigenvalues, eigenvectors, and diagonalization

I've been studying for my linear algebra final and was going through the review sheet the professor gave us. Most of the content was easy to understand, but I couldn't get my head around a few ...
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How can I show that: $\text{Tr}(A^*A)=\sum_{\lambda\in\sigma(A^*A)}\lambda$?

Let $A\in \mathbb{K}^{n\times n}$, how can we show that: $$ ||A||^2_F=\text{Tr}(A^*A)=\sum_{\lambda\in\sigma(A^*A)}\lambda$$ Where $A^*=\bar A^T.$ Thanks in advance for any help!
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Eigenvalues of $N^T M N$

let the matrices $M, N \in \mathbb{R}^{n\times n}$ be positive definite, and $M$ is a lower block diagonal matrix. Consider the matrix $T \in \mathbb{R}^{n\times n}$ $$ T = N^T M N .$$ Can I say ...
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1answer
41 views

Expressing cubic power of an eigenfunction? [closed]

Mapping between polynomials and column matrices So any polynomial can be mapped to a vector. For example $$ y(x) = x + x^2 \to y_M= \begin{pmatrix} 0 \\ 1 \\ 1 \\ \...
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1answer
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Mathematical problem related to spectral method for time evolution

I have a mathematical problem related to spectral method for time evolution in QM. Considering the standard evolution for a generic quantum state $\psi(t) \in \mathbb{C}^N $ which is not express in a ...
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1answer
36 views

BA diagonalisable show that AB diagonalisable ( A and B are not square) [duplicate]

let A $\in \mathcal{M}_{4,3}(\mathbb{R})$ and B $\in \mathcal{M}_{3,4}(\mathbb{R})$ such as BA$ = \begin{pmatrix} 0&1&1 \\ 1&0&1 \\ 1&1&0 \end{pmatrix}$ show that AB is ...
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1answer
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Eigenvector of the product of a rank-one matrix and an arbitrary matrix

Suppose $A$ is a rank-one (real) matrix ($n \times n$) of the form $u v^t$ and $B$ is an any real square matrix ($n \times n$) with a dominant eigenvalue. Can it be shown that $A$ and $AB$ have the ...
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Is it true that the dimension of an eigenspace of a square matrix is at most the multiplicity of the corresponding eigenvalue?

By "the multiplicity of the corresponding eigenvalue", I mean the multiplicity of the eigenvalue as a root of the characteristic polynomial $p(x)=\det(xI-A)$, i. e. the maximum integer $k$ ...
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1answer
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Help approximating the leading eigenvalue of this Jacobian

I have a system of 4 ordinary differential equations with positive parameters, which has two stable fixed points. The Jacobian matrix evaluated at the first equilibrium leads easily to finding the ...
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Find the axis of reflection which is described a matrix

I need help with the following example. Find the axis of reflection which is described by a matrix $\frac{1}{10}\begin{pmatrix} 3 & 1 \\ -1 & 3 \end{pmatrix}$ Should I use eigenvectors ...
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Eigenvalues equation and differential equations [closed]

Can we express the Hermite's differential equation $$\frac{d^2y}{dx^2}-2x\frac{dy}{dx}+2ny=0$$ as an eigenvalue equation? If possible, then how? Can someone elucidate a little bit?
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Proving that $a_kA^k+a_{k-1}A^{k-1}+\cdots +a_1A+a_0I = 0_n$

I've been studying linear algebra and came across this question that I have no general idea how to solve. The question is as follows. Let $A$ be an $n×n$ diagonalizable matrix with distinct ...
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2answers
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Find eigenvalues of $I - uv^T$

I want to show that the eigenvalues of $I - uv^T$, where $u,v \in \mathbb{R}^n$ are given by 1 with multiplicity $n-1$ and $1-v^Tu$ with multiplicity 1. I have tried setting up the eigenvalue equation ...
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Eigenvalues when perturbed along anti-diagonal.

Given a $n \times n$ matrix $A$ with eigenvalues $\lambda_k$ for $k = 1, 2, \dots, n$ we know the relationship between $\{\lambda_k\}$ and the eigenvalues of $A + t I$ where $I$ is the identity matrix ...
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Change of eigenvalues and eigenvectors caused by matrix expansion

I have a set of $k$ data pieces, each data piece is an $n$-d vector. Each data piece here could be an image block or a piece of music. The dataset could be expressed as an $n\times k$ matrix $A$, that ...
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1answer
22 views

Comparing two Diagonalisable matrices with identical specific features

Hello, I was wondering if anyone could help me with this Linear Algebra question that I'm stuck on. So far this is what I've worked out: I know that if a Matrix is Diagonalisable then it's diagonal ...
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Linear map questions in association with eigenvalues, eigenspace and diagonalisablility.

Hi I was wondering if anyone could help me on this question I have become stuck on. This is what I know so far: For c) I know in general the algebraic multiplicity and geometric multiplicity of an ...
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Any good/simple references for degenerate perturbation theory

Let $A$ be Hermitian matrix and $\lambda_1(A)\ge \cdots \ge \lambda_d(A)$ be the corresponding eigenvalues in descending order. If $A$ is simple ($\lambda_i$ are distinct), then we see that $f(\lambda,...
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How to solve the eigenvector equation using an inverse?

For a Matrix $$A=\begin{bmatrix}-2 & 1\\6 & 3\end{bmatrix}$$ The eigenvalues are $$(A-\lambda I)x=0$$ $$det(A- \lambda I)= (-2-\lambda)(3-\lambda) - 6=0$$ $$\lambda^2 - \lambda -12=0$$ $$\...
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A geometrical interpretation of the product of a positive diagonal matrix and a matrix with positive eigenvalues

I'm trying to shed some light on a recurrent problem I find while studying control systems. In many of the systems I work with, their stability depends on the eigenvalues of a matrix $B = U^{-1}A$, ...
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1answer
26 views

What will be the eigen vector of $A^t$

$Ax = ax$ and $x$ is an eigen vector of $A$ corresponding to the eigen value $a$ . What will be the eigen vector of $A^t$ corresponding to the eigen value $a$. Can anyone please give me a hint ?
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How to find eigen values of such kind of matrices ( Matrices which are bigger than $ 3\times 3$)

The following question is part of a masters exam for which I am preparing and I don't have any methodology on how this type of matrices are solved. This was question asked: I don't know how to find ...
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Transforming generalized eigenvalue problems

I'm working on generalized eigenvalue problems (GEPs) which is to find the so called eigenvalue $\lambda$ such that $Av-\lambda Bv=0$ some $v \neq 0$. Denote the spectrum of a GEP associated with A ...
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How does a linear transformation effect areas in 2-dimensional Euclidian plane?

If I have a 2 by 2 matrix A (the only info I have on A is that it's singular). How can I describe the effect of the linear transformation corresponding to A? I can do this with specific A but I am not ...
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Eigenvalue of random matrix

Assume, $G \in \mathbf{R}^{p \times p} , B \in \mathbf{R}^{n \times n} \succ 0$ are positive define matrices. $X \in \mathbf{R}^{p \times n}$ is a random matrix. Define, $Y = GXB$. Consider the ...
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1answer
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Interpretation of eigenvalues and associated eigenvalues

I am given a 4x4 matrix such and a basis for $R^4$, $\{u_1,u_2,u_3,u_4\}$. Suppose the following conditions: $$Bu_1 = 2u_1 \\Bu_2 = 0\\Bu_3 = u_4\\Bu_4 = u_3$$ First, I am asked to find the ...
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Finding $x^{k+1}$ (gradient-descent)

I want so apply the formula $x^{k+1}=x^k-\eta A_\sigma^{-1}\nabla f(x^k)$ introduced on https://arxiv.org/pdf/1901.06827.pdf in 2.2.1 for a matrix. Let $x^0 = \begin{pmatrix} 1\\0 \end{pmatrix} \in \...
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Bounding the smallest eigenvalue of a matrix from below

Let $\lambda_i, i =1,\dots,n$ be in $(-1,1)$, $\omega \in (-\pi,\pi]$ and consider the matrix $X = (X_{uv})$ with entries $$ X_{uv} = \frac{1}{1-\lambda_u e^{-i\omega} - \lambda_v e^{i\omega} + \...
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1answer
36 views

Find the eigenvalues of an identity matrix minus a rank 1 matrix [duplicate]

This problem comes from Linear Algebra and its Applications by David C Lay and Steven R Lay. Let $c$ be a unit vector in $\mathbb{R}^3$. Consider the matrix $A = I - 2cc^T$ a ) What are the ...
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1answer
51 views

Given matrices $A$ and $B$ solve for $P^{-1}AP=B$

Given $A = \begin{bmatrix} 0 & 1 & 0\\ 0 & 1 & 4 \\ 0 & 0 & 2 \end{bmatrix} $ and $B=\begin{bmatrix} 0 & -2 & 3\\ 0 & 1 & -1 \\ 0 & 0 & 2 \end{bmatrix}$...
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$A$ be $n \times n$ matrix over $\mathbb{C}$ such that every non-zero vector of ${\mathbb{C}}^n$ is an eigenvector of $A$

I am trying assignment questions in Linear Algebra and this question could not be solved by me. So, I thought of posting it here. Let A be an $n\times n$ matrix over $\mathbb{C}$ such that every non-...
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Estimation of a matrix by knowing some of its eigenvalues [closed]

I have the following equation: Lambda=Eig(inv(M)*K) where [M]nn is a known square matrix. [K]nn is the unknown matrix, but we know the sparsity of this matrix. K ...
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31 views

Relation between maximum eigenvalue and trace

Why $\max⁡_{1≤i≤n}(\lambda_i (A^T A))≥tr(A^T A)$? Lambda(i) is the eigenvalues of multiplication matrix A transpose and A. Thanks guys :) SOURCE: Part of proof norm2 by definition
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Properties of SDP solution Lovasz Theta function

I have the following SDP program for a graph $G=(V,E)$, $|V| = n$, $k \in \mathbb{N}$ and $k<n$, $J$ the matrix of all ones. \begin{equation} \begin{aligned} & {\underset{A,B \in \mathbb{S}^n}{ ...
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1answer
29 views

Comparison of rank and eigenvalues of matrix AB and BA

I was unable to solve this question of linear algebra and hence I am posting it here. Let A and B be n$\times $ n matrices over $\mathbb{C}$.Then which of the following are true: A AB and BA have ...
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Can A Real Matrix having complex eigenvalues Have Real Eigenvector [closed]

Let A be a n×n matrix with real entries If it has Complex eigenvalues values . Can the eigenvectors be real??
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35 views

Operator Norm of an augmented matrix vs unaugmented matrix

This problem comes from Linear Algebra and its Applications by David C Lay and Steven R Lay. Show that $$||[A B]|| \geq ||[A]||$$ Where $[A B]$ is a a matrix $A$ augmented with another matrix $B$ So ...
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1answer
22 views

Greatest common divisor of Characteristic Polynomials

I am stuck with the following proof about matrices: Let $A\in \textbf{M}_{n}(k),B\in \textbf{M}_{m}(k)$ and $M\in \textbf{M}_{n\times m}(k)$ such that $AM=MB$. Prove that the degree of gcd{$ P_{A} ,P_{...
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1answer
27 views

Let $T:V \rightarrow V$ such that $T^2 = \frac{1}{2}T$. Find its characteristic polynomial.

I am used to dealing with transforms such as $T:P_2(\mathbb{R}) \rightarrow P_2(\mathbb{R})$ where $T(ax^2 + bx + c) = (a+b)x^2+cx$. In this case you would just use a basis of $P_2(\mathbb{R})$ and ...
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3answers
51 views

Geometric multiplicity for non zero eigen values of matrices $AB$ and $BA$.

As lot of information is given in this site about eigen values of $AB$ and $BA$ for square matrices $A$ and $B$. As characteristics polynomial of $AB$ and $BA$ are same so both have same set of eigen ...
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0answers
31 views

Can Newton's Law of Cooling be viewed as an eigenvalue problem?

Newton's Law of Cooling states that the rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its surroundings. Written as: $$\frac{\partial{u}...

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