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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1answer
25 views

Are all eigenvalues of $((C^TQ_fC)^{-1} (C^TQ_gC))$ lie in $(0,1]$?

Consider a matrix $X_f = (C^TQ_fC)^{-1}$ and $X_g = (C^TQ_gC)^{-1}$, where $C \in \mathbb{R}^{n \times m}$ is a full column tall matrix ($m < n$). $Q$ ($\in \mathbb{R}^{n \times n}$) is a diagonal ...
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1answer
19 views

Find Square root of a matrix from its spectral decomposition

Let $\rho$ be $n\times n$ symmetric matrix, thus the spectral decomposition of $\rho$ is $$ \rho=\sum _{i=1}^n e_i\left|\psi _i\right\rangle \left\langle \psi _i \right| $$ where $e_i$ and $\psi _i$ ...
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1answer
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Eigenvalues of the sum of a positive and a positive semidefinite matrix

Let $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times n}$ respectively a positive and a positive semi-definite matrix. Is it possible to establish an upper bound for the minimum eigenvalue of the ...
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1answer
13 views

“Diagonalisation” of second order linear PDEs

Define the operator $$ Pu=\sum_{ij} a_{ij}(x) \partial^2_{ij} u+\sum_k b_k(x) \partial_k u+c(x) u. $$ My question is, in what situation can we somehow "diagonalise" the principle part of $P$,...
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0answers
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Is there a geometric interpretation of eigenvalues of integer matrices?

In some instances, like physics, you may find that quantities you are after are eigenvalues of matrices. However, for example, explaining that "the mass of a muon is an eigenvalue of a matrix&...
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1answer
63 views

Commuting Matrices with Complex Eigenvectors

Main Question Say that $K$ and $S$ are two commuting matrices with a full set of real eigenvectors, where the dimension of the eigenspaces of the matrices may be greater than one. $$KS=SK$$ It is a ...
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2answers
46 views

Show that $T \in \mathcal{L}(\mathbb{R}^4)$ with det $T<0$ has at least two distinct eigenvalues.

a) Suppose that $T \in \mathcal{L}(\mathbb{R}^4)$ satisfies det $T < 0$. Show that $T$ has at least two distinct eigenvalues. b) Find a $T \in \mathcal{L}(\mathbb{C}^4)$ where det $T < 0$ such ...
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2answers
58 views

Eigenvector of a matrix of all 1's

Consider the matrix $A \in \mathbb{R}^{n \times n}$ of all ones. Because there is only 1 linearly independent column, there are $n-1$ zero eigenvalues and 1 non-zero eigenvalue which is $n$. So one ...
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1answer
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What is a simple definition of Lanczos iteration that is understandable?

I have seen that Lanczos's Algorithm can be used to tri-diagonalize a matrix but all of the definitions I have seen of it have been very complicated to understand.
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1answer
20 views

Redundant eigenfunctions of an operator

In quantum mechanics, the eigenvalues and eigenfunctions of the operator $\hat{L}_z$ can be calculated by solving the differential equation $$ -i\hbar\dfrac{\partial\Phi}{\partial \varphi} = \lambda\...
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27 views

Dimension of difference of eigenspace and span of a eigenvector.

Let $\lambda$ be an eigenvalue of $A$. Let $r$ be the geometric multiplicity of $\lambda$. Show that the dimension of the space of the left eigenvectors of $A$ corresponding to $\lambda$ is also $r$. ...
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1answer
20 views

Support of a generalized eigenvector

Let $A$ be a square matrix with eigenvalue $\lambda$. Let $\mathbf{c}$ be such that $\mathbf{v} := (A - \lambda I) \mathbf{c}$ is an eigenvector, but $\mathbf{c}$ is not an eigenvector. That is, $\...
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85 views

What does $\| W \|$ mean in math?

I've been reading a paper to replicate an experiment and it says that $\| W \| = 1$, where $W$ is the value of a vector that has eigenvalues and eigenvectors. I'm a bit rusty, but I know that $| W |$ ...
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1answer
23 views

Proving that $\det(I-T_i) >0$ where $T$ is a primitive stochastic matrix and $T_i$ is a principal submatrix

Let $T$ be an $n \times n$ row-stochastic matrix which is primitive (i.e. there is a positive integer $k$ such that all entries of $T^k$ are strictly positive). Let $T_i$ denote the matrix obtained by ...
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48 views

Derivative of eigenvalues and eigenvectors of real symmetric, positive definite matrix?

Given a system of linear differential equations specifying the time evolution of a covariance matrix, how to rewrite it in terms of differential equations for that matrix's SVD? You are given a ...
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37 views

What does this notation mean? Up-arrow on eigenvalues

I am reading this paper, and they use this up-arrow notation on each eigenvalue. What is the meaning of this? (Screenshot attached below)
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27 views

Finite difference of eigenvalue problem with complex discrete eigenvalues

Suppose we consider solving the following eigenvalue problem $$ -\Delta u=\lambda u $$ with Neummann boundary condition $\frac{\partial u}{\partial \nu}=0$ (on $[0,1]$ for example). Then the analytic ...
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Solving $\partial P(x,t)/\partial t = \left( D \partial^2 /\partial x^2-\mu \partial /\partial x \right) P(x,t)$ with no-flux boundary conditions

In a mathematical physical problem, I would like to solve the following partial differential equation of type Fokker-Planck: $$ \frac{\partial P(x,t)}{\partial t} = \hat{H} P(x,t) \, , $$ with the ...
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Axler on Invariant Subspaces IFF There is an Eigenvalue

On page 134 of the 3rd edition of Linear Algebra Done Right, Professor Axler makes the following statement (where "The comments above" refers to a brief and trivial proof of the assertion, ...
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Can eigenvectors be scaled and still be eigenvectors?

Given the matrix: $$ A = \begin{bmatrix} 2 & 0 \\ 1 & 4 \end{bmatrix}. $$ The eigenvalues are: $$λ_1 = 2,$$ $$λ_2 = 4.$$ To find the eigenvectors: $$v_1 = \operatorname{nullity}(A - λ_1I),$$ $...
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1answer
38 views

Analytic solution of the eigenvalue problem of a special matrix

Let $$A=(i\wedge j)_{1\leq i,j\leq n}$$ This is the covariance matrix of $(W_1,...,W_n)$ where $W_t$ is the standard Brownian motion. What is the eigenvalue and eigenvector of $A$? I believe that ...
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1answer
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Show an orthonormal basis is a set of eigenvectors

Consider the orthonormal basis $B= \{ {\bf u}, {\bf v}, {\bf w} \}$ for $\mathbb{R}^3$ and the transformation $T: \mathbb{R}^3 \to \mathbb{R}^3$ defined by $$T({\bf x}) = {\bf x} - 2 ({\bf x} \cdot {\...
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23 views

Relationship between eigenvectors of a covariance matrix and its component covariance matrices

I have a covariance matrix $\mathbf{S} = \mathbf{X}^T \mathbf{X}$. I then divide the matrix $\mathbf{X} \in \mathbb{R}^d$ that has $N$ rows into $n$ portions $\mathbf{x_i} \in \mathbb{R}^d$ each ...
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48 views

Question about a proof of the Perron Frobenius Theorem

I am going through a proof of the Perron-Frobenius theorem that I found online, but I'm having trouble understanding some of the steps. Here is the relevant info: Let $T$ be a primitive row-stochastic ...
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0answers
23 views

Eigenfunctions and eigenvalues of bidimensional operator in $Z^+$

I'm dealing with some discrete operators and I'm having some difficulties to find an expression or a method to find a set of eigenfunctions and eigenvalues for the bidimensional operator $S$ defined ...
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2answers
40 views

Are all strictly positive semi-definite matrices singular?

If I have some matrix A with an eigenvalue of 0, what makes this matrix singular? and I am assuming All positive definite matrices are non singular so all strictly positive-semidefinite matrices would ...
2
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2answers
59 views

Eigenvalues with positive real part for a matrix product

Let $M \in \mathbb{R}^{d\times d}$ be an invertible real matrix (not necessarily symmetric), and assume $M$ is positive semi-definite in the sense that $$ v^T M v \geq 0 $$ for all $v \in \mathbb{R}^d$...
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1answer
40 views

A question based on linear transformation from $\mathbb{R}^{2}$ to $\mathbb{R}^{2}$ with its 2 eigenvalues given

This question was asked in my linear algebra quiz and i am unable to solve it . Let $A: \mathbb{R}^{2} \to \mathbb{R}^{2}$ be a linear transformation with eigenvalues $2/3$ and $9/5$ . Then , show ...
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3answers
197 views

Complex eigenvalues of a matrix in conjugate pairs (or not)

I have learnt that in a matrix, if there are complex eigenvalues, they should come as conjugate pairs. Also, I know that, in a diagonal matrix, eigenvalues are the diagonal elements. So how about the ...
2
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1answer
45 views

Sum of row entires of a real symmetric matrix

So let $A$ be an $n\times n$ real symmetric matrix with eigenvalues $\lambda_1\geq...\geq\lambda_n$, and the sum of all entries is $s$. I'd like to prove that: If either 1) $\lambda_1=s/n$ or 2) $\...
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1answer
66 views
+50

Characterization for invariant subspace

I have the following problem consisting of $3$ parts of which I'm not being able to figure out the last. Notation: $T^*$ is the adjoint operator of $T$. $$\text{im}(T) = \{T(v) : v \in V\} \quad \text{...
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0answers
26 views

Eigenvectors for eigenvalues in algebraic closure

For a matrix $A$ acting on a $n$-dimensional vector space $V$ over arbitrary field $F$, by considering the characteristic polynomial, we can find its eigenvectors $\lambda_1,\dots,\lambda_n\in\bar{F}$ ...
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1answer
15 views

Left eigenvector for eigenvalue < 1 for a square stochastic matrix: coordinates of eigenvector sum to zero.

If $v$ is a left eigenvector of stochastic matrix $P$ with $vP = \lambda v$ for $\lambda <1$, can you show that $\sum_{i = 1}^{N} v_{i} = 0$. You can assume that $v$ is normalized.
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2answers
32 views

prove if $[A,B] = 0$ and $Af=\lambda f$ then $f$ is an eigenfunction both for $A$ and $B$

let commutator of two operators A, B be $[A, B] = AB - BA$ prove: if $[A,B] = 0$ and $Af=\lambda f$ and there are no repeated eigenvalues, then $f$ is eigenfunction both for $A$ and $B$ My book gives ...
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3answers
56 views

Need help to understand a solution to a polynomial problem.

Let $r$ and $s$ be roots of $x^2-(a+d)x+(ad-bc)=0$. Prove that $r^3$ and $s^3$ are the roots of $y^2-(a^3+d^3+3abc+3bcd)y+(ad-bc)^3=0$. The solution given : The solution didn't give full details. It ...
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1answer
27 views

Reference: the first eigenvalue of $-\Delta_p$ operator

I look for a book or a paper about the first eigenvalue of the $-\Delta_p$ operator. To be more precise, I am interested in understanding what is the relation between the first eigenvalue of the $-\...
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1answer
40 views

Prove that there exists a positive integer $m$ such that $\left\|T^m(v)\right\| \le \epsilon\left\|v\right\|$ for every $v \in V$

I've been stuck on a linear algebra problem from Axler's Linear Algebra Done Right. The problem statement is as follows: Suppose $F=\mathbb{C}$, $V$ is finite-dimensional, $T \in \mathcal{L}(V)$, all ...
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1answer
20 views

A lower bound on the largest eigenvalue of a symmetric matrix

I am trying to prove the following: Let $A$ be an $n \times n$ real symmetric matrix with eigenvalues $\lambda_1 \geq \cdots \geq \lambda_n$; the sum all entries in $A$ is $s$. Prove that $\lambda_1\...
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2answers
46 views

A $2\times 2$ matrix $A$ has eigenvalues $e^{i \frac{\pi}{4}}$ and $e^{i \frac{\pi}{5}}$ such that $A^n = I$. Find the smallest possible value of $n$.

A $2\times 2$ matrix $A$ has eigenvalues $e^{i \frac{\pi}{4}}$ and $e^{i \frac{\pi}{5}}$ such that $A^n = I$. Find the smallest possible value of $n$. How to find the smallest value of $n$?
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0answers
46 views

Finding the associated unit eigenvectors from $A=\left(\begin{smallmatrix}4&0&-2\\0&2&2\\-2&2&3\end{smallmatrix}\right)$

My work so far The eigenvalues are $\lambda_1=0,\lambda_2=3,\lambda_3=6$ And, the eigenvectors are $$\lambda_1=\begin{pmatrix}\frac{1}{2}\\-1\\1\end{pmatrix},\lambda_2=\begin{pmatrix}2\\2\\1\end{...
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0answers
10 views

How to control second eigenvalue of X, given second eigenvalue of Y and spectral norm of X-Y?

Let's say X and Y are real symmetric matrices. Assume $\lambda_1 \le \ldots \le \lambda_n$ for all matrices involved. I need to estimate $\lambda_2(X)$, and some of the information I have are bounds ...
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0answers
29 views

Numerically Computing the eigenvector of a large sparse matrix when the eigenvalue is known

I am running numerical simulations on large, sparse, Markov matrices $P$ and want to understand the steady state equilibrium of said Markov matrices. Since these matrices are Markov, all the columns ...
4
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1answer
39 views

Confused with this SVD problem: Does it matter which singular vectors you choose?

I am trying to decompose the following matrix using the Singular Value Decomposition (SVD): $$A = \begin{bmatrix} 4 & 4\\ -3 & 3\\ \end{bmatrix} = U\Sigma V^T$$ Here is my work (I ...
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0answers
39 views

Prove that this matrix cannot have complex eigenvalues

This question was asked in my Linear Algebra assignment and I was unable to solve it. Let $a, b, c \in \Bbb R_{>0}$ such that $b^{2} + c^{2} < a < 1$. Consider the following $3 \times 3$ ...
0
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1answer
28 views

Normalising eigenvector to length 1

For this matrix: $\begin{bmatrix}2 & -4\\-4 & 8\end{bmatrix}$ The eigenvalues are 0 and 10. The first eigenvector is then: 0 $\begin{bmatrix} 2 \\ 1 \end{bmatrix}$ The normalised eigenvector ...
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4answers
45 views

A simple question on eigenvalues.

I started linear algebra this semester and I just had a thought. A real number λ is an eigenvalue of A if and only if there is a non-zero vector x where (A-λ)x=0 Does that mean we can imply A=λ?
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1answer
42 views

What is the intuition behind the outer product of two eigenvectors?

I know that the outer product of every two eigenvector forms a 2-D basis for the 2-D matrices. For example, when we write a matrix based on its eigenvectos, we have: $$ X = \sum_{i,j} \lambda_{i,j}...
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0answers
18 views

Write a 2-D matrix based on the sum of some 2-D basis

I have a 2-D matrix ($P \in \mathbb{R}^{N\times N}$) and I have $N$ basis with size $N\times N$ called $B_i$. I want to write $P$ based on a weighted sum of $B_i$s as follow: $P = \sum_i c_i B_i$ I ...
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2answers
66 views

About rank of matrix

Let $A \in M_{6}(\mathbb{R})$ and $A^{3}-2 A^{2}-15 A=0$. If $\operatorname{tr}(A)=4,$ find $\operatorname{rank}(A)$. How we can solve this? I think we have $x(x^2-2x-15)=0$ so $x=-3 ,0, 5$ so ...
0
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1answer
24 views

Finding sign of unsolvable eigenvalues and eigenvectors

Given the following system of differential equations $$ \begin{bmatrix} x'\\ y' \end{bmatrix}=\begin{bmatrix} -a & b\\ r&-(b+c)) \end{bmatrix}\begin{bmatrix} x\\ y \end{bmatrix} $$ I need ...

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