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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15 views

Number of negative eigenvalues and Min-Max Principle

Let $H=(H, (\cdot, \cdot)_H)$ be a Hilbert space. Let $T_1,T_2:D \subset H \longrightarrow H$ be a self-adjoint operators (not necessarily bounded). It's well-know that the spectrum $\sigma(T_i)$ of $...
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Finding the lowest Eigenvalue of a direct product of two linear transformations

Suppose we have a linear transformation $H$ acting on some vector space $V$, specified by the direct product of linear transformations $H_1$ and $H_2$, acting on $V_1$ and $V_2$, respectively. Now ...
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To find the eigen values

I am going to find out the Eigen values of the matrix $$\begin{bmatrix} 6 & 5 \\ -8 & -6 \end{bmatrix}$$ and end up finding $\lambda = \sqrt{-4}$. Is this possible to get value like this?
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What does need to be considered for getting a non-trivial eigenspace?

Compute the non-trivial eigenspaces of $T∈ L(\mathbb{R}^2)$ where $T$ is defined by, $T(x,y) = (x+y,−y)$. My solution (without knowing what exactly is the non-trivial eigenspace): Consider for $...
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Reference Request: Measure convergence for spectrum of Stationary time series

Let $\{Y_t\}_{t\in \mathbb{Z}_{>=0}}$ be a centered, stationary random process, with autocovariance $R(t)=Cov(Y_0Y_t)$. The spectral density $S$ is defined as the Fourier transform of $R$, $S(\...
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23 views

Hermitian matrix and meromorphic function [closed]

Let $H\in \mathbb{C}^{n\times n}$ an hermitian matrix. I want to demonstrate that for $z\in \mathbb{C}, v\in \mathbb{C}^n$ the components of the vector $(H-zI)^{-1}v$ are meromorphic functions and ...
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49 views

Uniqueness of Eigenvalues

Given the $2\ \times\ 2 $ matrix $$\boldsymbol{A}=\begin{pmatrix}4 & -2\\\ 1 & 1\end{pmatrix}$$ I am aware that eigenvalues of matrices can be solved from the usual $p(\lambda)=|\boldsymbol{A}-...
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Eigenvalues of the sum of a matrix such that all its principal submatrices are stable, and a diagonal matrix with non-positive entries

I have a real square matrix $A$ (not necessarily symmetric) with all its principal submatrices (including $A$) having eigenvalues with a negative real part. On the other hand, I have a matrix $D$ ...
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31 views

What is the relationship between Rank of square matrix and its higher orders?

Suppose $A$ is a square matrix, and $\operatorname{Rank}(A^{2})=3$, then can we establish any relationship to determine the rank of $A$? If it helps, $A$ has three distinct eigenvalues and it is a $4\...
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How to choose a p.d. matrix $\Sigma$ s.t. $x^T\Sigma Mx>0$ for all $x$?

Assume $M \in \mathbb{R}^{n\times n}$ is a real square matrix, which is fixed and we also assume that its eigenvalues all have positive real parts. In particular, it is invertible. Then my question is ...
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83 views

$A^2 = AB, |(A-B)|$ and eigenvalues [closed]

Let $A, B$ be square matrices. $A \ne 0$ (as a matrix) and $A^2 = AB$. How can I prove that $ |(A-B)| = 0$? I think that key is a chain $|(A-B)|=\dots = |(B-A)|$ but not sure if it's the right way. ...
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Simultaneously Symmetric Matrices

I have two diagonalisable matrices $M$ and $N$ and would like to show that the eigenvalues (in order) of $M$ are greater than those of $N$. Now, $M-N$ has only non-negative eigenvalues, so if the two ...
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24 views

Show square form of symmetrical Matrix is $\geq 1$ as long as the smalles Eigenvalue of the Matrix is 1

Let $A$ be a symmetrical Matrix in $\mathbb{R}^{n*n}$. Also note that $q_A(x) := x^\top Ax$. We know that $\|x\| = 1$. Show that $q_A(x) \geq 1 $ if the smallest Eigenvalue of $A$ is $1$. What I know ...
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What is the problem of solving Infinite Square Well using finite difference methods? [closed]

I am learning resolve Schrodinger equation using finite difference methods according to a paper (https://arxiv.org/abs/0704.1622). %% code $L = 2\pi$; $N=500$; x=linspace(0,L,N)'; $dx = x(2) - x(1)$; ...
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Semi-stable fixed points in plane

Suppose that we have a system of 2 difference equations depending on some parameter and that $(x^*,y^*)$ is non-hyperbolic equilibrium point with one eigenvalue equal to 1 (which appears for the ...
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58 views

If $X^6+X^5-2X^4-X^2-X+2=0$, what values of $\mbox{tr}(X)$ are admissible?

I was asked the following question: Let matrix $X \in \Bbb R^{3 \times 3}$ be such that $$X^6+X^5-2X^4-X^2-X+2=0$$ Which of these is not a possible value of the trace of $X$? a. $-4$ b. $-2$ c. $0$ d....
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Is there is any relationship between number of zero rows in row echelon form of a matrix and multiplicity of eigenvalue $0$?

I have two questions. Questions: (1) Is there is any relationship between number of zero rows in row echelon form of a matrix and multiplicity of eigenvalue $0$? (2) Easy method for finding ...
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1answer
100 views

Infinite Product of 2 Matrices

Let us assume that I have the $n \times n$ matrices $A$ and $B$ where $n$ is finite. I construct the matrix infinite matrix product $P = \lim_{N \rightarrow \infty}\prod_{i=1}^{N}X_{i}$ and, for each $...
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An inequality of eigenvalues of $A$ and $U^TAU$, where $A$ is symmetric, $U^TU = I$.

The question is as follows: $A$ is a $J\times J$ symmetric matrix, $U$ is a $J\times K$ matrix where $K\le J$. Suppose that $U^TU=I_K$. Please show that $\lambda_{j}(U^TAU)\le\lambda_{j}(A)$ I have ...
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Eigenvalues of a triangular matrix from one base to another

@Gerry Myerson in the comments bellow offered a better formulation to my question ...
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Eigenvalue that depends on eigenvector

Consider the problem $$(\mathbf{1}^TAx) x=Ax,\tag{*}$$ where $A$ is a square matrix and $\mathbf{1}$ is a column vector of ones. It does not need to be of exactly this form, the main point is that we ...
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Two eigenvalues and an eigenvector walk into a bar…

Suppose I have the transformation $T(v) = Av = \lambda v$. If two of the eigenvalues are $\lambda_1$ and $\lambda_2$ where $\lambda_1=-\lambda_2$, is there a way to quickly find the eigenvector(s) for ...
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A question about the proof of Weyl's inequalities

If $A,B$ are $n\times n$ Hermitian matrices, then a version of Weyl's inequalities states that $$\lambda_k(A)+\lambda_n(B)\leq\lambda_k(A+B)\leq \lambda_k(A)+\lambda_1(B) \quad 1\leq k\leq n,$$ where $...
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FInding the number of eigenvalues of the given $\mathbb{C}$ linear transformation.

$T: \mathbb{C}[x] → \mathbb{C}[x]$ be the $\mathbb{C}$-linear transformation defined on the complex vector space $\mathbb{C}[x]$ of one variable complex polynomials by $T (f(x)) = f(x + 1)$. How many ...
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Prove AD+DA is positive definite

Suppose $A$ is a positive definite matrix such that $\lambda_{min}(A)\ge\frac12$ and $\lambda_{max}(A)\le 2$, and $D=A\circ I$ is the diagonal entries of $A$. I want to prove that $AD+DA$ is positive ...
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38 views

Identifying a set of real numbers which is associated with a square matrix A

Let $A=\begin{bmatrix} 3 & 1\\2&4\end{bmatrix}$ Find the eigen values of $A$. Then identify the set $\{a\in\mathbb R : \lim_{n\to\infty}a^nA^n$ exists and different from zero}. I have find the ...
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1answer
60 views

Algebraic multiplicity 1 of the maximal eigenvalue of matrix

When $α_j$ are real numbers positive for each $1 ≤ j ≤ n$ and $λ_{\max}\in \mathbb C $ maximal modulus eigenvalue of matrix A. I.e. $|\lambda_{\max}|=\max \{|\lambda_i|\}$ for $i=1,2,\cdots,n$ We want ...
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1answer
26 views

diagonalizability of a matrix depending on a parameter

I am trying to understand for which $a\in\mathbb{R}$ the matrix $\begin{bmatrix} a & 2 & -1\\ 1 & a & -1\\ 0 & 0 & 2\end{bmatrix}$ is diagonalizable and, in that case, which ...
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25 views

Matrix is orthogonally similar to a diagonal matrix

I have a problem understanding the following problem: Let $U$ be a non-zero vector of $\mathcal{M}_{n, 1}(\mathbb {R})$, of components $u_1,...,u_n$. We set $M = U^TU$. The eigenvalues and the ...
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Solving an $a \times (a-1)$ matrix of the $a-1$ eigenvectors. How?

I am considering $\mathrm{I_a}-a^{-1}\mathrm{J_a}=\mathrm{Q}\cdot\mathrm{I_{a-1}}\cdot\mathrm{Q^{\top}}$, where $\mathrm{I_a}$ is the identity matrix of size $a$, $\mathrm{J_a}$ is the all-ones matrix ...
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Eigen decomposition and Diagonalization [closed]

I am writing to ask if the eigen decomposition and diagonalization are the same concept. Also, how eigen decomposition is used in the PCA analysis,can anyone share the link about the explanation?
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1answer
40 views

Spectral radius of discrete Laplacian

We discretize the domain $\Omega = [0,1]^2$ with the step size $\Delta x = \frac{1}{n-1}$. Then, for the largest eigenvalue of the discrete Laplacian it holds $$\lambda_{\max}(-\Delta)\leq 8(n-1)^2.$$ ...
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IF absolute square of all eigenvalues of $A$ is smaller than 1, is absolute square of all eigenvalues of $A-A^T$ also smaller than 1? [closed]

Given the matrix $A$, whose all eigenvalues lie in the unit circle, or absolute square of all eigenvalues of $A$ is smaller than 1. Can I show that absolute square of all eigenvalues of $A-A^T$ also ...
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$a\geq 0$ is a eigenvalue of $A\bar A$ iff there exists $X\neq 0$, such that $A\bar X=\sqrt{a}X$.

$a\geq 0$ is a eigenvalue of $A\bar A$ iff there exists $X\neq 0$, such that $A\bar X=\sqrt{a}X$. Here $A$ is a square matrix, $\bar A$ is the conjugate of $A$. It sounds like that the singular ...
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1answer
31 views

$[A, B] = 0 \Rightarrow E_\lambda(A) \cap E_\mu(B) = E_{\lambda\mu}(AB)$?

$E_\lambda(A)$ stands for the eigenspace of an operator (say, a matrix) with eigenvalue $\lambda$. It probably doesn't matter whether the antecedent stands for the commutator or a general Lie bracket, ...
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What is my error in computing the singular value decomposition

I wrote a small python program to compute the singular value decomposition of a matrix $A$. It is based on the idea of computing the eigenvectors of $A^TA$ and $AA^T$ (for details see here). However ...
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1answer
37 views

Minimum of $x^TWx$

I have one question on the proof of the problem below. For any symmetric matrix $W$, the minimum of $x^TWx$ when $||x||_2 = 1$ is achieved at the eigenvector $v_n$ corresponding to the minimum eigen ...
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Min max inequality for dependent matrices

Consider the positive semidefinite matrix $R(x)\geq0$, $\forall x\in\mathbb R^n$. Furthermore, consider the pos. semidefinite matrix $A\geq0$. From the min-max theorem it is known that $\lambda_{\min}...
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Characteristic Equation for Traceless Matrices

The characteristic polynomial for traceless $2\times 2, 3\times 3, 4\times 4$ matrices $A$ are \begin{align} x^2+&\det A \\ x^3-\frac{1}{2}{\rm Tr}A^2 x - &\det A \\ x^4-\frac{1}{2}{\rm Tr}A^...
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A “straightfoward” statement about eigenvalues

I'm reading a paper about clustering techniques and I found the following easy construction Let $S = \{s_1,\dots , s_n\}$ be a ser of $n$ points where each $s_i \in \mathbb{R}^2$. Suppose that $S = ...
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What is the smallest eigenvalue of this infinite, symmetric, tridiagonal matrix?

$A_{nm}$ ($n,m = 0, 1, 2, \ldots$) is a symmetric, tridiagonal matrix. The diagonal elements are $A_{nn} = a_n = n + 1$, and the off-diagonal elements are $A_{n,n+1} = A_{n+1,n} = b_n = \lambda \sqrt{\...
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What does it mean for Katz Centralities to “diverge”?

In Mark Newman's Networks book, 2010 edition, page 173, he explains some mathematical details behind the Katz Centrality measure: In matrix terms, Eq. (7.8) can be written x = αAx + β1, (7.9) where 1 ...
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73 views

Show the eigenvalues of a square matrix are between zero and one.

Let $A$ and $B$ be two positive definite matrices of orders $n$ and $d,$ respectively. Let $X$ be a matrix of type $(n,d)$ ($n$ rows and $d$ columns). Is it true that $A^{-1} X B^{-1} X^\intercal$ ...
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1answer
42 views

A proof about generalized eigenvalue problem

I'm considering the following model. Given $x_i\in\mathbb{R}^n$, $i\in \{1,2,\ldots,N\}$, where $n\leq N$. $X=[x_1,x_2,\ldots,x_N]$ and a full-rank $W\in \mathbb{R}^{N\times N}$ $P=[p_1,p_2,\ldots,...
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Lower & upper bounds on $\|Av\|$? [closed]

Suppose that $A$ is a positive definite matrix of dimension $n\times n$, and $v$ is a vector in $\mathbb R^n$. Exploiting the positive definite properties of $A$, what are the lower and upper bounds ...
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4answers
113 views

Find trace and determinant of matrix $A$ such that $A^2 = I$.

I have a $ 2\times 2$ matrix $A$, where $A^2 = I$. So the eigenvalues are $\lambda= \pm1$ . I need to find its trace and determinant. There's no mention of upper or lower triangular matrix, therefore,...
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2answers
59 views

Cyclic Vector and Diagonalizable operator

I've been working on this exercise: Prove that a diagonalizable operator $T$ on an $n$-dimensional vector space has a cyclic vector iff it has $n$-distinct eigenvalues. I tried to do the following: ...
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1answer
124 views

If A is the 4 by 4 matrix of ones, find the eigenvalues and the determinant of A−I

So I want to find the eigen values and eigen vectors of a matrix with all 1's \begin{bmatrix}1&1&1&1\\1&1&1&1\\1&1&1&1\\1&1&1&1\end{bmatrix} Only 1 ...
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3answers
72 views

$\phi$ is diagonalizable iff $\lambda_1, \lambda_2,\lambda_3$ are distinct

We have the matrix $A=\begin{pmatrix}\lambda_1 & 1 & 0 \\ 0 & \lambda_2 & 1 \\ 0 & 0 & \lambda_3\end{pmatrix}$ and $\phi (x)=Ax$, where $\lambda_1, \lambda_2,\lambda_3\in \...
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2answers
92 views

Why is $\phi$ diagonalizable if $\phi \circ \phi =id_V$?

V is a finite-dimensional $\mathbb{Q}$- vector space with $\phi: V \rightarrow V$ Why does it follow that $\phi$ is diagonalizable if $\phi \circ \phi = id_{V}$? My ideas so far: I do know that if i ...

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