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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Eigenvalue and spectral condition

Let $A=\begin{pmatrix} 1& 1 \\ a^2 &1 \end{pmatrix} \text{ with } a\in (0,\frac{1}{2}]$. Show $$cond_2(A)=||A||_2 \cdot ||A^{-1}||_2\leq 4(1-a^2)^{-1}$$ by first showing $||A||^2_2\leq||A||_1||...
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3 votes
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Is the minimum eigenvalue of "$-|H|$" ever larger than that of $H$?

Consider a Hermitian matrix $H = (h_{ij})$ and its smallest eigenvalue $\lambda_1$. Construct the matrix $H' = (-|h_{ij}|)$ and consider its smallest eigenvalue $\lambda'_1$. All of its eigenvalues ...
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Eigenvalues after multiplication with permutation matrix.

Let A be a diagonalizable matrix, and P be permutation matrix of same size. Does A and PAP have the same eigenvalues (or characteristic polynomial)?
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2 votes
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Absolute value of all the elements of Hermitian matrix - when can the lowest eigenvalue shrink?

Consider some $N \times N$ Hermitian matrix $H = (h_{ij})$ and order its eigenvalues from least to greatest, $\lambda_1, ..., \lambda_N$. When does the matrix $H' = (|h_{ij}|)$ have a smaller smallest ...
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On Weyl's majorant inequality

In this paper, the author uses the Weyl majorant inequality... (page 8, lemma 4.5) I don't know how to prove this theorem and I can't find a simple proof. Can someone explain me a simple proof ?
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31 views

Express a normal linear transformation as a linear combination of projections.

There's the following theorem: Let $T:V\rightarrow V$ be a normal linear transformation $(TT^{*}=T^{*}T)$. Prove there exists $E_1,...,E_k$ such that: $E_i^{2}=E_i$ $E_iE_j=0$ for all $i\neq j$ $\...
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4 votes
4 answers
97 views

Eigenvectors & eigenvalues of "nearby" matrices

Suppose that I have a square matrix $A$ and another square matrix $B$ whose entries differ by $\varepsilon>0$. Is there any way to bound the differences in their eigenvalues and unit eigenvectors? ...
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2 votes
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Show that $\lambda_j(Q'_nA_nQ_n)-\lambda_j(A_n)\to 0 \quad \text{as}\quad n\to \infty$

Let $(A_n),(Q_n)$ be sequences of $k\times k$ real matrices. Moreover suppose that $(A_n)$ is symmetric and bounded, and that $Q_nQ'_n\to I_k$ as $n\to \infty$, where convergence is with respect to ...
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Any thought on that rank similar object?

Context : This problem arises on some exploration around Wyner's common information in information theory and the related minimization problem. Problem : Let $A$ be a $m\times n$ real matrix. Its ...
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If a skew-symmetric real matrix has all eigenvalues zero, must it be the zero matrix?

This can be easily verified for $2\times2$ and $3\times3$ matrices, but can the result be generalised?
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Systems of differential equations and spectral theory.

Say that we have the system of differential equations in matrix formulation: $$ \begin{bmatrix} C(t) \\ C_p(t) \end{bmatrix}' = \begin{bmatrix} -k_{cp}-k_{ce} & k_{pc} \\ k_{cp} &...
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Invertible linear operators without eigenvalues [duplicate]

Suppose $V$ is a vector space over the complex numbers, and let $\alpha: V \mapsto V$ be an invertible linear operator. If the dimension of $V$ is finite, we know that $\alpha$ has (nonzero) ...
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6 votes
2 answers
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Smallest eigenvalue of a nearest neighbor matrix in $2$ dimensions.

Consider a 2D square lattice with $n \times n$ lattice sites. A matrix $M_n$ of size $n^2 \times n^2$ is constructed by setting $M_{ij} = u$ (where $0 \leq u \leq 1$) if sites $i$ and $j$ are nearest ...
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3 votes
3 answers
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Is there a method to calculate the eigenvalues for this a n x n symmetrical matrix

I'm working with a mechanics problem where I try to find the eigenmodes of the system. The system contains of $n$ masses all connected with springs to one another (same spring constant $k$), the outer ...
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1 vote
1 answer
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Are the determinants of a matrix and the diagonal matrix obtained after diagonalization equal? [closed]

This question is related to a derivation step needed to find an n-dimensional generalization of the Gaussian integral, derived here: reference for multidimensional gaussian integral Is it true that ...
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Proof this Endomorphism has an eigenvalue

Let $\mathbb{F}=\mathbb{R}, \Phi:\mathbb{R}^7\rightarrow\mathbb{R}^7$ be an endomorphism, Proof that there exists an eigenvalue of $\Phi$ I know that every function with an odd degree takes on the ...
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The eigenvalues of a matrix composed of powers

During my research work related to the convergence of estimates, I needed to calculate the eigenvalues of the following symmetric matrix. $$\Sigma_{n\times n} := \begin{pmatrix} 1 & \frac{a}...
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0 votes
1 answer
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Maximizing generalized Rayleigh quotient with constraints

Let $A$ and $B$ be $n \times n$ symmetric matrices with real entries and let $k \geq 2$ be an integer. I want to find the maximum of $$ \frac{\sum_{i=1}^k X_i^{\mathrm T}\,A\,X_i}{\sum_{i=1}^k X_i^{\...
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-2 votes
0 answers
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Show that applying a function $f(x)$ to the matrix $A$ gives $f(A) = V f(D)V^{−1}$ where $A = V DV^{−1}$. [closed]

Recall that a matrix $A$ has an eigen-decomposition $A = V DV^{−1}$, where $D$ is diagonal and $V$ is the matrix of eigenvectors. (a) Applying a function $f(x)$ to the matrix $A$ gives $$f(A) = V f(D)...
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Routh-Hurwitz criterion without computer algebra

Let $M:=\begin{pmatrix} -\lambda_1 &0&0&0&0&0&\lambda_{10} \\ \lambda_1 & -\lambda_2-\lambda_4-\lambda_7 & \lambda_3 &0&0&0& \lambda_{9} \\ 0 & \...
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1 vote
0 answers
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Relating Trace and determinant of dynamic matrices to stability of ODE in high dimension

Given a n-dimensional linear system of ODEs $y' = Ay + b$ where $A$ is an $n \times n$ square matrix of coefficients dependent on model parameters. Can we deduce anything about the stability and ...
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-1 votes
0 answers
28 views

Find real-valued closed formulas for the trajectory x(t+1)=Ax(t), where

Find real-valued closed formulas for the trajectory $x(t+1)=Ax(t)$, where $$ A=\pmatrix{−4 & 3 \\ -3 & −4}\quad \text{and}\quad x(0)=\pmatrix{1 \\ 0} $$ (image link)
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Is the relation $\det M_n = \prod_i \lambda_i$ general? [duplicate]

I read in my book that the eigenvalues of a matrix $M_n$ which size is $n$ are linked by: $$\det M_n = \prod_i^n \lambda_i$$ This is quite usufull when $n = 2$ and $\det M = 1$. I wondered if this ...
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3 votes
2 answers
83 views

Eigenvalues of a sparse 8x8 matrix

I have the following $ 8 \times 8 $ sparse matrix $ P=\begin{bmatrix} 0.5 & 0.5 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.5 & 0.5 & 0.0 & 0....
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0 votes
0 answers
17 views

Finding the maximum of quadratic forms

For some real vector $x \in \mathbb{R}^p$ and $p \times p$ positive definite real matrices $A_1, A_2, \dots, A _n$, consider another vector of quadratic forms: $$ (x^\top A_1 x, \dots, x^\top A_n x). $...
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0 votes
3 answers
88 views

Do you get a Eigen value without the respective Eigenvector?

I was solving some problems on diagonalization of matrixes, and I came across a particular question which seemed a bit odd. $$\left[\begin{array}{rrr} -8 & -6 & 2\\ -6 & 7 & -4\\ 2 &...
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2 votes
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What characterizes the set of matrices that admit non-negative eigenvectors (w. positive eigenvalue)?

Let us define the sets $$S = \{ M \in \mathbb{R}^{n \times n} | \exists \lambda, v \in \mathbb{R}_{\geq 0}^n : Mv = \lambda v\}$$ $$P = \{ M \in \mathbb{R}^{n \times n} | \exists \lambda \in \mathbb{R}...
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1 vote
2 answers
25 views

Showing an equation is solved by unit eigenvectors

Show that, for a symmetric matrix $A$, the equation $$(x^T x) A x + \left( \left( x^T x \right)^2 - x^T A x - 1 \right) x = 0$$ is solved by unit eigenvectors of $A$. A simple rearrangement shows ...
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2 votes
1 answer
82 views

Eigenvalue problem of time dependent Hamiltonian

I would like to solve an eigenvalue problem of a Hamiltonian. I was able to find the lowest eigenvalue by converting the Hamiltonian into a matrix and applying linear algebra eigenvalue techniques. ...
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1 vote
1 answer
23 views

Can we say anything about the eigenvalues/eigenvectors of a matrix composed by submatrices in a 'circulant' way?

Circulant matrices, that is matrices in $\mathbb{R}^{n \times n}$ of the form $$\begin{pmatrix} c_0 & c_1 & c_2 & ... & c_n \\ c_n & c_0 & c_1 &... & c_{n-1} \\ c_{n-1} ...
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2 votes
1 answer
46 views

Proving $\lim_{n \to \infty} A^n=0$ iff $\rho(A)<1$ for a square matrix $A$.

I recently came across a theorem which says: Let $A$ be a square matrix. Then, $\lim_{n \to \infty} A^n=0$ iff $\rho(A)<1$ I get that for a diagonizable matrix, we can find an invertible matrix $...
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1 vote
0 answers
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Sufficient conditions for an indefinite real symmetric matrix

Given a real symmetric matrix $S$ of order $n$, assuming $\mathbf{x} \equiv \mathbf{e}_i$ with $i \in [1,\,n]$ we have: $$ \mathbf{x}^t\,S\,\mathbf{x} = s_{i,i} $$ so if $s_{i,i}=0$ for at least one $...
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0 votes
1 answer
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Time complexity of getting $r$-largest eigenvalues and vectors of a symmetric matrix.

$A$ is a $n \times n$ symmetric matrix. I would like to know the time complexity of calculating $r$-largest eigenvalues and vectors. When we need all eigenvalues and eigenvectors, it means $r=n$, I ...
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0 votes
1 answer
33 views

Finding Matrix Corresponding to Ellipses of Cost Function

For a given cost function $$J(w)=(w-w_0)^TA(w-w_o)$$ it is known that contours of J(w) are ellipses with principal directions have angle of 45° and -45° with the horizontal axis. If eigenvalues of A ...
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0 answers
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Graph Laplacian for weight matrix with negative edges

How can I normalize my weight matrix to get a positive semi-definite Laplacian, if I am using a weight matrix with negative edges?
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0 answers
18 views

An eigenfunction inequality for integral operators

I have an integral operator $T$ defined with respect to a positive semidefinite kernel function $k: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ probability measure $\mu(dx) = p(x) dx$ defined ...
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1 vote
1 answer
53 views

Eigenvalues of sum $\lambda_i(X(D_1 + D_2)X^T) = \lambda_{\sigma_1(i)}(XD_1X^T) + \lambda_{\sigma_2(i)}(XD_2X^T)$ where $D_1, D_2$ are diagonal.

Let $D_1, D_2\in\mathbb{R}^{d}$ be positive definite diagonal matrices. Let $X\in\mathbb{R}^{n\times d}$ satisfy $\text{rank}(X) = n < d$. Is it true that $$\lambda_i(X(D_1+D_2)X^T) = \lambda_{\...
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3 votes
1 answer
51 views
+50

How to compute principal components for a curvature found given XYZ points?

I have a certain XYZ set of points that make up an object. I chose a random point and make the nearest radius analysis and find the neighbors. From these neighbors, I get the green pointcloud curve ...
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1 vote
0 answers
48 views

Restrict mapping of symmetric matrix to eigenvectors to make eigenvectors differentiable

For computational reasons I must map a $3 \times 3$ symmetric, traceless matrix into $\mathbb{R}^2 \times \mathbb{R}^3$ where the components of $\mathbb{R}^2$ are two of the eigenvalues (the third is ...
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0 votes
1 answer
22 views

Does changing the order of rows and columns (in the same way) of a block diagonal matrix change its eigenvalues?

If we speak of any block diagonal matrix, simply switching its rows, the answer is generally yes. Since I can give an example: just take $I$ the $2 \times 2$ identity matrix (eigenvalues 1 and 1) and ...
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2 votes
2 answers
48 views

How come this matrix with non-degenerate eigenvalues has two sets of possible eigenvectors?

I expect that when we have a $n \times n$ matrix with non-degenerate eigenvalues, that is to say a matrix for which none of its $n$ eigenvalues have the same value, that there is a unique set of n ...
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0 answers
21 views

Eigenvalues of symetric matrix and eigenvalues of symetric matrix plus rank one matrix

If $M$ is a symetric matrix with eigenvalues $m_1 \geq m_2 ... \geq m_n$ is there any connection between those eigenvalues and the eigenvalues $m'_1 \geq m'_2 ... \geq m'_n$ of the matrix $M' = M + xx^...
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1 vote
2 answers
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Can a small perturbation of a diagonal matrix increase its smallest eigenvalue to any arbitrarily large value?

Let $S\in\mathbb{R}^{n\times n}$ be a diagonal positive semidefinite matrix with exactly $k$ positive entries in its diagonal, where $k<n$. Let $\epsilon$ be any arbitrarily small positive real ...
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1 vote
1 answer
107 views

Maximal and minimal eigenvalues of a symmetric tridiagonal Toeplitz matrix

Given $m \times m$ symmetric tridiagonal Toeplitz matrices $$M=\begin{pmatrix} 4 & 1 & & \\ 1 & 4 & \ddots & \\ & \ddots & \ddots & 1\\ & & 1 & 4\...
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1 vote
0 answers
53 views

Is the assumption that $V$ is finite-dimensional really necessary in Exercise 5.A.28? (Sheldon Axler "Linear Algebra Done Right 3rd Edition")

I am reading "Linear Algebra Done Right 3rd Edition" by Sheldon Axler. The author assumed that $V$ is finite-dimensional in Exercise 5.A.28. But I don't think that this assumption is ...
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0 answers
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Singular vectors from true and sample covariance do not align

Suppose we have a covariance matrix $C\in\mathbb{R}^{p\times p}$ which is diagonal with exponential decaying values. This covariance matrix is used to sample $n$ data points $x\sim\mathcal{N}(0,C)$ ...
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0 votes
1 answer
65 views

Confusing statement regarding Routh-Hurwitz criterion

I'm currently reading "On the Solutions and the Steady States of a Master Equation" by Joel Keizer. Keizer introduces a matrix $\Lambda$ with the following properties: $-\Lambda_{ij} \geq 0$ ...
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0 votes
1 answer
35 views

Change of eigenvalues under near orthogonal matrix multiplication

Let $A,B\in\mathbb{R}^{d\times d}$ be positive definite diagonal matrices. Let $\Phi\in\mathbb{R}^{n\times d}$ satisfy $\text{rank}(\Phi) = n \leq d$ and be such that $\Phi\Phi^T = I_n$. When $d = n$, ...
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  • 33
1 vote
1 answer
38 views

Showing 1 cannot be an eigenvalue after rotating an operator

I have a compact self-adjoint positive integral operator $Q:L^2(0, \infty) \to L^2(0, \infty)$ with operator norm $\| Q\| =1$. By the assumptions, we know $1$ is an eigenvalue of $Q$. Let $y\ne 0$ and ...
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1 vote
1 answer
24 views

Determinant of circulant $(0,1)$ matrices of certain form

I am interested in computing the determinant of the following circulant matrices: let $n=p^k$ for $p$ a prime and $k\in \mathbb{N}$, take $a\in \mathbb{N}$ to be such that $a<p$ and $(a,p)=1$. ...
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