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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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Sum of sqrt of eigenvalues without computing all eigenvalues

Let $A$ be a positive-definite matrix with eigenvalues $e_1, ..., e_n$. I want to compute $\sum\limits_{i=1}^{n} \sqrt{e_i}$ without calculating all eigenvalues first (or rather: with a method faster ...
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1answer
28 views

Is this simple symmetric matrix positive semi-definite?

Let the $n\times n$ symmetric matrix $A$, where $n\geq 9$ be given by \begin{equation} A_{i,j}= \begin{cases} 1.4, &\text{for } 1\leq i=j\leq 9\\ (0.9)^{|i-j|},&\text{for } 1\leq i\neq j\leq 9\...
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Show that a linear map on a finite dimensional complex vector space always have an eigenvalue.

What is an alternative proof that a linear map $T$ on a finite dimensional complex vector space $V$ with dimension $n$ always has an eigenvalue? Here is the original proof idea: We take a no zero ...
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3answers
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Eigenvalues and eigenvectors for the moment of inertia matrix

Find the eigenvalues and eigenvectors for the moment of inertia matrix given by $$I={m\over 2}\left(\begin{matrix} 1 & -1 & 0\\ -1 & 1 & 0\\ 0 & 0 & 2\end{matrix}\right)$$ ...
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1answer
76 views

Eigenvector of two rotation matrices

I am having difficulty in understanding a geometry problem which contains geometric-transformation, rotation and reflection. Background In this image, a camera with camera center $O_c$ is presented. ...
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1answer
9 views

Eigendecomposition proof

Let $X$ be an $(n \times n)$ matrix. Let $V$ be the $(n \times n-k)$ be the matrix of eigenvectors of $X$ which correspond to non-zero eigenvalues of $X$. Let $E$ be the $(n-k \times n-k)$ diagonal ...
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1answer
25 views

Eigenvalues and eigenvectors of $A_{1}$ and $ A_{2}=A_{1}^{T}$

We have a positive integer $n$ and two $n\times n$ matrices of real numbers, $A_{1}$ and $A_{2}$. For $j=1, 2$, we have the eigenvalues and eigenvectors $\lambda _{j}$ and $x_{j}$ of $A_{j}$. Show ...
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1answer
43 views

Characteristic polynomial and eigenvector of Frobenius matrix

Consider the following $n \times n$ matrix (I believe this is similar to companion matrix): $$ A = \begin{pmatrix} 0 & -1 & 0 & \cdots & 0 & 0 & 0 \\ 0 & 0 & -...
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20 views

Matrix perturbation and eigenvector

$Ae=\lambda e$ $e^Te=1$ $A$ is a real matrix. $\lambda, e$ are real. $(A+\Delta A)(e+\Delta e)=(\lambda+\Delta \lambda)(e+\Delta e)$ Neglecting small terms, $\Delta Ae +A \Delta e=\Delta \lambda ...
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1answer
45 views

Why are singular values of “complex” matrices always real and non-negative?

I've already read the following related questions on math.SE: Why can't singular values be complex numbers? Clarification on the SVD of a complex matrix Why are singular values always non-negative? ...
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12 views

What can be said about the definiteness of the following inequality?

Given a Hurwitz matrix $R\in\Re^n$ which has all the eigenvalues located in the closed left-half plane. For a positive-definite matrix $Q$, we know that there exists a unique solution $P$ to the ...
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15 views

QR-algorithm complexity on a symmetric tridiagonal matrix

Why does the QR algorithm (for calculating eigenvalues) only require O(m) calculations per step when performed on a symmetric tridiagonal matrix?
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2answers
90 views

How to prove that a $3\times 3$ matrix has only $2$ eigenvectors?

I am working through a problem in Riley, Hobson and Bence (Mathematical Methods for Physics and Engineering) that revolves around the following matrix: $$ A= \begin{pmatrix} 2 & 0 & 0 ...
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0answers
6 views

Way of predicting change in gradient of line following matrix transformation?

I recently learnt about dominant and repulsive eigenvectors. I noticed that the farther a line is initially from the dominant (though still closer than to the repulsive eigenvector), the more dramatic ...
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1answer
38 views

Let $V$ be a real vector space and $E$ be an idempotent linear operator on $V$. Prove that $I + E$ is invertible.

Let $V$ be a real vector space and $E$ be an idempotent linear operator on $V$, that is a projection. Prove that $I + E$ is invertible. Find $(I + E) ^{-1}$ My teacher taught me the following proof ...
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1answer
37 views

How to project a vector $x$ onto the eigen-space, i.e. $x = \sum_{i=1}^{n} \langle w_i, x \rangle v_i$?

Assume $A$ is a complex-valued square matrix, i.e. $A\in \mathbb{C}^{n\times n}$, and $A$ has a full set of eigenvectors denoted as $V=[v_1, v_2, \cdots, v_n]$. Then we known the following facts \...
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1answer
23 views

Matrix Normalization

From the eigenvectors matrix: I did normalization but I think there's an error I could not find.
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1answer
17 views

Matrix Orthogonality

I have the eigenvector matrix like this $\begin{bmatrix} 1 & 1 & 1 \\ 0 & \frac{-b + \sqrt{(b^{2} + 8a^2)}}{2a} & -\frac{b + \sqrt{(b^2 + 8a^2)}}{2a} \\ -1 & 1 & 1 \end{...
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2answers
37 views

If $P(x)$ is characteristic polynomial of $A$ then is $P(A) = 0$?

I'm a student and I've just read the Characteristic polynomial on Wiki. I have a feeling that: If $P(x)$ is characteristic polynomial of $A$ then is $P(A) = 0$ thanks to the Matrix calculator I've ...
1
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1answer
74 views

Eigenvalues decrease with power

Take $n \in \mathbb{N}$, and consider a square matrix $A$ of size $n \times n$, with real and positive entries, and such that $\|A\|_2 \leq 1$. I think the following statement holds from simulation, ...
0
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1answer
33 views

Can the eigenvectors of a linear operator in an infinite-dimensional space span the space and be linearly dependent at the same time?

Consider a vector space $V$ over the complex field which is infinite-dimensional with a Euclidean inner-product. Let $L$ be a linear operator on $V$. Say a subset of eigenvectors of $L$ forms a ...
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0answers
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Relation between leading eigenvlaue and eigenvector of individual blocks with the leading eigenvector of non-negative symmetric block matrices

For a 2 by 2 block non-negative symmetric matrix $\mathcal{M}$, $\mathcal{M}= \left[ \begin{array}{c|c} \mathcal{A} & \mathcal{C} \\ \hline \mathcal{C}^T & \mathcal{B} \end{array} \right]$ ...
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2answers
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Calculate the eigenvalues and eigenvectors of 2 x 2 matrix

Its $2 \times 2$ matrix and having square-root value
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1answer
64 views

Can an orthogonal non-symmetric 3x3 matrix have 3 real eigenvalues?

I was wondering if a non-symmetric orthogonal matrix can have his 3 eigenvalues in the real numbers. All the 3 real eigenvalues orthogonal matrix i've found are symmetric. Can someone give me a ...
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0answers
13 views

Find eigen vectors of a “subsetted” matrix

I have a symmetric matrix, $A$, with dimension 100 x 100, of which I know the eigen vectors and eigen values ($A = U'VU$ ). Now I want to know the eigen vectors and eigen values of $B$, which is just ...
0
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0answers
19 views

Spectral radius of a hollow symmetric block matrix

Let $B$ be a $2 \times 2$ matrix. Suppose we have a $4 \times 4$ real symmetric matrix of the following form $$A = \begin{bmatrix} O_2 & B \\ B^T & O_2\\ \end{bmatrix}$$...
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1answer
31 views

How to find the eigen values of the following matrix:

Is there any way to find the eigen values of the following matrix: $A_{2n\times 2n}=$ \begin{bmatrix}\textbf{0} & E_{n\times n}\\E^T&\textbf{0}\end{bmatrix} where $E=$ \begin{...
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1answer
24 views

Eigenvalue multiplication endomorphism

When considering the multplication endomorphism \begin{equation*} \begin{split} [\times z]_{K/Q}: & \:\: K \rightarrow K \\ & \:\: x \:\mapsto xz \end{split} \end{equation*} for $Q$ a field ...
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0answers
28 views

Find first eigenvector of Hadamard division of $AA^T$ and $BB^T$ using power method

For an $m \times n$ matrix $A$, it is possible using the power method to find the eigenvector corresponding to the largest eigenvalue of $AA^T$ by factoring it into a matrix vector product $(AA^T)v = ...
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3answers
255 views

Show that the characteristic polynomial is the same as the minimal polynomial

Let $$A =\begin{pmatrix}0 & 0 & c \\1 & 0 & b \\ 0& 1 & a\end{pmatrix}$$ Show that the characteristic and minimal polynomials of $A$ are the same. I have already computated ...
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2answers
31 views

Proving a matrix iteration converges

Consider the sequence $x_{n + 1} = Mx_{n} + b$. Suppose the matrix $M$ is symmetric and for any $x \neq 0$, $$-1 < \frac{x^{T}Mx}{x^{T}x} < 1$$ holds. Prove that $x_{k}$ converges. ...
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1answer
71 views

Maximize the smallest nonzero singular value

I want to maximize the smallest nonzero singular value of (non-square) matrix $X$. This is equivalent to maximizing $\lambda_{\min}(X^\top X)$, which can be reformulated as follows $$\begin{array}{ll}...
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1answer
21 views

How to create matrix with high ratio of first two eigenvectors and equal row sums

To create a matrix where the ratio between the first two eigenvectors $\frac{\lambda_1}{\lambda_2}$ is large, I can set a matrix $A = \lambda_1 u u^T + \lambda_2 v v^T$ with orthonormal $u,v$ and the $...
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1answer
36 views

eigenvalue of a graph

What does the eigenvalue of a graph mean? I know how to compute the eigenvalues from the adjacency matrix representation of a graph but am interested in its physical significance. If two graphs have ...
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0answers
31 views

How to measure repulsion between numbers

How is repulsion measured between two eigenvalues or any two numbers for that matter? Assume that repulsion is $1,$ ($100$ percent) when the two numbers have zero space between them, and repulsion is $...
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2answers
41 views

Eigenvalues of $Q=I+2P$

I have tried to do it evaluated option (a). I think it is correct.Can not get the other options.Please help me.
3
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1answer
32 views

How to show for a normal matrix $A$, $(A,\lambda,x) \Leftrightarrow (A^*,\bar{\lambda},x)$?

A matrix $A$ is normal if $AA^*=A^*A$. Suppose $(\lambda,x)$ is an eigenpair of $A$, i.e., $Ax = \lambda x$. Proof for a normal matrix $A$, $(\lambda,x)$ is an eigenpair of $A$ if and only if $(\bar{\...
0
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1answer
35 views

How can we decompose determinant of an upper triangular matrix? [closed]

Let $T= \begin{bmatrix}A & B \\ 0 & C \end{bmatrix}$. How to show $\text{det}(T-\lambda I)=\text{det}(A-\lambda I) \text{det}{(C-\lambda I)}$ for square $A$ and $C$?
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2answers
69 views

Showing $| \lambda_i | < 1 \implies \operatorname{tr}(A) - 1 < \det(A) <1$

Prove that every eigenvalue of $A \in \mathbb{R}^{2 \times 2}$ has modulus less than 1 if and only if $$ \operatorname{tr}(A)-1 < \det(A) <1. $$ I can prove the forward direction, but not ...
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0answers
16 views

Quantum Mechanics - why should Probability of a degenerate eigenvalue be independent of the choice of an eigenvector in Euclidean space En?

Given $$|\psi_n> = \sum_{i=1}^{g_n}|u_{n}^{i}><u_{n}^{i}|\psi>$$ we get a projection $P_n|\psi>$. Incidentally, the square of this statement gives us the probability of finding a ...
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0answers
22 views

How do I solve the eigenvalue problem by matlab

My question is how to solve the eigenvalue equation $$(\frac{d^2}{dx^2}+\frac{d^2}{dy^2})u(x,y)+b\frac{d}{dx}u(x,y)+c\frac{d}{dy}u(x,y)+sin(x) cos(y)u(x,y)=\lambda u(x,y)$$ with periodic boundary ...
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0answers
23 views

Eigenpairs of a sparse symmetric block tridiagonal matrix with diagonal blocks on the outer diagonals

I'm looking at methods to find the eigenpairs of symmetric block tridiagonal matrices, with sparse blocks on the main diagonal and diagonal blocks on the outer diagonals. Has any research been done on ...
1
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0answers
37 views

Canonical Jordan form contradiction

I am faced with the following problem: Given endomorphism $f$ whose characteristic polynomial is $$P_c(x) = (x+1)^{10} (x-1)^{10} x^{10}$$ and whose minimal polynomial is $$P_m (x) = (x+1)^5 (x-1)^...
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1answer
23 views

Proof for a property of graph laplacian and its complement?

Here it says Let $L(G)$ be the Laplacian of an undirected graph. $L(G)+L(G^c)=n I_n-J_n$ Where $J_n$ is a matrix with all entries $1$. Then how do we prove this (page $224$; eq. $5$): $$\...
2
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0answers
44 views

What is the meaning of $\frac{(Ay,x)}{(y,x)}$?

$x^TAy$ is the inner product of a matrix A. If x,y are unit vectors, then what is the meaning of $\frac{x^TAy}{x^Ty}$? What does it do to the inner product? The orthogonal component of y wrt x is ...
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0answers
25 views

eigenvalues of a direct sum of matrices

According to Bacher's article, the eigenvalues of the adjacency matrix of Cayley graph of the symmetric group of order $n$ are $2-2\cos(\pi/n)$; my question is: If we know that this adjacency matrix ...
0
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1answer
32 views

(real symmetric matrices) does orthogonality of eigenvectors (distinct eigenvalues) depend on choice of basis?

For some reason I cannot wrap my head around this one. My schoolbook begins the chapter on eigendecomposition of real symmetric matrices by stating that eigenvectors from distinct eigenspaces are ...
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0answers
15 views

Invariant subspace SO(n)

Suppose $v\in\mathbb{C}$ is an eigenvector of $R\in SO(n)$ with nonreal eigenvalue $\lambda$. Let $V\subset \mathbb{R}^n$ be the two dimensional space spanned by $(v+\bar{v})/2$ and $(v+\bar{v})/(2i)$....
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2answers
32 views

Sturm-Liouville differential equation eigenvalue problem

If we have a Sturm-Liouville differential equation of the form $$ \frac{d}{dx}[p(x)\frac{dy}{dx}]+q(x)y=-\lambda w(x)y $$ and define the linear operator $L$ as $$L(u) = \frac{d}{dx}[p(x)\frac{du}{...
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2answers
39 views

Eigenvectors and invariant subspaces

If I have a 3 eigenvectors within $\mathbb{R}^3$ and I wanted 2 dimensional invariant subspaces of $\mathbb{R}^3$, could I just take the span of any 2 of of the 3 eigenvectors and they would give me ...