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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Question regarding Eigen Value Decomposition and Singular Value Decomposition

I have a product of matrices that have the following form $$ {\bf A} ^H {\bf A}$$ where subscript $H$ means hermitian transpose. I am trying to find the eigen value decomposition (EVD) of ${\bf ...
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How are closed form solutions for eigenvalues constructed from sines and cosines?

For example a size N 3-point finite difference scheme has eigenvalues $\lambda_j = 2+cos(j\pi/(N+1))$. How is this determined? I know the Gershgorin circle theorem, but this is not what I am looking ...
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Textbook incorrect? - Eigenvalues [on hold]

I took this screenshot of my textbook, and have been struggling to understand whether i am doing something wrong or the textbook is. So they are performing a spectral decomposition, of an matrix, at ...
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Using Jacobi eigenvalue decomposition for decomposition into non-eigenvalue matrix?

I am a student in computer vision struggling with the problem of camera calibration. I am having trouble decomposing a matrix, Q, according to the formula: ...
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Matrices with left and right singular vectors being vandermonde matrices

Assume we have matrices ${\bf H_i}$ for $i\in[1:K]$ and that the Singluar Value Decomposition (SVD) of ${\bf H_i}$ is such that $${\bf H_i = A_{bi} D_iA_{si}^*}$$ where ${\bf A_{bi}}$ and $ {\bf ...
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1answer
19 views

Matrix and eigenvalues question hints?

This is the homework I have done part a, b, but I don t have any idea how to do the rest $y = 5$ and $z = 12 $ Those are the eigenvalues of matrix $A$ For part c, and d, I've tried to put some ...
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0answers
7 views

Eigen-decomposition of augmented block rectangular matrix

I have a rectangular matrix $\mathbf{X}_{n\times p}$ where the eigenvector decomposition of its inner product with itself is $$ \mathbf{X}^T\mathbf{X} = \mathbf{P}^T\mathbf{\Lambda P} $$ where ...
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2answers
39 views

Finding eigenvalues and eigenvectors of $2 \times 2$ matix

I having a few issues finding the eigenvectors for the following matrix: $$ \begin{bmatrix} -1 & -1\\ 0 & -2 \\ \end{bmatrix}$$ I calculated the eigenvalues to be ...
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1answer
16 views

Do all Stochastic matrix have a stationary probability vector?

I know that a stochastic matrix will have 1 as one of its eigenvalues. But do the stochastic matrices all have a stationary probability vector? Basically, could there be a case where the eigen ...
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2answers
51 views

Distinct eigenvalues and matrices problem

Let $V$ be a real vector space and $T: V \rightarrow V$ be a linear transformation. It is given that if $v_1, . . . , v_n$ are eigenvectors for distinct eigenvalues $λ_1, . . . λ_n$ then $\{v_1, . . ...
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0answers
28 views

Eigen vectors of a matrix multiplied with its transpose [on hold]

Do the eigen vectors of $A A^T$ and $AA^T$ belong to the row, column, null or left null spaces of the matrix $A$?
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2answers
21 views

Finding eigenvalues from characteristic polynomial

I am finding it extremely hard to find the eigenvalues after finding the characteristic polynomial. For example (instead of $\lambda$ I will use $x$) I have: $-x^3+x^2+16x+20=0$, how do i find the ...
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1answer
27 views

The relationship between diagonal entries and eigenvalues of a diagonalizable matrix

Let $\mathbf{C}$ be an $n\times n$ Hermitian matrix. Let $\dagger$ indicate a matrix conjugate-transpose. Let $\mathbf{V}\mathbf{D}\mathbf{V}^\dagger$ be the eigendecomposition of $\mathbf{C}$, where ...
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0answers
22 views

Von Neumann stability analysis of non-linear systems

The von-neumann stability analysis is based on the time and space discretisation schemes, what if the schemes are non-linear and too complicated to analyse. Is there a way to look at the matrices of ...
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1answer
17 views

Computing the characteristic polynomial

Consider the following matrix A over the field $F_7$ $$ \left(\begin{array}{rrr} 3 & 4 & 4 \\ 2 & 5 & 2 \\ 1 & 2 & 5 \end{array}\right) . $$ I'm asked to ...
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1answer
51 views

Intuition: why distinct eigenvalues -> linearly independent eigenvectors?

Suppose you have an n x n matrix with n distinct (not repeated) eigenvalues. There is a theorem telling us that the eigenvectors corresponding to these eigenvalues must be linearly independent. I can ...
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0answers
29 views

Eigenvalues of Moore–Penrose Pseudo-Inverse of a Symmetric Matrix

I was wondering if there is any bound or inequality for the eigenvalues of Moore–Penrose pseudo-inverse of a real $n\times n$ symmetric matrix $A$ in terms of eigenvalues of $A$, namely $\lambda_i$'s ...
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2answers
46 views

Eigenvectors and Kronecker product

Let us define $$ v:=v_A\otimes v_B\quad (*) $$ where $v_A$ is a fixed vector in $\mathbb{R}^{d_A}$, $v_B$ is any vector in $\mathbb{R}^{d_B}$ and $\otimes$ denotes the Kronecker product. To rule out ...
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2answers
40 views

What is an Eigenbasis and how do I calculate it with the information below.

I have the matrix $$A = \begin{bmatrix} 4 & 2 & 2\\ 2 & 4 & 2\\ 2 & 2 & 4 \end{bmatrix}$$ I've calculated the Eigenvalues and Eigenvectors as follows with help in a previous ...
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1answer
12 views

If the Rref(A) of a 3x3 matrix is I(A), is this a valid eigenvector?

For the vector A: EDIT: I had originally multiplied the matrix by -1. Apologies. $$ \begin{bmatrix} 4 & 2 & 2 \\ 2 & 4 & 2 \\ 2 & 2 & 4 \\ ...
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0answers
36 views

minimize smallest eigenvalue

Assume $P_A,P_B$ are probability transition matrices (each element is nonnegative and row sum is 1) and $v$ is probability row vector (each element is nonnegative and sum of elements is 1). How to ...
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1answer
22 views

Jacobian Eigenvalue Algorithm and Positive definiteness of Eigenvalue matrix

For a real symmetric matrix A of size n x n, the Jacobian Eigenvalue Algorithm produces n - Eigen values of A in the form of a Square Diagonal Eigenvalue Matrix of order n n - Eigen vectors of A ...
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1answer
17 views

Improving the performance of eigs for a large spd Problem

I have two large (think around $100.000\times 100.000$), sparse, real symmetric and positive definite matrices $A$ and $B$ and I want to find the smallest generalized eigenvalue $$Ax = \lambda_{\min} ...
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1answer
27 views

A question on matrix's eigenvalue problem from Eberhard Zeidler's first volume of Nonlinear Functional Analysis.

I have a question from Eberhard Zeidler's book on Non-Linear Functional Analysis, question 1.5a, he gives as a reference for this question the book by Wilkinson called "The Algebraic Eigenvalue ...
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1answer
37 views

If $n$ is odd, do the eigenvalues need to by multiplied by $-1$ if I use $\det(\lambda I-A)$ instead of $\det(A - \lambda I)$?

I have seen the characteristic polynomial written as $f(\lambda)=\det(\lambda I-A)$ or $f(\lambda)=\det(A-\lambda I)$. By determinant rules $\det(\lambda I-A)\iff (-1)^n\cdot\det(A-\lambda I)$. That ...
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Effects of Eigenvalue decrease of a PSD matrix or vice versa [closed]

Is there any theoretical proof or intuition behind what is happening to a PSD matrix when an operation is operating on its entries individually in a way such that all of its eigenvalues are decreasing ...
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65 views
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Bounding 2nd-smallest eigenvalue of the Laplacian of the binary tree

I am reading on my own the notes of this lecture series from 2012: http://www.cs.yale.edu/homes/spielman/561/2012/lect04-12.pdf. In section 4.7.2 (page 8) it's mentioned that we can prove a lower ...
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1answer
18 views

Eigenvectors of the square of an operator

Let $V$ be a vector space and $D:V\rightarrow V$ be linear. Let $s$ be an eigenvector of $D^2$. Can we always express $s$ as a linear combination of eigenvectors of $D$? If not, what conditions might ...
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1answer
35 views

How to find the symmetric matrix if its eigenvalues and eigenvectors are given?? [closed]

Find a $2 \times 2$ symmetric matrix if its eigenvalues are $1$ and $3$ and its corresponding eigenvectors are $(1,-1)$ and $(1,1)$.
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1answer
36 views

How to compute the matrix $S$ in Sylvester's law of inertia

Sylvester's law of inertia states that for any symmetric matrix $A$ there exist an invertible matrix S such that, $S^T A S = D$, where $D$ is a diagonal matrix which has only entries 0, +1 and −1 ...
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1answer
23 views

Finding the Jordan Normal Form for a General Linear Transformation

Hey everyone here's the problem: Let V be a vector space with dim(V)=n For a particular linear transformation,f, we are given that there are two distinct eigenvalues, λ1 and λ2, with corresponding ...
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Eigenvector length close to zero

I have a Fortran subroutine that calculates eigenvalues and eigenvectors of a symmetric $3 \times 3$ matrix. The eigenvector corresponding to an eigenvalue are calculated and then they are normalized ...
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27 views

Diagnalization of block matrix with circulat blocks

I have the following Matrix $A = \begin{pmatrix} X \\ Y \end{pmatrix}$ Where X, and Y are circulant Matrices. I want to diaganlize $AA^T$. I tried the following: $AA^T = \begin{pmatrix} ...
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24 views

finding the shortest distance of a hermitian matrix to a set of hermitian matricies with specific eigenvalues 2-norm

The title is more general, and all that I require is to show an inequality that I already have verified using random matrices in matlab. Let $\lambda_1 \leq ... \leq \lambda$ and $\mu_1 \leq ... \leq ...
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1answer
61 views

About the Jordan Form

So i have a few questions about the Jordan form. Say we have a matrix $A$ and has λ1 λ2..λκ eigenvalues.Why is it Usefull to know the index of the matrices $A-λιI$ ? Also i have seen jordan forms ...
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2answers
65 views

Find trace of linear operator

Let $A$ be a linear operator which acts on the vector space $V=\langle x_1,x_2, \ldots,x_n\rangle$ by permutation of the basis vectors. Suppose we know its eigenvalues ( some roots of unity ): ...
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2answers
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Why does positive definite matrix have strictly positive eigenvalue?

We say $A$ is a positive definite matrix if and only if $x^T A x > 0$ for all nonzero vectors $x$. Then why does every positive definite matrix have strictly positive eigenvalues?
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Jordan Form of generic matrix

Say $ A\in\mathbb{C}^{6\times6} $ and has eigenvalues $\lambda_1$ and $\lambda_2$ of multiplicity $ 3$ both of them. And for $\kappa=1,2,3$ the echelon form of the matrix $$ (A-\lambda_1I)^\kappa $$ ...
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1answer
59 views

Second-smallest eigenvalue as $\displaystyle \min_x \frac{x^TAx}{x^Tx}$

In Mining Massive Datasets, page 365, the following theorem is stated without proof: Let A be a symmetric matrix. then the second-smallest eigenvalue of A is equal to $\displaystyle \min_{x} ...
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Question about “Third existence theorem for weak solutions” in Evans - Partial Differential equations

I'm currently studying Evans excellent book "Partial Differential Equations" and I'm a bit stuck in the proof of Theorem 5 in Section 6.2.3 (P. 305, "Third Existence Theorem for weak solutions). What ...
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15 views

Spectrum of the circulant graph

How to prove that the eigenvalue of cycle $C_n=\lambda_r=2 cos(2\pi r/n)$?where $r=0,1,...n-1$, which is proved for the circulant matrix with first row $(v_0=0,v_1=1,v_2=0, ...v_{n-2}=0,v_{n-1}=1)$, ...
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1answer
23 views

a question how to compute the eigenvalues of a matrix [duplicate]

I have a question: Suppose I have a $n\times n$ matrix: $$ \begin{bmatrix} 1 & 1 &...& 1 \\ 1 & 1 &...&1 \\ \vdots&\vdots &\ddots & ...
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1answer
42 views

Bounds for inner product of $Ax$ and $x$

Reading a math text, I found, with no proof given, the following assertion. Suppose $A$ is a real $n \times n$ matrix, and suppose the real part of its spectrum lies between $a$ and $b$; i.e., the ...
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2answers
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Calculate matrix A from null space basis of $A-4I$

How to find a matrix $A$ when you are given some parameters and the basis for the null space? The problem I've been scratching my head over is this. The basis for the null space of $A-4I$ is ...
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1answer
50 views

Find eigenvalues of operator

Let $A$ be a linear operator which acts on the vector space $V=\langle x_1,x_2, \ldots,x_n\rangle$. Suppose we know its eigenvalues - $\lambda_1, \lambda_2, \ldots, \lambda_n.$ Now consider the ...
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$A_{n\times n}$ that implies : $A^2-2A+I=0$ Proof $1$ is an eigevalue of $A$

I have the following question : Let $A_{n \times n}$ that implies : $A^2-2A+I=0$ Proof $1$ is an eigevalue of $A$ I don't really know how to approach this this what I manage to do (its not much ...
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Can more pertubations in eigenvalues/vectors lead to smaller changes?

Say i have a $n$ x $n$ matrix $M$, and i change it's smallest eigenvalue from a small negative value $v$ to a small positive value $t$ to obtain $M^*$: $$M^* = VE^*V'$$ $E^*$ is a diagonal matrix of ...
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1answer
48 views

Why do eigenvalues exclusively form the main diagonal in a diagonalizable matrix?

So, why do eigenvalues exclusively form the main diagonal in a diagonalizable matrix? If we have $n\times n$ matrix ($n$ being a natural number) that is diagonalizable, why is it eigenvalues ...
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1answer
22 views

On the eigenvectors of difference of positive semi-definite matrices

For a given positive semi-definite matrix $A$, expressed using singular value decomposition as: $A=UD^2U'$ (subject to the orthonormality conditions), any positive semi-definite matrix $X$ that ...
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1answer
29 views

General question about eigenvalue, eigenvectors.

I have the following question : $A$ is a $n \times n$ matrix, and this is the characteristic polynom $$p(x)=(x+3)^2(x-1)(x-5)$$ Then I can conclude that $n=4$ since the number of the roots is $4$, ...