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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Von Neumann stability analysis of spatial models

I am interested to perform a Von Neumann stability analysis applied to the finite volume formulations of the wave equation. The reason I am doing the analysis is to assess the spatial discretisation ...
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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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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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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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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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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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31 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
21 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
16 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
36 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 [on hold]

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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42 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
31 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
31 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
22 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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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
59 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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35 views

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
58 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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18 views

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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0answers
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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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
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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
21 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
27 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 $4$, ...
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1answer
20 views

Algorithmic solultion for eigenproblem over finite field

i am looking for the standard algorithms for solving eigenvalue problems over finite fields. (For example the algorithm implemented in GAP). I googled a lot but did not come to a conclusion. I saw ...
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Prove that the identity $\sum_{i=1}^n\underline u_i\underline u_i^{*T}=I_{n\times n}$ is valid for $n\times n$ hermitian matrices

Consider the hermitian matrix: $$\begin{bmatrix}5&10&0\\10&25&5\\0&5&5\end{bmatrix}$$ Its eigenvalues is $\lambda_1=30\;,\lambda_2=5\;,\lambda_3=0$ and its corresponding ...
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4answers
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Eigenvector proof for repeated eigenvalues

I am stuck trying to solve the following problem: In diagonalizing a symmetric matrix $S$, we find that two of the eigenvalues ($\lambda_1$ and $\lambda_2$) are equal but the third ($\lambda_3$) is ...
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1answer
27 views

Spectral Radius of a Sum of Permuted Matrices

Consider $n$ real, symmetric, and positive semi-definite matrices as: $A_1,A_2,\cdots,A_n$. These matrices are convertible to each other under appropriate permutation ($A_i(p_i,p_i)=A_j$). Moreover, ...
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1answer
20 views

Signs in the orthonormal bases

What I am trying to figure out is how they got the negative sign in the red circle, because from my calculations (blue circle) I get the first one to negative but the second one, that is the red ...
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Complete eigen-vector basis from non invertible linear application

Consider a non-invertible linear application $O$ acting on a Hilbert space (quantum mechanics). Is there still any chance to find a complete basis of $O$ eigen-vectors or no?
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Proof if $I+AB$ invertible then $I+BA$ invertible and $(I+BA)^{-1}=I-B(I+AB)^{-1}A$

I have the following question : Proof if $I+AB$ invertible then $I+BA$ invertible and $(I+BA)^{-1}=I-B(I+AB)^{-1}A$ I managed to proof that $I+BA$ invertible My proof : We know that $AB$ and $BA$ ...
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Eigenvalues of Hermitian matrix with off-diagonal blocks

I have a strange problem with the calculation of the eigenvalues of a $4 \times 4$ Hermitian Matrix: \begin{align} H = \begin{bmatrix} \lambda & a & 0 & -U S \\ \overline{a} ...
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Question about eigenvectors of real matrix with real eigenvalues

I have two related questions: Can a real matrix with real eigenvalues have complex eigenvectors? Is it always the case that a real matrix with real eigenvalues is diagonalisable?
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Determining whether two matrices are similar without calculating eigenvalues

How would you prove that $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ and $\begin{bmatrix} d & c \\ b & a \end{bmatrix}$ are similar? Their characteristic polynomials are identical, ...
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Questions about a Sparse Diagonal Matrix with other nonzero Diagonal Bands.

Working on a problem I've found that the matrices take on a rather unique form, and I was wondering if there's a name/classification for this type of matrix? (So I could look up if there are any ...
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1answer
54 views

Find $a$ in the following matrix

I have the following question : matrix $A$ isn't diagonalizable while $a \in R$ $$A = \begin{pmatrix} 3 & 0 & 0 \\ 0 & a & a-2 \\ 0 & -2 & 0 \end{pmatrix}$$ Find $a$. I ...
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1answer
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Real eigenvalues, similar symmetric matrix

I know that symmetric matrices have real eigenvalues, and that non-symmetric matrices that are similar to symmetric matrices must also have real eigenvalues, but is the converse true? That is, if a ...