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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A Basis for a Jordan Normal Form

In my assignment I have to find a Jordan normal form for this matrix: Thank you for your help, and I'm sorry the question is pronunced with Latex.
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36 views

Knowing the eigenvalues for A find the matrix A

I know the eigenvalues for a matrix. Let's say they are 2 and 1. How can I find the matrix A for them (all members of A are not null) ?
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4answers
93 views

Show that a given matrix always has an eigenvector in $\mathbb{R}$ Can somebody give a hint?

The given exercise is, for all $\theta$ in $\mathbb{R}$, show that the matrix always has an eigenvector in $\mathbb{R^2}$ $$ A = \begin{pmatrix} \cos(\theta) & \sin(\theta) \\ \sin(\theta) & ...
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33 views

Eigenvector Problem

Given a matrix $X$, let $eigvec(X)$ be its eigenvector associated with the largest eigenvalue. Is there a relationship among $eigvec(X+X^T)$, $eigvec(X)$ and $eigvec(X^T)$? In other words, can I use ...
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1answer
29 views

Finding the corresponding Perron eigenvalue

Find the Perron root and the corresponding Perron eigenvector of A. $\begin{bmatrix} 0 &1 &1 \\ 1&0&1 \\ 1&1&0 \end{bmatrix}$ I figured out the Perron root which happens to ...
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26 views

Algebraic multiplicity of an eigen value

Let $T$ be an operator on a complex Vector space $V$. Then, the algebraic multiplicity of an eigen value is equal to $\dim ~null~ (T - \lambda I)^{\dim V}$ Which means, if we obtain the upper ...
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23 views

Is there a formula for the sum of absolute eigenvalues in terms of matrix elements?

Given a symmetric matrix $X \in \mathbb{R}^{n \times n}$. We know the following: trace$(X) = \sum_{i=1}^n x_{ii} = \sum_{i=1}^n \lambda_i$ where $x_{ii}$ is the $i$th element on the diagonal of $X$, ...
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Prove that a Graph is connected using eigen values [on hold]

This question relates with expander graphs Prove that for a graph is connected if and only if $\lambda_{max}$ > $\lambda_{1}$ Prove that for a $d$-regular graph $\lambda_{\max} = \lambda_1 = \cdots ...
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1answer
26 views

extended PCA (tangled matrices)

Given an $m$ by $n$ matrix $A$ and the constant $r$, the principal component analysis allows us to find matrices $W$ and $H$ so that the $WH$ gives a lower rank approximation of $A$. In other words, ...
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1answer
48 views

Real matrix with the property that every nonzero vector in $\mathbb{R}^n$ is an eigenvector of $A$. [duplicate]

so I'm supposed to let $A$ be a square real matrix with the property that every nonzero vector in $\mathbb{R}^n$ is an eigenvector of $A$. And I'm supposed to show that $A=\lambda I$ for a constant ...
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1answer
23 views

Eigenvalues of operator on $S_n$'s group algebra

Take the group algebra of the symmetric group $S_n$ (or equivalently consider $S_n$'s regular representation) - I guess over $\mathbb{C}$. If $e_{i,j} \in S_n$ denotes the element which swaps only ...
2
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1answer
57 views

Eigenvalues of symmetric matrices are real without (!) complex numbers

Is there any proof of the fact that the eigenvalues of symmetric matrices (i.e. $A\in\mathbb{R}^{n\times n}$ with $A^t=A$) are real without the use of the concept of complex numbers?
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18 views

Eigenvalues of discrete one dimensional wave equation converge to eigenvalues of the continuous problem

consider the following approach to a solution of the one dimensional wave equation: $-U''(x) = \lambda U(x), \quad \lambda := \frac{\omega^2}{c^2}, \quad 0 < x < 1, \quad U(0) = U(1) = 0$ I ...
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1answer
31 views

Eigenvalues of a Product of two matrices A and B inside trace operator expressed in terms of any eigenvalue of A or B?

This question has been in asked in a few varieties here but not in this one. If we have a real, symmetric, positive-definite matrix $A$ and a real, symmetric, positive-definite matrix $B$ and we know ...
2
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1answer
20 views

Does negative definiteness imply anything about ALL principal minors?

Unfortunately I haven't received any response for my previous question, so I'm trying to solve it in a different way. I know that iff matrix $H$ is negative definite, its leading principal minors ...
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1answer
24 views

Let $f: \mathbb{R^3} \to \mathbb{R^3}$ with characteristic polynomial $-\lambda(\lambda-3)^2$ and $f(1,0,0)=(1,0,-1)$ and $f(0,1,0)=(2,3,1)$

Let $f: \mathbb{R^3} \to \mathbb{R^3}$ be a diagonalizable endomorphism with characteristic polynomial $-\lambda(\lambda-3)^2$ such that $f(1,0,0)=(1,0,-1)$ and $f(0,1,0)=(2,3,1)$. Given these data, ...
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1answer
13 views

Spectral gap vs. algebraic connectivity

Can someone please clarify how the spectral gap of a graph relates to its algebraic connectivity (aka Fiedler value) and whether these use the adjacency matrix or laplacian matrix?
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1answer
21 views

Finding the real irrational root of a cubic polynomial?

I just wanted to check if anyone can see a simpler way to solve this. Because I am not looking forward to using the cubic formula to solve it! $$ det(\lambda-AI) = \left| \begin{array}{ccc} \lambda + ...
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2answers
50 views

Finding Eigenvalues of given linear operator

Find the eigenvalues and the eigenvectors of the linear operator $T:C^\infty(0, 1)\to C^\infty(0, 1)$ $T(f)(x) = \frac{f'(x)}{x}, x \in (0,1) $ Using the definition : $TF = \lambda F \iff ...
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1answer
33 views

How to bound the biggest eigenvalue of $\sum_{i=1}^{n}x_ix_i^T$?

My question is to bound the biggest eigenvalue of $A=\sum_{i=1}^{n}x_ix_i^T$, where $x_i\in\mathbb{R}^d$ is a column vector. My idea is, to bound the biggest eigenvalue of $A$, i.e. $\|A\|_2$. I can ...
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0answers
24 views

How do I find the common invariant subspaces of a span of matrices?

Let $G_1, \ldots, G_n$ be a set of $m\times m$ linearly-independent complex matrices. Let $\mathcal{G} = \operatorname{span}\left\{ G_1, \ldots , G_n\right\}$ be the vector space that spans the set ...
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57 views

Can a matrix have eigenvalue with infinite multiplicity?

Suppose we have matrix of the form $$ A= \begin{bmatrix} a & -1 \\ 0 & a \\ \end{bmatrix} $$ and we would like to analyze its diagonalizability. By taking the ...
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What is a basic description of how diffusion mapping works?

I have been trying to get a basic understanding of diffusion mapping, and I think I understand the concept, but I am having trouble understanding the math behind it (I have knowledge of advanced math ...
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26 views

Eigenvalues of a transpose multiplication

Say I have a matrix $\mathbf B \in \mathbb R^{m\times n}$. Is it correct to say that the eigenvalues of $\mathbf B^T\cdot\mathbf B$ are always positive?
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1answer
29 views

Eigen value system? solution

I have the following system. $AW = \lambda B W$ Where $A,B,W$ are matrices and $\lambda$ is a scalar. The values of $A,B$ and $\lambda$ are known. $B$ is invertible. This is a solution to an ...
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1answer
35 views

Finding a linear transformation such that $T^{3} = T $

I have to show that there exists a linear transformation such that $T^{3} = T $ i can see that from here that T has eigen values $0.1.-1$ .But how do i find linear transformation .Also for v and q ...
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46 views

Relationship between Eigenvalues

I am looking at a matrix $$\mathbf{M} = \left(\mathbf{I}+k\theta\mathbf{B}^{-1}\mathbf{A}\right)^{-1}\left(\mathbf{I}-k(1-\theta)\mathbf{B}^{-1}\mathbf{A}\right) $$ where $\mathbf{I}$ is the identity ...
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1answer
55 views

Eigenspaces and jordan normal form

I have a question here regarding the jordan normal form of two matrices where the eigenspace is one is contained in the other. Let $A,B$ be two $nxn$ matrices s.t $AB=BA$. I firstly proved that the ...
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Dimension of $n \times n$ diagonal matrix with characteristic polynomial $(x-a)^{p}(x-b)^{ q}$?

What is the dimension of an $n \times n$ diagonal matrix with characteristic polynomial $(x-a)^{p}(x-b)^{ q}$? Do I have to make distinct cases with as $p + q < n$ and equal to $n$? And if their ...
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3answers
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Determine a matrix knowing its eigenvalues and eigenvectors

I read through similar questions, but I couldn't find an answer to this: How do you determine the symmetric matrix A if you know: $\lambda_1 = 1, \ eigenvector_1 = \pmatrix{1& 0&-1}^T;$ ...
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0answers
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How to Simplify/Rewrite this Expression into a Generalized Eigenvalue Problem - via Similarity perhaps?

I have the following optimization problem: \begin{eqnarray} min~b' y' Z (Z' \Omega Z)^{-1} Z' y b \end{eqnarray} such that $b'b=1$. The matrices are $Z \in R^{n \times k}$, matrices $y \in R^{n \times ...
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Why is a linear autonomous system asymptotically stable iff for all eigenvalues $\lambda$ of $A$, $Re(\lambda) < 0$

I'm trying to understand asymptotic stability of linear antonymous systems. I'm not sure if for the system $x' = Ax$, $x(t) = 0$ is the only fixed point that can be stable. In any case, I can ...
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1answer
24 views

A connection between a matrix norm and a related matrix's largest eigen-value

I have been asked to prove that for $A\in M_n(\mathbb{C})$, with $||A||:=\sup_{x\in\mathbb{C}^n,|x|=1}|Ax|$, $$||A||=\sqrt{\lambda}$$ where $\lambda$ is the eigen value of largest modulus of $A^*A$. ...
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1answer
32 views

How to determine the signs of the eigenvalues of a symmetric $3\times 3$ matrix?

This is a homework problem: Let $a,b,c$ be positive real numbers such that $b^2+c^2<a<1$. If $A=\begin{pmatrix} 1&b&c\\b&a&0\\c&0&1\end{pmatrix}$, then which of the ...
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11 views

Algebraic multiplicity of an eigenvalue for abstract operators

How does one define algebraic multiplicity of an eigenvalue for an abstract operator? (for a matrix the definition is clear). E.g. Consider $\partial_x^2$ on $H^2_{per}(0,1)$ then $\partial_x^2 ...
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1answer
32 views

Courant minimax principle on block matrix

in going through some books about numerical mathematics I found the following exercise: Let $A,B \in \mathbb{R}^{n\times n}$ with $A$ symmetrical and rank($A$) = rank(B) = n. Define $M = ...
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1answer
43 views

Eigenvalues and eigenfunctions of fourth order ODE

Find the eigenvalues and eigenfunctions of the problem $$y^{(4)} − λy = 0$$ with the boundary conditions (i) $\quad y(0) = y'' (0) = y(β) = y'' (β) = 0$ (ii) $\quad y(0) = y' (0) = y'' (β) = y''' ...
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2answers
33 views

Proof with orthogonal matrix

I stuck at this problem: I need to prove that for linear transformation $$T:R^n\to R^n$$ defined by $$T(x)=Px$$ such that $$P^T=T^{-1}$$ for any $x,y$ $$T(x) \cdot T(y)=x\cdot y$$ and also that $T$ ...
3
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25 views

Eigenvalues of Overlapping block diagonal matrices

I look for eigenvalues of general overlapping block diagonal matrices. e.g. $$\left[ \begin{matrix} 1 & 4 & 0 & 0 & 0 & 0\\ 4 & 2 & 3 & 2 & 0 & 0\\ 0 & 3 ...
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28 views

Commuting operators

Let's consider a number of linear operators, defined on a finite dimensional complex vector space, which two by two commutes with each other. (the amount of them can be infinite). How to prove that ...
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1answer
25 views

One eigenvalue and eigensystem

Matrix $A \in \mathbb{K}^{n,n}$ has one engenvalue $\lambda \in \mathbb{K}$ and its engensystem $V_{\lambda}$ has dimension that equals to $n$. How to show that $A = \lambda I_{n}$?
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1answer
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If two matrices have the same characteristic polynomials, determinant and trace, are they similar?

If two $n \times n$ matrices have the same characteristic polynomials, determinant and trace, are they similar, EVEN if ($ \lnot \#Spec= 0$)?
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26 views

Approximate the second largest eigenvalue (and corresponding eigenvector) given the largest

Given a real-valued matrix $A$, one can obtain its largest eigenvalue $\lambda_1$ plus the corresponding eigenvector $v_1$ by choosing a random vector $r$ and repeatedly multiplying it by $A$ (and ...
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24 views

Spectrum of convolution operator

I was trying to find the spectrum of the convolution operator $$ J \ast u = \int_D J(x-y) u(y) dy $$ for bounded domain $D \subset \mathbb{R}$. Does anybody know it or have a reference for me? ...
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1answer
37 views

How do I show that $\inf\limits_{\det(X)\neq0}\|X^{-1}AX\|^{2}_{F}=\sum\limits_{\lambda\in{\Lambda}}|\lambda|^{2}$?

Show that $$\inf\limits_{\det(X)\neq 0}\|X^{-1}AX\|^2_F=\sum_{\lambda\in\Lambda}|\lambda|^{2}$$ holds, where $\Lambda(A)$ is the set containing all eigenvalues of A, and $\|\cdot\|_{F}$ is the ...
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(Fast) eigen decomposition of $DXD$ where $D$ is diagonal, $X$ is symmetric with known eigen decomposition

Assuming that I already know the eigen-decomposition of a real symmetric matrix $X$, is there any way to use it to retrieve efficiently the eigen-decomposition of $DXD$, where $D$ is a diagonal ...
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2answers
26 views

Using eigenvalues of a hessian matrix vs D operation to classify critical points.

Having recently covered using the discriminant, $D(x_0,y_0)$, for classifying critical points of equations of two variables. For example: $$R(x,y)=-x^2+4x+2xy+8y-2y^2$$ to find that $(6,8)$ is the ...
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1answer
27 views

Show that for every three linear maps $A, B, C: V → V$ we have $rk(ABC) ≤ rk(B)$.

Let $V$ be a vector space. Show that for every three linear maps $$A, B, C: V → V$$ we have $$rk(ABC) ≤ rk(B)$$ My only idea is to try and show something like $rk(ABC) ≤ rk(BC) ≤ rk(B)$, but ...
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19 views

finding proections with certain eigenvalues

Find a projection with following eigenvalues: $\sigma$($\emptyset$) $\sigma$(1) $\sigma$(0) I know that a shift matrix has no eigenvalues and that a zero matrix has the eigenvalue 0, but are these ...
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Show A is not similar to a Diagonal Matrix

Find the characteristic polynomial, eigenvalues and eigenvectors of the matrix $A = \begin{bmatrix} 4 & 0 & 0 &0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & -2 & -3 \\ 0 & -1 ...