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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Solve the eigenvalue problem : [on hold]

Solve this eigenvalue problem: $$ x^2 y'' + x y' = \lambda y, \quad y(e^\pi)= y(e^{2 \pi}) = 0$$
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2answers
24 views

Does the lowest diagonal element of a real symmetric matrix form an upper bound to the lowest eigenvalue?

If I have a real symmetric matrix, is it possible to look at the lowest diagonal element and then claim that the lowest eigenvalue of the matrix must be less than or equal to that diagonal element? I ...
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12 views

Algorithm for getting Markov chain given the complex eigenvalues

Given real and complex eigenvalues (occurring in conjugate pairs) how to get a single instance of a Markov Chain which has these eigenvalues. I know the Markov chain is not unique as eigenvectors are ...
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2answers
26 views

Algorithm for real matrix given the complex eigenvalues

Given complex eigenvalues (occurring in conjugate pairs) how to get a single instance of a real matrix which has these eigenvalues. I know the matrix is not unique as eigenvectors are not fixed but in ...
5
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2answers
91 views

Eigenvalues of linear operator $F(A) = AB + BA$

Let $B$ be the $n \times n$ square matrix; $\lambda_1, \lambda_2, \dots, \lambda_n$ are its pairwise distinct eigenvalues. For all $n \times n$ matrix $A$ let me define $F(A) = AB + BA$. We can ...
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1answer
14 views

Constructing matrices with eigenvalues equal to roots of a given polynomial

Suppose we are given a polynomial e.g. $$x^4+Ax^3+Bx^2+Cx+D,\tag1$$ and we need to construct a matrix, whose eigenvalues would be equal to roots of this polynomial. One way, rather inelegant, is to ...
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32 views

clarification on eigendecomposition of a matrix

looking for some clarification on a couple things related to the eigendecomposition of a square matrix. Suppose we have a square n x n matrix, A, and we are interested in finding its eigenvectors and ...
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35 views

Impact of random numbers on the eigen-values

How do the eigen-values of the following tridiagonal matrix ($A$) change when adding random numbers $R_i$ (with the condition $|R_i| < m$) to its diagonal. A is a square matrix defined as follows: ...
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2answers
26 views

Orthonormal basis for the null space of almost-Householder matrix

A matrix $H$ is defined as: $$H = I - vv^T$$ where $v$ is a unit vector. What is the rank of $H$? What would be an orthonormal basis for the null space of $H$? How do we find the number of zero ...
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3answers
33 views

Eigenvalues of Householder matrix

What would be the eigenvalues for a Householder matrix defined as: $H = I - 2 u u^T$? Can someone explain it to me intuitively or with a simple proof?
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1answer
20 views

Eigenvalues of an upper Hessenberg matrix

I'm interested in calculating the roots of an 11th degree polynom. To do so, I calculated the 10x10 companion matrix which eigenvalues are the roots of the polynom. ...
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1answer
21 views

Eigenvalue Deflation (Wielandt or Hotelling)

I am doing a project on eigenvalue deflation techniques and I wanted to include some examples of deflation giving poor results (results with high accumulated error). Ideally the examples would be ...
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1answer
38 views

Matrix with entries equal to $1$ and $-1$ (Sign Matrix)

What can we say about the determinant and (or) maximum eigenvalue of a matrix with entries equal to $1$ and $-1$. Further assume that the rows and columns are linearly independent. Are there special ...
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1answer
63 views

Characteristic polynomial of A, if $\det(\operatorname{adj}(\operatorname{adj}(A))) = 81$?

Let $A$ be a square real matrix whose eigenvalues are positive integers, with $$\det(\operatorname{adj}(\operatorname{adj}(A))) = 81 \, .$$ What is the characteristic polynomial of A? Any hints? ...
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3answers
67 views

Optimal approximation of quadratic form

Let $\mathbf{x}\in\Bbb{R}^n$ and $A\in\Bbb{S}_{++}^n$, where $\Bbb{S}_{++}^n$ denotes the space of symmetric positive definite $n\times n$ real matrices. Also, let $Q\colon\Bbb{R}^n\to\Bbb{R}_{+}$ be ...
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3answers
85 views

Find the necessary and sufficient condition for $A^m\to0$

Let $A$ be $n\times n$ matrix on $\mathbb{C}$. Find a necessary and sufficient condition for $A^m\to0$ as $m\to\infty$. My thought: I think it should be that eigenvalues of $A$ are less than $1$. ...
0
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1answer
48 views

Decide the range of eigenvalues for $A+B$

Let $A,B$ be $n\times n$ Hermitian matrices on $\mathbb{C}$ such that all eigenvalues of $A$ lie in $[a,a']$ and all eigenvalues of $B$ lie in $[b,b']$. Show that all eigenvalues of $A+B$ lie in ...
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1answer
74 views

Show that $\frac{\int_\Omega|\nabla u|^2+\int_\Omega\alpha|u|^2}{\int_\Omega|u|^2}$ attains a minimum in $W_0^{1,2}(\Omega)$

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain $H:=W_0^{1,2}(\Omega)$ be the Sobolev space $|\;\cdot\;|_p$ be the seminorm $$|u|_p^p:=\int_\Omega|\nabla u|^p\;d\lambda^n\;\;\;\text{for ...
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1answer
14 views

If complex matrix 2*2 has a real eigenvalue then matrix of its conjugate elements has a real eigenvalue too

If $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ $\in$ $\mathbb C^{2x2}$ has a real eigenvalue then $\begin{pmatrix} \overline a& \overline b\\ \overline c&\overline d\end{pmatrix}$ $\in$ ...
3
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1answer
34 views

Is it possible, that the fist two weak eigenvalues of $-\Delta$ in a bounded domain are equal?

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain $\lambda_1$ be the first weak eigenvalue of $-\Delta$ in $\Omega$ $\varphi_1$ be the weak eigenfunction associated with $\lambda_1$ ...
2
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1answer
28 views

If $l_i$ is the first weak eigenvalue of $-\Delta$ in a domain $G_i$ and $G_1\subseteq G_2$, then $l_1\ge l_2$ and equality is possible

Let $\Omega_i\subseteq\mathbb{R}^n$ be a domain $\lambda_i$ be the first weak eigenvalue of $-\Delta$ in $\Omega_i$ It's easy to verify that $\Omega_1\subseteq\Omega_2$ implies $\lambda_1\ge ...
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1answer
23 views

existance of a solution to quadratic form equation

Let $\lambda$ is an unknown scalar and; $Q=Q_1 - \lambda*Q_2$ where $Q_1, Q_2$ are $NxN$ positive symetric matrices, $B=B_1 - \lambda*B_2$, where $B_1, B_2$ are $Nx1$ vectors, $m=m_1 - ...
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33 views

Impact of perturbation on the eigen-values of 3 diagonal matrix [on hold]

Lets consider a 3-diagonal matrix as following: $$ A(i,i) = 2 $$ $$ A(i,i+1) = -1 $$ $$ A(i,i-1) = -1 $$ The eigen-values of this system is known easily. How eigen-values would change if we add ...
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11 views

How to optimize a generalized trace problem in dimensionality reduction

I know how to solve this problem in dimensionality reduction. $argmax_{X}$ $Trace[XLX^T]$ with $XX^T=I$ ,where $L$ is symmetric, $X$ is unitary, and $I$ is identity matrix. But I'd like to know how ...
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1answer
29 views

If $S:(x_0,x_1,x_2,\ldots) \to (0,x_0,x_1,x_2,\ldots)$ Why does $0 \in \sigma (S)$? and what is $\dim(R(S))$ in $\ell^2$?

Consider the unilateral shift $S$ on $\ell^2$ defined by $$(x_0,x_1,x_2,\ldots) \to (0,x_0,x_1,x_2,\ldots)$$ Why dose $0$ is eigenvalue of $S$? and what is $\dim(R(S))$ in $\ell^2$?( Range of $S$)
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23 views

FFT Hyperbolic Distribution R

This is my first posting so forgive me if it is not 100% in line with this forum's best practices. I am completing an analysis using ICA as the decomposition technique. I am keeping 4 of the 10 ...
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18 views

Computing the Log-Euclidean distance efficiently by using eigen-analysis.

Let $A,B\in\Bbb{S}_{++}^n$ be two symmetric positive definite $n\times n$ matrices with real entries. The Log-Euclidean distance between these matrices is defined as follows $$ d = \lVert \log(A) - ...
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1answer
13 views

Relationship between eigenvectors and singular vectors of a Hermitian matrix?

What is the relationship between the eigenvectors and singular vectors of a Hermitian matrix? Intuitively, I would expect them to be the same (modulo scaling). However, this doesn't seem to be the ...
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Convex functionals, min-max of linear functionals, etc

I was reading in this book on the well-known Courant-Fischer min-max Theorem, which states that if $\lambda_1(A)\geq \lambda_2(A)\geq \lambda_n(A)$ are the $n$ eigenvalues of a $n\times n$ Hermitian ...
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What can we say about the graph when many eigenvalues of the Laplacian are equal to 1?

The Laplacian of the graph has all the eigenvalues real and non-negative, the smallest being 0. I have a graph where the second smallest eigenvalue (the so called algebraic connectivity) is equal to ...
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1answer
26 views

prove the similar matrices have the same rank

I've seen some proofs on the Internet, which make use of the transformation map. But I couldn't understand the methods since What I learnt about the transformation map is so superficial. Can you use a ...
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1answer
20 views

Show that spectral radius is lower than 1

Consider the following matrix: $I - \frac{1}{h^2}\mu \Delta t A$. Where $A$ is an NxN matrix. The eigenvalues of A $\lambda_j$ are given by $4sin^2(\frac{j\pi}{2(N+1)})$ for $j=1,...,N$. And $\mu , ...
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Shifting eigenvalues to the left by shifting diagonal elements

Let $D$ be a real diagonal matrix with non-positive elements, not all zero. For any real or complex square matrix $A$, it is true that the eigenvalues of $A+D$ are the eigenvalues of $A$ shifted ...
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Is there any structure in matrices computed from eigen vectors?

Let $\mathbf{A}\in \mathbb{R}^{nn}$ be a positive definite matrix with eigen value decomposition so that \begin{equation} \mathbf{A}=\mathbf{V}\mathbf{\Lambda}\mathbf{V}^T=\sum_{k=1}^{n} ...
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$L^\infty$-bounds on eigenfunctions of Laplace-Beltrami opeator

Let $w_k$ be the eigenfunctions of the Laplace-Beltrami operator on a compact manifold $M$ without boundary. We assume that $\{w_k\}$ are orthonormal, thus $\|w_k \|_{L^2} = 1$. We know $w_k$ are ...
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1answer
45 views

Prove that the product of 2 vectors Normally distributed converges for large dimensions to the full zero matrix

Let $\mathbf{x}, \mathbf{y}$ $\in C^{M \times 1}$ are two i.i.d. vectors with distribution $\mathcal{CN(0,1)}$. How we can prove by the strong law of large numbers that: $\lim_{M\rightarrow \infty} ...
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pseudo-Wishart distribution with shifted rows

I have a problem and I don't know where to start finding a solution. The problem is that I have a vector of i.i.d normal random variables such that, ...
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14 views

Calculation of second derivative of Rayleigh quotient

I have an eigenvalue problem of the form [(A-kB)V=0] and I calculate the eigenvalues k and left (Vl) and right (Vr) eigenvectors using the qz command in matlab. For verification reasons, I also ...
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1answer
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If a $n \times n$ complex matrix $A$ satisfies $A^k=I_n$ and does not have eigenvalue $1$, then which of the following are necessarily true… [closed]

An $n\times n$ complex matrix $A$ satisfies $A^k=I_n$ where $k> 1$. Suppose $1$ is not an eigenvalue of $A$. Then which of the following are necessarily true- $A$ is diagonalizable. $A+A^2 \dots ...
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1answer
29 views

What happens if the power method is applied with a starting vector $q=c_2 v_2+…+c_n v_n$ in the presence of roundoff errors?

Supose $\{v_1,...,v_n\}$ is an eigenvector basis and $|\lambda_1|>|\lambda_2|>\ldots >|\lambda_n|>0$, so, my question is, if our starting vector $q \in span\{v_2,\ldots,v_n\}$ and in the ...
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If $A$ is a $3 \times 3$ matrix. and $B=A'A$, then what can be said about the eigenvalues of $AB$?

If $A$ is a $3 \times 3$ matrix and $B=A'A$, then what can be said about the eigenvalues of $AB$? No form of $A$ is given; then how to proceed ? Can this problem be at all solved? If anyone can ...
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2answers
53 views

How to modify a matrix to push all of its eigenvalues into the unit circle?

Let $A$ be a strictly positive $n \times n$ matrix. That is, $a_{ij} >0, \ \forall i,j \in \{1,...n\}$. If some of eigenvalues of $A$ are outside or on the unit circle, I was wondering if I can ...
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1answer
32 views

Eigenvalues within the unit circle

Let $P$ be a positive $n×n$ matrix. That is $P_{ij} > 0,\ \forall i,j \in \{1,...,n\}$. I am aware that if all row sums of $P$ are smaller than 1, then the Perron–Frobenius eigenvalue, the largest ...
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Generalized eigenvectors of the identity matrix or any other matrix

I am learning generalized eigenvectors at the moment, and I am not totally sure I get them yet. $v\in V$ is a generalized eigenvector if $v\ne 0$ and $v\in null(T-\lambda I)^j$ for some $j\gt 0$. ...
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Eigenvalues and eigenvectors after a congruence transformation

Say I have a symmetric matrix $A$ and a symmetric matrix $B$ such that $B$ is congruent with $A$, i.e. there exists a non-singular matrix $X$ such that $B = X^TAX$. Is there a general relation between ...
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1answer
34 views

Finding normalized eigen vectors

I want to find normalized eigen vectors for: $$ \begin{pmatrix} 1 & -2 & 0 \\ -2 & 5 & 0 \\ 0 & 0 & 2 \\ \end{pmatrix} $$ The eigen ...
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1answer
27 views

constructing a matrix such its square is not '0' but its cube is.

i have been asked to construct a matrix A such that $A^2$ is not equal to '0' but, $A^3=0$. how should i proceed. i can only understand that all the eigenvalues for A , $A^2$ and $A^3$ will be ...
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2answers
32 views

what are the general form of eigenvalues of an orthogonal matrix

could anyone help me to understand which fatal mistake is there..? I mean, I can understand , that orthogonal matrices can have complex eigenvalues. but then, what fallacy is there in that proof ? ...
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2answers
29 views

Isometries and Preservation of Eigenvalues

Does conjugation by an isometry preserve eigenvalues? If not, are there certain (non-trivial) situations where it does?
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3answers
32 views

Trouble finding eigenvectors

Let $A= \begin{bmatrix} 2 & 0 & 3 \\ 3 & -3 & -2\\ 2 & 0 & -2\\ \end{bmatrix}$ I have trouble finding the correct eigenvector for the eigenvalue $\lambda=-3$. $[A - ...