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 $\lambda x - Tx = 0$.

learn more… | top users | synonyms (2)

1
vote
1answer
9 views

How does this form of Poincare's inequality for self-adjoint matrices hold?

I'm reading "Introduction to Matrix Analysis and Applications" by Hiai and Petz, and they state Theorem 1.26 ("Poincare's Inequality") as follows: Let $A\in B(H)$ be a self-adjoint operator with ...
0
votes
1answer
13 views

The Maximum Eigenvalue of $F\mathrm{max(B)}F^T - FBF^T$

$F$ is a $b \times n$ real matrix. $B$ is a $n \times n$ real matrix, constructed by $B = w^T w$, where $w$ is a row vector with strictly positive real numbers, and clearly $B$ is a rank 1 matrix. ...
3
votes
0answers
48 views

Show that $A^k$ has eigenvalues $\lambda^k$ and eigenvectors $v$.

I want to prove the following statement: Let $A \in \Bbb R^{n\times n}$ with eigenvalues $\lambda$ and eigenvectors $v$. Show that $A^k$ has eigenvalues $\lambda^k$ and eigenvectors $v$. ...
12
votes
2answers
389 views

Are eigenvalues of the limit of a sequence of matrices limits of eigenvalue sequences?

Let $\{A_n\}\in \mathbb{R}^{m\times m}$ be a sequence of symmetric matrices such that $A_n\to A$ as $n\to \infty$, i.e. $\lim_{n\to \infty}a_{ij}(n)=a_{ij}\ \forall 1\le i,j\le m$ where ...
0
votes
0answers
26 views

Is the converse of the Spectral Theorem true?

In the book by Friedberg, Insel and Spence, symmetric matrices are orthogonally diagonalizable, and over the complex number field, normal matrices are orthogonally diagonalizable -- this is all from ...
2
votes
4answers
259 views

If a matrix has positive, real eigenvalues, is it always symmetric?

We know that symmetric matrices are orthogonally diagonalizable and have real eigenvalues. Is the converse true? Does a matrix with real eigenvalues have to be symmetric? A class of symmetric ...
-5
votes
0answers
27 views

Solve the eigenvalue problem : [on hold]

Solve this eigenvalue problem: $$ x^2 y'' + x y' = \lambda y, \quad y(e^\pi)= y(e^{2 \pi}) = 0$$
2
votes
2answers
32 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 ...
0
votes
0answers
15 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 ...
0
votes
2answers
27 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
votes
2answers
101 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 ...
1
vote
1answer
15 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 ...
1
vote
0answers
38 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 ...
1
vote
0answers
51 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: ...
1
vote
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 ...
1
vote
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?
1
vote
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. ...
1
vote
1answer
22 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 ...
0
votes
1answer
40 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 ...
0
votes
1answer
64 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? ...
5
votes
3answers
72 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 ...
2
votes
3answers
86 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
votes
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 ...
2
votes
1answer
77 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 ...
0
votes
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
votes
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
votes
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 ...
0
votes
1answer
25 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 - ...
-1
votes
0answers
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 ...
0
votes
0answers
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 ...
1
vote
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 $S$)
0
votes
0answers
24 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 ...
0
votes
0answers
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) - ...
0
votes
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 ...
0
votes
0answers
18 views

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 ...
9
votes
0answers
48 views

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 ...
0
votes
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 ...
1
vote
1answer
21 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 , ...
0
votes
1answer
46 views

Generalized eigen vectors of real symmetric matrix

From generalized characteristic equation, $Av=DBv$, where $A$ is a real symmetric matrix, $D$ is a diagonal matrix containing the eigenvalues as diagonal elements, and $v$ is the eigen vector ...
0
votes
0answers
20 views

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 ...
1
vote
0answers
26 views

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} ...
3
votes
0answers
55 views

$L^\infty$-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 ...
2
votes
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} ...
0
votes
0answers
11 views

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, ...
0
votes
0answers
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 ...
-2
votes
1answer
46 views

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 ...
1
vote
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 ...
1
vote
0answers
101 views

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 ...
4
votes
2answers
55 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 ...
0
votes
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 ...