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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Bound on Signal Amplitude for subspace methods (MUSIC, ESPRIT)

MUSIC and ESPRIT are methods that use subspace decomposition to identify signal Parameters. Subspace decomposition is achieved either by SVD or Eigen Value Decomposition. Subspace decomposition ...
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23 views

matrix diagonalization without eigen decomposition, what other ways available?

I have a matrix, $A$ (it may be symmetric or asymmetric). I need to have a diagonal matrix without eigenvalue decomposition, please suggest what others ways are possible? Any new idea would be much ...
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20 views

If I have a real symmetric matrix and one of the diagonal elements are zero, does that say anything about the eigenvalues?

In the two dimensional case, if I have a matrix $\left[\begin{array}{cc} 0 & a\\ a & b \end{array}\right]$ or $\left[\begin{array}{cc} b & a\\ a & 0 \end{array}\right]$ then I have a ...
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1answer
21 views

Why does this expression for the minimum eigenvalue work?

I read that we can have, for a column vector $d \in R^N$, \begin{eqnarray} min_d = \frac{d' A d}{d'd} = \lambda_{min} \end{eqnarray} , where $\lambda_{min}$ is the smallest eigenvalue of the ...
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2answers
43 views

Finding the eigenvalues of a given Markov matrix

Let $$A = \begin{pmatrix} 0.6 & 0.1 & 0.1\\ 0.1 & 0.8 & 0.2\\ 0.3 & 0.1 & 0.7 \end{pmatrix}$$ I want to find the eigenvalues of this matrix. Because this is a markov matrix, ...
2
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3answers
49 views

To prove that $A$ has a one-dimensional eigenspace , where $A \in SO(3)$ , $A \ne I$

Let $A\ne I$ be a $3\times3$ real orthogonal matrix with determinant $1$ , then how to prove that $A$ has a one-dimensional eigenspace ?
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0answers
16 views

proving bounds by bounding eigenvalues?

I have an expression that takes the form:$\mu_n= \displaystyle (I_d+\sum_{i=1}^n f_i f_i^T)^{-1}(\displaystyle\sum_{i=1}^n f_i)$ where $f_i$'s are d dimensional vectors with bounded $L^2$ norm. $I_d$ ...
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0answers
15 views

Eigensystem of direct sum of matrices with diagonal elements of different order of magnitude

I have got a problem with matrices like, for example: $\left( \begin{array}{cccccc} 1 & 1 & 2 & 1 & 1 & 2 \\ 1 & 1 & 1 & 1 & 3 & 1 \\ 2 & 1 & 1 & ...
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0answers
12 views

Two views on a matrix

An n times n matrix $M$ may be viewed as a linear operator on $\mathbb{R}^n$ to itself, but it is also a linear operator on the function space $L^2 (\mathbb{R}^n)$, which is infinite dimensional: ...
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1answer
37 views

Eigenvalues & eigenvectors of a matrix

I have a couple of questions regarding eigenvalues and eigenvectors. Let $A=\begin{pmatrix}4 & 2 \\ 5 & 1\end{pmatrix}$, $\mathbf{u}=\begin{pmatrix}2\\-5\end{pmatrix},\mathbf{v}=-2\mathbf{u}$ ...
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40 views

Help with Particular Solution of DE?

So I am assigned the task of finding the particular solution for the given problem: $$ y' = 1/t \begin{bmatrix} 1 & t\\ -t & 1\\ \end{bmatrix} + t \begin{bmatrix} cost\\ sint\\ ...
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1answer
15 views

Use eig and svd syntax in matlab to find complex eigenvalues of a matrix

For matrix $A= \left( \begin{array}{c} 1 & 1 \\ -1 & 1 \\ \end{array} \right) $ when I calculate the eigenvalues (without matlab) , I find $\lambda_1=1+1i$ and $\lambda_1=1-1i$ and when I ...
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0answers
43 views

General Solution of DE?

I've got the following ODE, and I'm just having trouble coming up with the form of the general solution. I'm really trying to find the particular solution, but in order to do that, I need to know the ...
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3answers
55 views

If $A$ is a $2\times 2$ matrix with a repeated eigenvalue $r$, then $\mathrm{e}^{At}=\mathrm{e}^{rt}\left[I+(A-rI)t\right]$

If $A$ is a $2\times 2$ matrix with a repeated eigenvalue $r$, show that $\mathrm{e}^{At}=\mathrm{e}^{rt}\left[I+(A-rI)t\right]$. I have already been able to show that if $A$ is a general ...
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0answers
6 views

Has the degree to which a partial eigensystem of a large sparse matrix approximates the complete eigensystem been determined?

Does anyone know of any studies or results regarding the degree of approximation or the error in estimating the complete spectrum of a large sparse matrix by means of its first $n$ eigenvalues and ...
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3answers
60 views

To prove that the sum of the roots of the characteristic polynomial of a square matrix is equal to the trace of the matrix

How do we prove that the sum of the roots of the characteristic polynomial of a square matrix is equal to the trace of the matrix ? I want a proof which does not use much computation or determinants ; ...
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1answer
24 views

Minimal polynomial of f restricted to its image

Let $f:V\to V$ be a $F$-linear map, $V$ an $n$-dimensional vector space over $F$, $\operatorname{rank} E=r$, $W=\operatorname{Im} f$, $\tilde f:=f|_W:W\to W$. Let $\mu$ be the minimal polynomial of ...
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1answer
26 views

Finding roots of characteristic polynomial of 3x3 matrix

I have never learned how to solve cubic equations and unfortunately need to do it in an upcoming exam for finding eigenvalues. I have been searching on the web for good resources, but whenever I find ...
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1answer
41 views

Unit vector that maximizes or minimizes

I know by the Taylor expansion $f(x,y)$ that in order for the origin to be a minimum point, $f_{xx}$ and $f_{yy}$ have to be both positive. Which I know how to prove. I also know other methods like ...
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0answers
17 views

computing leftmost eigenpair of positive-definite matrix

Let $A$ be an $n\times n$ real symmetric positive-definite matrix. Assume that $n$ is large and that $A$ is dense (i.e. it is not sparse). Question: What is the state-of-the-art algorithmically for ...
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1answer
39 views

Generalized eigenvalues of overdetermined systems

I have a system of equations that can be written as ${(\bf{A}} + \lambda{\bf{B}}){\bf{x}} = 0$ Where ${\bf{A}}$ and ${\bf{B}}$ are $n \times m$, integer matrices. I know that there are several ...
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1answer
33 views

Hilbert Spaces - an application of the minimax principle.

Let $A$ be a compact, self-adjoint operator, $A \geq 0$. We need to prove that for any orthonormal system $\{e_i\}_1^{\infty}$ and for any $N$, $$\sum_1^N \langle Ae_i,e_i \rangle \leq \sum_1^N ...
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1answer
18 views

find the eigenvalue and eigenvector of a matrix over complex field [closed]

Hi im working on a eigenvalue question of a matrix over complex field. The matrix is: 1 -1 2 -1 For eigenvalue i got i,-i, but i cant get any non-zero ...
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23 views

Eigenvector and 2D Rotation

I have a problem where i have a 2x2 matrix and need to rotate the coordinate system to make it a diagonal matrix. The solution involves calculating the eigenvector of this matrix. Considering that a ...
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0answers
36 views

Test for powers method

I have been told that for a normal matrix $A$, the powers method (i.e. computing the succession of Rayleigh quotients for a succession of vectors $z_k=A\cdot z_{k-1}$) can use the following stop ...
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1answer
20 views

Hilbert Spaces; eigenvalues of $PBP$ vs. $B$ for $B$ compact selfadjoint and $P$ orthoprojection.

An exercise I have come upon while studying Hilbert Spaces: Let $A$ be a compact operator, and $P \in L(H)$ be an orthoprojection. Prove that $$\lambda_n (PA^*AP) \leq \lambda_n (A^*A)$$ (Where ...
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1answer
27 views

Is the largest eigenvalue a unique weighted sum of the linear combination of the elements of a matrix?

Let $\lambda$ be the largest eigenvalue of $\boldsymbol{A}\in\mathbb{C}^{n\times n}$ ($\boldsymbol{A}$ is hermitian). Is $$\lambda = ...
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1answer
17 views

Effect of the nature of noise on the spectrum of a random matrix

Consider the following two equations $X = M + \eta_1$ $Y = M + \eta_2$ where, $X\in\mathrm{R}^{n\times n}$, ia a real random matrix with mean $M\in\mathrm{R}^{n\times n}$. $\eta_1$ is Gaussian ...
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0answers
12 views

What is the effect on the eigenvalues of reducing a column of a stochastic matrix.

The following is for any 2 right stochastic matrices $A_x$ & $A_y$ of equal size $n$x$n$ with known eigenvalues $\lambda_{x1}-\lambda_{xn}$ and $\lambda_{y1}-\lambda_{yn}$ respectively. Also given ...
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0answers
10 views

Issues with connecting the SVD and Eigenvalues for block matrix

In class, we have talked about the singular value decomposition and its connection to Eigenvalues. Specifically, for a matrix A, if the columns of a matrix contain linearly independent eigenvectors, ...
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2answers
36 views

Eigenvalues and Eigenvector of matrix

I try to find the eigenvectors and eigenvalues from the matrix: $$M =\pmatrix{1/5 & 2/5 \\ 2/5 & 4 /5}$$ I started like this: $$M = \pmatrix{1/5 - \Delta & 2/5 \\ 2/5 & 4 /5 ...
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0answers
15 views

Question about eigenvalue problem of a selfadjoint operator.

Let $x=(x_1,x_2)$, and let $X_m$ denote the space of homogeneous polynomial vector fields on $\mathbb{R}^2$ of degree $m$. For example if $m=2$ a vector field $U\in X_2$ is of the form $$ ...
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1answer
10 views

Covariance matrix computed based on a covariance function

I am reading Chapter 4 of Gaussian Processes for Machine Learning. It says that a matrix $K$ whose entries are computed as $k_{ij} = k(x_i, x_j)$ where $k$ is a covariance function is a positive ...
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35 views

matrices and eigen values [NBHM-2014] [closed]

In each of the following cases, describe the smallest subset of $\Bbb{C}$ which contains all the eigenvalues of every member of the set $S$. a. $S = \{A ∈ M_n(\Bbb{C}) | A = BB^*, B \in ...
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1answer
34 views

Eigen vectors of symmetric matrix

The eigen values of following matrix are $4, -2,-2$ and corresponding eigen vectors are $(1,1,1), (-1,1,0), (-1,0,1)$. But as the matrix is symmetric the eigen vectors has to be orthogonal, where as ...
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1answer
52 views

Derivation of power method

POWER METHOD Let $x_0$ be an initial approximation to the eigenvector. For $k=1,2,3,\ldots$ do Compute $x_k=Ax_{k-1}$, Normalize $x_k=x_k/\|x_k\|_\infty$. Then $\|x_k\|_\infty$ approaches the ...
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4answers
46 views

Finding the diagonalizing matrix.

Find a nonsingular matrix $C$ such that $C^{-1}AC$ is a diagonal matrix. $$ A=\begin{pmatrix} 1 & 0 \\ 1 & 3 \\ \end{pmatrix} $$ I have found the eigenvalues to be 1 ...
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1answer
29 views

Square root of a matrix proof

Let $B$ be a real symmetric $2 \times2$ matrix which satisfies: $$\sqrt{B}v_1=\lambda_1v_1$$ $$\sqrt{B}v_2=\lambda_2v_2,$$ where $v_1,v_2$ are eigenvectors of matrix $B$ and ...
3
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2answers
39 views

Eigenvalues of Differential Equation with Boundary Condition

Here is a problem from my homework assignment that I am struggling with: Consider the differential equation $\frac{d^2\phi}{dx^2}+\lambda\phi=0 $. Determine the eigenvalues $\lambda$ if $\phi$ ...
3
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1answer
50 views

Why is PageRank an eigenvector problem?

You hear all the time that PageRank uses eigenvectors. But for those who don't really understand what eigenvectors are, it is unclear why Pagerank needed to invoke eigenvectors and eigenvalues in ...
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1answer
25 views

Critical points characterization of real function

Providing a real multi-variable function $f(\bar{x})$ twice differentiable with respect of all its variables. Looking for critical points is equivalent to solve $\nabla f = \vec{0}$. And to ...
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1answer
93 views

How prove this matrix $B^{-1}-A^{-1}$ is positive-semidefinite matrix,if $A-B$ is positive matrix

Question: Let $A,B$ be positive $n\times n$ matrices, and assume that $A-B$ is also a positive definite matrix. Show that $$B^{-1}-A^{-1}$$ is a positive definite matrix too. My idea: ...
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3answers
52 views

Basis of eigenfunctions in Banach space

I have a question about proving the existence of a basis of eigenfunctions. Assume we have a compact operator $L:\mathcal{B}\rightarrow\mathcal{B}$, where $\mathcal{B}$ is a Banach space of analytic ...
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2answers
58 views

Prove that S is diagonal

Let $S: V\rightarrow\ V$ be an operator on an $n$-dimensional real vector space with an eigenvalue that has geometric multiplicity equal to $n-1$. Prove that $S$ is diagonal. Give an example of such ...
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0answers
19 views

Eigenvalue of Heun's function and its computation

It is known that the Heun's differential equation: \begin{equation} \frac{d^2 w}{dz^2} + (\frac{\gamma}{z}+\frac{\delta}{z-1}+\frac{\epsilon}{z-a})\frac{dw}{dz}+\frac{\alpha \beta z -q}{z(z-1)(z-a)} ...
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2answers
33 views

Eigenvalues of the product of two matrices

Let $A$ and $B$ be $m \times n$ and $n \times m$ real matrices. I was asked to prove that if $\lambda$ is a nonzero eigenvalue of the $m \times m$ matrix $AB$ then it is also an eigenvalue of the $n ...
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1answer
39 views

Linear Algebra Problem Proof

I have been stuck on this problem for quite some time now and, unfortunately, appear to have given up. Perhaps the minds on this page will help me out. Given an $n\times n$ matrix D, where ...
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2answers
26 views

Find solutions for an differential equation system

I have a differential equation system $x_1'(t) = -x_2(t)$ $x_2'(t) = -x_1(t)$ I see that I can write $\dot{x} = Ax$ where $A = \begin{pmatrix}0 & -1 \\ -1 & 0\end{pmatrix}$ The complete ...
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0answers
19 views

Find real solution for an inhomogene system

I have an inhomogene differential equation system $\begin{pmatrix}\dot{x}_1 \\ \dot{x}_2\end{pmatrix} = \begin{pmatrix}-1 & 3 \\ -3 & -1\end{pmatrix} \begin{pmatrix}x_1 \\ x_2\end{pmatrix} + ...
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0answers
25 views

What is the proof/show that the post of linear transformation generated by LDA is at most k-1

What is the proof/show that the matrix $Sw$ generated by LDA is at most rank $p-k$, where $p$ is the dimension of the data and $k$ is the number of classes. LDA: ...