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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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
16 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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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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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
34 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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13 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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7 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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25 views

matrices and eigen values [NBHM-2014] [on hold]

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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31 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
50 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
45 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 ...
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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 $\phi$ ...
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1answer
49 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
91 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
51 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
55 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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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
32 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
37 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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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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24 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: ...
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52 views

Question about condition number $k$ of a matrix over a finite field

If $\lambda_{max}$, and $\lambda_{min}$ denote the maximum and minimum values of the eigenvalues of a normal square matrix repectively- are there any explicit bounds to the eigenvalues of such a ...
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29 views

Rayleigh quotient iteration and root finding

I'm trying to find the roots of a polynomial by finding the eigenvalues of its companion matrix. I understand that it is possible to use QR algorithm as the matrix happens to be in Hessenberg form ...
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1answer
23 views

properties of largest eignvalue of product of two matrices

I'm searching for the proof of this lemma it's about largest eignvalue of product of two matrices. one of them is positive definete and the other one is symmetric. B is symmetric matrix, A is Positive ...
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1answer
25 views

Cholesky Decomposition and Orthogonalization

I recently came across a methodology for orthogonalizing variables that are collinear, that uses Cholesky Decomposition, but I am not entirely grasping the intuition of it. Let' assume we have three ...
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1answer
31 views

Showing that $M$ and $N$ will have same eigenvalues.

Today I came accross to this problem. And after some study, I have derived the following solution. Request to the experts, kindly let me know if I have made any mistakes. The question is: if $M, N$ ...
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111 views

How do I show that $1$ is not an eigenvalue for $A$, by showing that there are no eigenvectors for $\lambda = 1$.

Consider the matrix $$A=\begin{pmatrix}-1 & 3& 3& 3\\ 3& 1& -1& 5\\ 3& -1& 7& -1\\ 3&5& -1&1\end{pmatrix}.$$ How do I show that $1$ is not an ...
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Prove one of the eigenvector entries has the smallest magnitude

Let $L\in \mathbb{R}^{n \times n}$ be the Laplacian matrix of a simple undirected graph and $D_i$ be the same size matrix with $i$th diagonal element $1$. Denote the smallest eigenvalue of $L+D_i$ as ...
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27 views

Find the limit of this matrix as its power approaches infinity

Find the matrix power, Ak, of A = (v1,v2) v1 = (p,1-p) v2 = (1-p',p') Where v1 and v2 are column vectors, and 0 <= p <= 1, 0 <= q <= 1, p /= q. ...
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2answers
19 views

Quantum mechanics, conmutative operators.

If two operators $A$ and $B$ commute then any eigenvector of $A$ is an eigenvector of $B$? I know that if that happens there is a basis in which the eigenvectors of $A$ and $B$ are equal, but I don't ...
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2answers
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If a matrix is non diagonalizable, what other method can I use to calculate the nth power?

First off, I have this matrix A: 1 0 3 1 0 2 0 5 0 I have calculated the eigenvalues, which are ...
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1answer
32 views

How to solve matrix eigenvalue equation which has a summation.

General problem: If I have some $n \times n$ matrices $\mathsf{M}^\tau$, and column vectors (with $n$ rows) $X^\tau$ is there some mathematical tricks I can do to solve the eigenvalue equation $ ...
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Sturm-Liouville equation with rational coefficient

I am trying to solve a regular Sturm-Liouville type Ordinary Differential Equation (ODE) with the following form: \begin{equation} \frac{d}{dy} [q(y) \frac{dW(y)}{dy}]+p(y) W(y) = -\lambda W(y) ...
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30 views

Eigenvalues that are functions

Let us have the Laplacian on a compact manifold $M$. Suppose I have some equation of the form $$-\Delta u(x) = f(x)u(x).$$ If $f \equiv c$ were a constant, this would be an eigenvalue problem ...
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2answers
25 views

Difference of Positive Semidefinite Matrices

Suppose I have two matrices: $$ A\succeq 0\\ B\succ 0 $$ and I know that $$ \langle v_i,Bv_i\rangle - \lambda_i \geq 0 $$ for every normalized eigenpair $(v_i,\lambda_i)$ of $A$. Is this enough to ...
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2answers
22 views

Argmax and eigenvectors

I am reading a paper (in biology) which performs a clustering algorithm. At one point in the paper, it is stated that: $$ \arg \max_{\lVert X\rVert=1} X^T S\, X $$ can be computed as the normalized ...
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2answers
53 views

Eigenvalues of a special $M \times M$ matrix

I could not obtain an explicit formula for the eigenvalues of matrix $$ \begin{pmatrix} a & b & 0 & 0 & 0 & \cdots & 0 \\ c & a & b & 0 & 0 & \cdots & 0 ...
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1answer
23 views

On matrices sharing the same smallest nonzero eigenvalue and related eigenvector

Suppose that $A$ and $B$ are square matrices with the proper size, then what kind of condition does $B$ have to satisfy such that $A$ and $AB$ share the same smallest(largest) nonzero eigenvalue and ...
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2answers
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Eigenvalue and proper subespace.

I have the follow problem: Suppose that $A,B\in{\cal M}_n(\mathbb{R})$ such that $AB = BA.$ Show that if $v$ is an eigenvector of $A$ associated to the eigenvalue $\lambda$, with $Bv\neq 0$ and ...
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8 views

equivalence of statements involving compact operators

Let T be a compact operator on a hilbert space H.I want to show that the following 2 statements are equivalent. For each $\lambda \in {\bf C} $, let $V_\lambda = \{ x \in H: Tx = \lambda x \}.$ ...
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smallest and largest eigenvalue of discretized operator $-d^2/dx^2$

In 1D, the second order derivative operator $-d^2/dx^2$ can be discretized as, using Matlab ...
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33 views

Determinant and generalized eigenvalues

Let A, B be two symmetric positive-definite matrices. Let $\lambda_i$ be the generalized eigenvalues of the pencil (A,B). Can we write function $\log\frac{|A|}{|B|}$ (where $|\cdot|$ stands for ...
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Computations for LDA: Eigendecomposition

While reading the book Elements of Statistical Learning p. 113, the author used eigendecomposition of the covariance matrix $\hat{\Sigma}_k =\mathbf{U}_k\mathbf{D}_k\mathbf{U}_k^T$ where ...
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1answer
37 views

How do you diagonalize this matrix and find P and D such that A = PDP^-1?

1 1 4 0 -4 0 -5 -1 -8 I3 = 3x3 identity matrix λ 0 0 λI3 = 0 λ 0 0 0 λ λ-1 -1 -4 = 0 λ+4 0 5 1 ...
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18 views

Ortonormal basis of unitary operator and its spectral decomposition - check my solution.

Dear fellow mathematicians, I'm trying to do a linear algebra exercise, but I have no idea whether I have a correct plan of solution. Here is the problem: Find orthonormal eigenbasis (not sure if ...
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Is $\mathrm{col}(\lambda I_n-A)\subseteq \mathrm{col}(B) $ for a complex $\lambda$?

Let $A\in\mathbb{R}^{n\times n}$, let $I_n$ denote the identity matrix of order $n$, and let $ \mathrm{col}$ denote column space. I'm interested in understanding for what values of $\lambda \in ...
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14 views

How can I expand a zonal polynomial in a sum of two matrices? C(A+B) = C(A)*C(B)

I am trying to solve an integral involving zonal polynomials. TO do that I need to somehow separate out the zonal polynomial C(A+B) where my A varies but B is constant. They're all square matrices ...