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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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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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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Linear Algebra Orthogonal Real Matrix [on hold]

Let O be an nxn orthogonal real matrix, i.e. OTO=In, where In is the identity matrix. Prove that 1) Any entry in O is between -1 and 1. 2)If gamma is an eigenvalue of O, then absolute value of gamma ...
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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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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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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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29 views

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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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 ...
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Leading eigenvalues of large sparse unsymmetric matrix

I have a matrix R which is sparse and all eigenvalues are -ve with a zero eigenvalue. Size of R is more than 1 million X 1 million. But I need to calculate only few large (by value not by magnitude) ...
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1answer
39 views

Normal Matrix Having all real eigen values is Hermitian

$A$ is a normal matrix (i.e. $AA^*=A^*A$, where * denotes the hermitian conjugate). If all its eigenvalues are real, prove that it is Hermitian (i.e. $A^*=A$). I have tried many things but could not ...
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34 views
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Eigenvalues for the Sturm-Liouville boundary value problem

Please show me how to calculate the eigenvalues for the following boundary value problem: $$x''+\lambda x=0\\x(0)=0\\x(\pi)=0\\x'(\pi)=0$$ This is what I did: let $\lambda=\mu^2$ $$X(x)=A\cos\mu ...
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2answers
25 views

Need verification - Given a Hermitian matrix and two eigenvectors corresponding to distinct eigenvalues, show x and y are orthogonal.

Claim: Let $A \in \mathbb{C}^{mxm}$ be hermitian ($A = A^*)$. If $x$ and $y$ are eigenvectors corresponding to distinct eigenvalues, then x and y are orthogonal. Proof: Let $x$ and $y$ correspond to ...
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2answers
15 views

Continuity in finding eigenvectors

I'm wondering whether there's "continuity" in the eigen vectors of different matrices corresponding to appropriate eigenvalues. For instance, if we change certain elements in a matrix, can we ...
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1answer
32 views

3-sigma Ellipse, why axis length scales with square root of eigenvalues of covariance-matrix

This is my first post on math.stackexchange and i am not a mathematician, but i took some undergrad math courses and some grad mathematical modelling courses, so i come with a basic understanding of ...
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42 views

How do you calculate this third eigenvector in this 3x3 matrix?

Scroll down to the bottom if you don't want to read how I arrived at my original two answers. My question is how are all the online calculators I check coming up with this third eigenvector (1, 1, ...
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32 views

About lemma $\rho(A) \leq \|A^k\|^{1/k}$

In the Spectral radius wikipedia article in section Matrices there is a lemma, what states that: Lemma. Let $A \in \mathbb{C}^{n \times n}$ be a complex-valued matrix, $\rho(A)$ its spectral ...
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61 views

Minimum eigenvalue of product of two matrices

Abstract description: Let $\mathbf{A}$ and $\mathbf{B}$ be two $n \times n$ real matrices. Let $\sigma( \mathbf{A B} )$ denote the spectrum of $\mathbf{A B}$. Assume that (A1) $\mathbf{A}$ is ...
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23 views

Define a positive dot product in $\mathbb{R^3}$

Consider the matrix $A= \begin{bmatrix} k & k-1 & 0 \\ 1-k & 2-k & 0 \\ 2k-3 & 2k-1 & 2 \end{bmatrix} $ with $k \in \mathbb{R}$ and let be $f_a: \mathbb{R^3} \rightarrow ...
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How to derive the spectral projection operator in finite element method?

For a compact self-adjoint operator $T$, the spectral operator can be defined as follows: $$E(\lambda) = \frac{1}{2\pi i}\int_{\Gamma}(z-T)^{-1}dz$$. For a finite dimensional approximate operator ...
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Which matrix norm gives the minimal variation of eigenvalues?

This is a follow-up of this question. The original question is intentionally as general as possible, because I was interested in the most general possible answer. I am now trying to understand its ...
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1answer
26 views

Finding eigenvalues and “eigenmatrices”.

On the space of $2\times 2$ matrices, let $T$ be the transformation that transposes every matrix. Find the eigenvalues and "eigenmatrices" for $A^T =\lambda A$. By taking determinants on the left and ...
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1answer
42 views

Is there a theorem about eigenvalues of sum of matrices?

Let's suppose I know $\lambda_i$ to be an eigenvalue of the real negative definite square matrix $A$ of size $n$. $\lambda_i$ can either be real or complex (and $\lambda_{i+1}$ will then be its ...
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40 views

Build up a not diagonalizable linear map

I need an hint for this problem. Let be $M = \begin{bmatrix}2 & 1 \\ -2 & 0\end{bmatrix} \in M_2(\mathbb{K})$ and $H=\{A \in M_2(\mathbb{K}) : AM=MA \} $ Build up a linear map $f: ...
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Finding the eigenvectors and the diagonal of a singular 2x2 matrix

i am trying to find the eigenvectors of a 2x2 singular matrix, A = [0 , 1 ; 0 , -3]. My problem is that i can't. I know the answer is, Q = [1 , 1 ; 0 , -3] (by using Matlab), but i don't understand ...
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Katz centrality and Node removal

I'm looking for any results about how Katz centrality changes when a given node is removed from a graph. For instance, if I define a function to be the average of the Katz centrality of the remaining ...
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2answers
44 views

How to find eigenvalues of the following block circulant matrix

I have a block matrix of size PN x PN of the form: Where A and C are P x P matrices. I would like to find the eigenvalues of the matrix B, that is where
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1answer
27 views

Finding a basis for eigenspace problem

I need to find the basis of the eigenspace for the matrix A for the eigenvalue of 4 $$A = \begin{bmatrix} 1 & 3 & 3\\ 3 & 1 &-3\\ -3 & 3& 7\\ \end{bmatrix}$$ ...
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3answers
44 views

Find the Eigenvector of a matrix

Find the eigenvectors of the matrix $$\displaystyle\begin{bmatrix} 0 &2 &3 \\ -2 &0 &5 \\ -3 &-5 &0 \end{bmatrix}.$$ So I start with $|A-\lambda I|=0$ ...
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1answer
48 views

A characterization of a certain family of matrices in terms of another matrix.

Consider a real matrix $A$ of dimension $n \times n$. Assume $k \leq n$ is given. I am looking for ways to describe the following set of matrices in terms of properties of $A$. $\mathcal{S}(A) = \{B ...
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Lower and upper bound for the largest eigenvalue

We will call a matrix positive matrix if all elements in the matrix are positive, and we will denote the largest eigenvalue with $\lambda_{\max}$, what is exist because of the Perron–Frobenius ...
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1answer
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Perron–Frobenius theorem

What is exactly the Perron–Frobenius theorem? In different books papers I read different statments, and I don't know what is the truth. In wikipedia there are also a lot of statements under this ...
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1answer
141 views
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Theorem about positive matrices

We will call a matrix positive matrix if all elements in the matrix are positive, and we will denote the largest eigenvalue with $\lambda_{\max}$, what is exist because of the Perron–Frobenius ...
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1answer
18 views

Example of a proper node?

I know that a proper node is a node with repeated eigenvalues, but with 2 linearly independent eigenvectors. An improper node has repeated eigenvalues, and 1 linearly independent eigenvector. I cannot ...
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1answer
43 views

Is it possible that a matrix depicts like this?

Is it possible for $A\in \mathbb{C}^{n\times n}$, that $$\frac{|Ax|}{|x|}>|\lambda_{max}|$$ where $\lambda_{max}$ is the biggest eigenvalue of A? I know this can not happen, if there is a basis of ...
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1answer
24 views

Finding eigenvectors for the largest eigenvalue vs one with the largest absolute value

If I want to solve a generalized eigenvalue problem such as: $$A x = \lambda x$$ The problem is to find eigenvectors corresponding to the largest eigenvalues (sometimes in an optimization problem ...
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1answer
34 views

Eigenvalues of ad (Adjoint action) in semisimple lie algebra?

Suppose $V=V_0\oplus V_1$ be a $Z_2$-graded semi-simple lie algebra and, $\xi\in V_1$. The maps $ad_\xi \circ ad_\xi :V_0\longrightarrow V_0$ and $ad_\xi \circ ad_\xi :V_1\longrightarrow V_1$ are ...
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33 views

Diagonalization of a matrix with change of basis

I was trying to diagonalize a not really nice matrix doing first a change of basis but I noticed that the two characteristic polynomials I get are different. Original matrix and its characteristic ...
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1answer
65 views

Continuity of the spectral radius

Let $M \in \mathbb{R}^{n\times n}$ be a nonnegative irreducible matrix with strictly positive entries on its main diagonal. Then $M$ is primitive and by the Perron-Frobenius Theorem we know that the ...
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What is known about the eigenvectors of random matrices?

Let $A$ be a real asymmetric $n \times n$ matrix with i.i.d. random, zero-mean elements. What results, if any, are there for the eigenvectors of $A$? In particular: How are the elements of the ...
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Same eigenvalue spectrum with different matrices

There are two matrices $D$ and $T$ which provides eigenvalue spectrum (dispersion relation) according to $$ E(K) = \operatorname{eig}(D + T \exp(iK) + T' \exp(-iK)) $$ $$ K = 0:dK:\pi $$ Where K is a ...
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1answer
46 views

Under what conditions are the eigenvalues of a matrix finite?

Suppose we have a square matrix $A$. Under what conditions on $A$ ensure that all eigenvalues of $A$ are finite?
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37 views

Positive definite matrix. [closed]

How to illustrate the positive definite matrix in vector space by using the eigen values and eigen vectors?
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73 views

Compute an upper bound on generalized eigenvalues (by using the coefficients)

Consider the generalized, symmetric eigenvalueproblem: \begin{equation} A x = \lambda B x, \end{equation} with $A, B$ symmetric and $B$ being positive definite. For some computations, i was trying ...
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Relationship between eigen-vector and adjacency matrix nodes

My question is short and simple. I am wondering the following: lets say I have a adjacency matrix of a graph lets say NxN and λ stands for the highest eigen-valueand u for the correspondant ...
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Convex combination of polynomials with roots on the unit circle and companion matrix

Given two $N^{th}$ order polynomials $P_0(z)$ and $P_1(z)$, let their roots be $w_k$ and $z_k$ respectively. All the roots of both polynomials lie on the unit circle $\mathcal{U}$. Also $w_i \neq z_j$ ...
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1answer
24 views

Eigenvalues and eigen vectors

Is it possible to have a matrix for which eigen vectors won't change by changing the eigen values? Please help me! I am seraching for the answer to this question.
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29 views

Show there exists a unique solution to $-u''(x)+\lambda \int^1_0 \sin(u(y))dy =f(x)$

Let $\lambda\in (-1,1)$. Show that for every $f\in C[0,1]$ there exists a unique solution $u\in C[0,1]$ to $$-u''(x)+\lambda \int^1_0 \sin(u(y))dy =f(x)$$ With $u(0)=u'(1)=0$. My work thus far: ...
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3answers
109 views

Eigen values of AB and BA

let A be a linear transformation from $R^n$ to $R^m$, and B be a linear transformation from $R^m$ to $R^n$, it's easy to show that AB and BA has same eigen-value(except $0$). But my question is how ...
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Diagonalization of sparse block matrix

I have a real symmetric matrix, \begin{equation} \left( \begin{array}{ccc} 0 & M & M' \\ M ^T & 0 & 0 \\ M ^{ \prime T} & 0 & 0 \end{array} \right) \end{equation} ...