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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Eigenvectors of two matrices whose sum is an identity matrix? [on hold]

What can I say about eigenvectors of two matrices whose sum is an identity matrix?
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34 views

Effective way of checking if all eigenvalues of a matrix are integers

Given A matrix with integer entries, it should be checked if all its eigenvalues are integers. Of course, the characteristic polynomial could be calculated, but is there any faster (or easier) ...
2
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0answers
26 views

Which n-tuples of positive integers can be the eigenvalues of some matrix with positive integer entries?

This question is closely related to some questions I already asked Given a tuple of positive integers (such as (1,2,5) ), is there a matrix A with positive integer entries such that the integers in ...
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1answer
42 views

Norm of symmetric positive semidefinite matrices

I have been researching a lot trying to find an answer to my question and didn't find any so I would appreciate it if anyone can help. If we have 2 symmetric, positive semi-definite matrices $A$ and ...
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2answers
74 views

When is the dominant eigenvalue of this matrix greater than one?

So I am trying to figure out when this matrix $\left[\begin{matrix} a_1 & 0 & b \\ a_2 & a_3 & 0 \\ a_4 & a_5 & a_6 \end{matrix}\right]$ $b, a_i\geq0$ for all $i$, and ...
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36 views

Calculating all potencies of a Matrix

I've stumbled across this problem while reading my textbook (chapter eigenvalues) Calculate all potencies of $A$ and $A+aE$ $ a \in K$ and $A \in K-Vectorspace$ $ A= \begin{pmatrix} 0 & 1 & ...
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1answer
23 views

Eigenvalues of real square matrix with non-negative off diagonal?

Suppose I have a matrix $A$ with real entries such that the off-diagonal entries of $A$ are positive or zero. (The diagonal entries may be positive, negative or zero.) Is this a sufficient condition ...
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0answers
15 views

Control principal eigenvector of a row stochastic matrix

I am just trying to consider the classical discrete-time Markov Chain problem. Consider the transition matrix P, which transforms state vector $x(k)$ to $x(k+1)$, satisfying: $x(k+1)$ = $P*x(k)$ It ...
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0answers
53 views

Criteria for a solvable septic equation

I have a certain 7x7 matrix whose elements are all symbolic, and I want to know the eigenvalues. I have to solve a septic equation, but it is generally impossible. However, I am only interested in the ...
2
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1answer
25 views

Lower bound for the spectralradius of a matrix

Any submultiplicative norm (for example the row-sum-norm) is an upper bound for the spectralradius of a matrix A. But is there a way to get a suitable LOWER bound for the spectralradius ? ...
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33 views

Shape operator and orthogonality of eigenvectors

When studying differential geometry (at a hobby level) I always run into problems when it comes to the varying notations and statements about the shape operator ...
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1answer
18 views

Difference between matrices with altered eigenvalues

Given two p.s.d. matrices $X_1$ and $X_2$ with eigen decomposition $X_1 = U_1V_1U_1^T$ and $X_2 = U_2V_2U_2^T$ and a constant $\lambda > 0$ Now consider an altered version of the eigenvalue ...
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25 views

Numerical Computation for K smallest eigenvalues of a large Real Symmetric Matrix with restricted methods

I'm writing some code on a distributed platform, using some programming language like Hadoop, and now I need to calculate the K smallest eigenvalues for a Large Matrix. K is a small constant at most ...
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2answers
29 views

vector question assistance

let there be 2 lines: $(2,-3,1) + s(3,-2,1)$ and $(2,-1,-3) +t(3,-2,1)$ which are parallel to each other. find the formula of the plane determined by them. my try: a vector perpendicular to ...
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1answer
11 views

Change of Eigenvalues of Ellipsoid Tensor during Rotation

I have an ellipsoid defined by the semiaxes $a,b,c$ and the orthonormal vectors $v_x, v_y, v_z$ describing the directions in which the axes point. The matrix ...
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2answers
24 views

Finding two matrices that permit a change of variable eliminating the crossed term.

Learning about eigenvalues/vectors. Here is an exercise which I guess is about that, but I am not really sure how to get started with it: For $5x_1^2-4x_1x_2+5x_2^2=21$ find a diagonal matrix $D$ ...
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1answer
37 views

Constructing regular integer matrices with distinct integer eigenvalues

How can I construct matrices with positive integer values and distinct integer eigenvalues (not necessarily positive, but 0 should not be an eigenvalue). The standard-method to construct matrices ...
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0answers
16 views

Triangularisation of a linear map

The Calculation of the Char Poly is wrong but it's the method I am not able to understand In this example why does the eigenvector of A give the required eigenvector that is contained in the basis? ...
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1answer
25 views

Triangularisation of a linear transformation

I understand that Upper triangular matrices must have at least one eigenvector, but why does this mean that the basis of $[T]_B$ must contain an eigenvector for $[T]_B$ to be upper triangular?
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1answer
43 views

$A,B$ matrices , prove $Bv = \Lambda v$

$A,B$ are $n \times n$ matrices and $AB = BA$ Also, there is an eigenvalue $\Lambda$ in $A$ which its geometric complexity is $1$. Also there is $ v \ne 0 $ that $v$ is an eigenvector of $A$. ...
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0answers
28 views

Order of eigenvectors in jblas?

I am using jblas to compute eigenvectors of a double symmetric matrix. Using symmetricEigenvectors(myMatrix)[0], I can get a matrix which columns are the eigenvectors of my matrix. However I need them ...
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0answers
84 views

Eigenvalues of $\pmatrix{1&1\\1&2}$

I use maxima for calculation eigenvalues of this matrix: $$ \begin{bmatrix} 1 & 1 \\ 1 & 2 \end{bmatrix} $$ and I get $\frac{3\pm\sqrt{5}}{2}$ and then $[1,1]$ for some reason. Namely: ...
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25 views

relation between eigen values

Let $W$ be a finite subgroup of $GL(V)$ and hence it acts on $V$. Now consider the contra gradient action of $W$ on $V^*$. Now how to show that the eigen value of this action is the reciprocals of the ...
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2answers
38 views

Question on Eigen values

Let $A$ be a square matrix and $A^*$ be its adjoint, show that the eigenvalues of matrices $AA^*$ and $A^*A$ are real. Further show that $\operatorname {trace}(AA^*)=\operatorname {trace}(A^*A)$.
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64 views

How to find the eigenvalue of matrix A?

We have: $$A\left(\begin{array}{l}\xi \\ \eta\end{array}\right) = \left(\begin{array}{l}a\xi+b\eta \\ a\xi-b\eta \end{array}\right)$$ How to find the eigenvalue of matrix $A$?
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36 views

How to show a total order is product order

Besides the definition of product order, is there any other way to show that a total order on two sets can induce a product order? Because I want to solve the problem below: For two graphs ...
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1answer
37 views

basis vectors of a 2D lattice plane in a 3D lattice

I know the basis vectors of the three-dimensional lattice $\Lambda = \{\mathbf{b_1}, \mathbf{b_2}, \mathbf{b_3} \}$. I also know the equation of the plane in this 3D lattice, suppose $Ax + By + Cz = ...
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2answers
52 views

Similar matrices that are not diagonalizable

Let $f:R^3 \to R^3$ be a function which matrix with respect to the standard basis is: $$ A = \begin{pmatrix} -4 & -8 & 8 \\ 1 & 2 & -2 \\ -1 & -2 & ...
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0answers
93 views

$n$ distinct real eigenvalues of an $n \times n$ matrix

What are the necessary and sufficient conditions for a real $n \times n$ matrix to have $n$ distinct real eigenvalues? Ideally I'm looking for a test that does not require (and is hopefully more ...
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2answers
130 views

Eigenfunctions of the Laplacian with imaginary eigenvalue

What are all the $\pm i$ eigenfunctions of the Laplacian on $\mathbb{R}^2$ (or on some domain in $\mathbb{R}^2$)? I know of a few: things like $e^{e^{i \frac{\pi}{4}}x} + e^{e^{i \frac{\pi}{4}}y}$ or ...
3
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2answers
47 views

Power iteration

If $A$ is a matrix you can calculate its largest eigenvalue $\lambda_1$. What are the exact conditions under which the power iteration converges? Power iteration Especially, I often see that we ...
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27 views

Orthogonal projections and matrix diagonalization

In the Euclidean space $R^4$ with the usual inner product, let $U$ be the subspace given by the solutions of $3x_1- x_2 -2x_3 = 0$ and $2x_1 + x_3 + x_4 = 0$ a) Find a base of $U$ and $U^{\bot}$ b) ...
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1answer
59 views

Why is this matrix diagonalizable?

Given the matrix $$A=\left( \begin{array}{ccc} 0 & -1 & -2 \\ -1 & 0 & -2 \\ -2 & -2 & -3 \\ \end{array} \right)$$ It has the following characteristic polynomial: ...
2
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1answer
32 views

What is the spectral radius of a non-diagonal matrix?

This is my first question in Math StackExchange. Assume that I know the spectral radius of matrix $A$. The matrix $\bar{A}$ is created from $A$ by removing all the $A$'s diagonal entries (i.e., ...
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How many matrices with integer eigenvalues are there?

Let m,n be natural numbers. How many mxm-matrices with integer entries from -n to n have the property that all eigenvalues (possibly multiple) are integers ? The following table calculated with PARI ...
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1answer
25 views

Calculating a limit power of a matrix

A = [ [ 4 , -2, 3] , [$\frac{1}{2}$ , 0 , $\frac{1}{2}$] , [-4,$\frac{5}{2}$,-3] ] Suppose, we know the eigenvalues $\frac{1}{2}$,$-\frac{1}{2}$ and 1 of A and a matrix T with $T^{-1}AT = D$, where D ...
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1answer
37 views

$T : M_{n \times n}(R) \rightarrow M_{n \times n}(R)$ and $T(A)= A^t$ and $ <A,B> = Tr(AB^t)$

Let $V = M_{n \times n}(R)$ with the inner product $ <A,B> = Tr(AB^t)$, and $T$ the linear operator given by $T : M_{n \times n}(R) \rightarrow M_{n \times n}(R)$ and $T(A)= A^t$ . How can i ...
2
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0answers
69 views

Is this the chord G Major I am hearing as base tones from interference of zeta zeros times eigenvalues of the von Mangoldt function matrix?

Mathematica knows that the logarithm of $n$ is: $$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)$$ The von Mangoldt function should then be: ...
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3answers
47 views

Questions on Jordan forms

I am studying Jordan form of a matrix from wiki. I am wondering how could two matrices have same eigenvalues with same multiplicities, but have different Jordan form? Also, if two matrices have ...
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0answers
24 views

Generalised eigenspace

I know that the ker(A-lambda)^2 (or any power of the kernal) and the generalised eigenspace should contain all eigenvectors of a matrix. However, for this matrix [2 -2 3;10 -4 5;5 -4 6] for the ...
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0answers
29 views

Linear independence of generalized eigenvectors

Let $V$ be a finite-dimensional complex vector space. Let $T\in \mathcal L(V)$ be an endomorphism. A vector $v\in V \setminus\{0\}$ is called a generalized eigenvector to an eigenvalue ...
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1answer
28 views

Predicting eigenvalues of bigger matrices

Consider the following $(3 \times 3)$ matrix: $K_3 = \left( \begin{array}{ccc} a & -1 & 0 \\ -1 & a+1 & -1 \\ 0 & -1 & a \end{array} \right)$ The question has a quantum ...
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1answer
20 views

The idea of eigenvectors and a transforming matrix

I'm reading this tutorial on PCA: http://nyx-www.informatik.uni-bremen.de/664/1/smith_tr_02.pdf I quote from it: It is the nature of the transformation that the eigenvectors arise from. Imagine a ...
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1answer
26 views

Constructing an eigenvector for a certain matrix representing a graph with a perfect code

Let $A$ be a symmetric $(0,1)$-matrix whose row sum is $r.$ Suppose I have a $(0,1)$ vector $v$ such that $$Av = \vec{1} - v.$$ By taking the vector $$u = \vec{1} - (r+1)v$$ we see that $$Au = A ...
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1answer
32 views

Circulant matrix

$A=\left(\begin{array}{cc} B & C\\ C & B \end{array} \right)$ Here $A$ is the block circulant matrix and B and C are $n \times n$ matrices which are circulant. How can write it as in roots ...
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Does a transition matrix has only real positive eigenvalues? [duplicate]

I am analyzing an ergodic Markov Chain of an $N\times N$ lattice grid and I have written down my Transition Matrix, i.e. all values are between $0$ and $1$ and the elements of each row sum up to $1$. ...
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2answers
73 views

$\det (A^2 - I) < 0 \Rightarrow \lambda \in (-1,1)$

Let A be real square matrix. If $\det (A^2 - I) < 0$, then A has eigenvalue $\lambda \in (-1,1)$. How to prove this?
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1answer
29 views

Show that if $A$ is a symmetric matrix with all eigenvalues greater than $0$, then it is positive definite.

Prove that if $A$ is a symmetric matrix with all eigenvalues greater than $0$, then it is positive definite. If $A$ is symmetric then there exists an orthogonal matrix $S$, such that $S^TAS$ is a ...
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2answers
70 views

Prove that $\det(xI_m-AB)=x^{m-n}\det(xI_n-BA)$

I want to prove that $\det(xI_m-AB)=x^{m-n}\det(xI_n-BA)$ If $A\in \mathbb{F}^{m\times n}$ and $B\in \mathbb{F}^{n\times m}$ It is easy to show that $0$ has algebraic multiplicity of at least $m-n$ ...
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1answer
36 views

diagonalisation unitary matrix

Let $A \in U(n) \subset \mathbb{C}^{n \times n}$ a unitary matrix. Show that: $\exists ~ S\in U(n)$ so that $\bar{S^t}AS=D:=\begin{pmatrix}\lambda_1&&0\\&\ddots & ...