2
votes
0answers
23 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 ...
3
votes
1answer
39 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 ...
1
vote
2answers
71 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 ...
0
votes
2answers
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 & ...
3
votes
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 ...
2
votes
1answer
23 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 ? ...
0
votes
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 ...
1
vote
0answers
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 ...
0
votes
0answers
15 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? ...
1
vote
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?
1
vote
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 $A$. ...
-1
votes
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 ...
1
vote
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: ...
0
votes
0answers
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 ...
-1
votes
1answer
34 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 = ...
-1
votes
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 & ...
3
votes
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 ...
1
vote
0answers
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) ...
1
vote
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: ...
12
votes
0answers
101 views

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 ...
1
vote
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 ...
0
votes
1answer
36 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 ...
4
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
1
vote
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?
1
vote
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 ...
0
votes
2answers
69 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 $m-n$ ...
1
vote
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 & ...
4
votes
1answer
61 views

Eigenvalues of a $4\times 4$ parameters matrix

Let $a,b,c,d\in\Bbb{C}$ and $B =\begin{bmatrix} a & b & c & d\\ d & a & b & c\\ c & d & a & b\\ b & c & d & a\\ \end{bmatrix}$ I ...
0
votes
0answers
17 views

PCA eigenvalues

When projecting the data set on the Eigen vectors of the co-variance matrix , the eigenvalues represent how much each example varies away from the mean of the data set in the projected direction , ...
0
votes
2answers
35 views

Eigenvectors of a hermitian matrix to the same eigenvalue

Probably, this question has already been answered, but I did not find an answer. If a matrix A is hermitian and an eigenvalue $\lambda$ has multiplicity k, are there always k pairwise orthogonal ...
1
vote
1answer
34 views

If A and B are real orthogonal matrices how to prove that either A-B or A+B is singular?

Degree of matrices is odd $n$-th degree. I figured out all eigenvalues of matrices A and B have to $1$ or $-1$. Now I assume I have to prove $\det((A-B)(A+B)) = 0$ and from that either $\det(A-B)$ or ...
1
vote
1answer
32 views

What's the connection between rank of matrix and $0$ eigenvalue?

My matrix B is nxn and know nothing about if diagonalizble, but I know that rank B = 1. Therefore the geometric multiplicity of λ=0 as an eigenvalue is n-1. But by knowing the rank is 1, can I say ...
0
votes
1answer
47 views

Do T and T* have the same eigenvalues with the same algebraic multiplicity?

I know that the eigenvalues of T* are the conjugates of T's eigenvalues , but how can I see each eigenvalue of T and it's conjugate , the eigenvalue of T*, have the same algebraic multiplicity?
6
votes
2answers
465 views

Why integration operator has no eigen values?

Let $V$ be the vector space of all functions from $\mathbb R$ into $\mathbb R$ which are continuous. Let $T$ be the linear operator on $V$ defined by $$(Tf)(x) = \int_0^x f(t) dt$$ Prove that ...
0
votes
2answers
43 views
0
votes
1answer
38 views

Is it true that every eigenvalue has at least one eigenvector?

As mentioned above: Is it true that every eigenvalue has at least one eigenvector? Or is it possible that while trying to find the basis of a specific eigenspace, i will get only the zero vector ...
0
votes
2answers
50 views

Eigenvectors and matrices

If A is a matrix of rank 1, show that any nonzero vector in the image of A is an eigenvector of A. I feel like this is easy but I am stuck. I am stuck on figuring out the image of A.
1
vote
0answers
25 views

Influence of preconditioning on degenerate eigenvectors

I'm using a hierarchical decomposition of a sparse matrix $A$ as suggested here. I find that the method essentially finds eigenvectors using the QR algorithm. $A$ has some eigenvalues with ...
2
votes
1answer
22 views

$U\in{Mat(\mathbb{C})} , $$U$ is unitary and $U+iI$ is self adjoint, prove: $U = -iI$

I'm preparing for a test and this question was a real pain to do. I get a bit confused by all the terms, so I'd appreciate if you guys peek at my attempt to solve this, and see if its correct. What I ...
0
votes
0answers
20 views

eigenvalues of block matrix

I would like to get an analytical expression for the eigenvalues of the block matrix $L$: $$ L=\begin{pmatrix}M(k) & Q(k)\\ -Q^{*}(-k) & -M^{*}(-k) \end{pmatrix} $$ where $M$ and $Q$ are ...
0
votes
1answer
38 views

real matrices $2\times 2$ and $3\times 3$ that are not similars to a diagonal matrix

Example of real matrices $2\times 2$ and $3\times 3$ that are not similars to a diagonal matrix. I find that $A =\begin{bmatrix} 2 & 1 \\ 0 & 1 \end{bmatrix} $ then i suppose that its ...
1
vote
3answers
40 views

Prove that if $A$ is invertible then $AA^\top$ is positive definite [duplicate]

I need to prove that if $A$ is a square invertible matrix then $AA^\top$ ($A$ multiply $A$ transpose) is positive definite. I tried to prove that all the eigenvalues are positive. I know that ...
1
vote
5answers
78 views

$A$ a $n\times n$ matrix with real entries such that $A^3 = I$ but $A \ne I$

Let $A$ a $n \times n$ matrix with real entries such that $A^3 = I$ but $A \ne I$. a) Give an example that satisfies this conditions. b) what are the eigenvalues ​​of $A$? Well for $a)$ i ...
1
vote
1answer
49 views

Efficient method for determining to the most positive eigenvalue of a matrix

I am trying to implement an algorithm that requires knowing the largest $\textbf{positive}$ eigenvalue of a $\textbf{real symmetric, non-sparse}$ matrix and the corresponding eigenvector. The actual ...
3
votes
1answer
22 views

Matrix $A$ with characteristic polynomial

Given: Matrix $A$ with characteristic polynomial $p(x) = (x+3)^2(x-1)(x-5)$ Also given: $\rho(A+2I) + \rho(A+3I) + \rho(A-5I) = 9$ (btw $\rho$ means rank of the matrix) Prove: $A$ is ...
0
votes
1answer
19 views

if T is normal - Im(T) orthogonal to Ker(T)

Hi I want to prove / contradict that if T is normal then Im(T) orthogonal to Ker(T). Above C it is clear, because each of the Eigenspace are orthogonal to each other and there exist a base of ...