All Questions
51 questions
15
votes
4
answers
7k
views
Eigenvectors and eigenvalues of a tridiagonal Toeplitz matrix
Is it possible to analytically evaluate the eigenvectors and eigenvalues of the following $n \times n$ tridiagonal matrix
$$
\mathcal{T}^{a}_n(p,q) = \begin{pmatrix}
0 & q & 0 & 0 &...
- 153
12
votes
2
answers
2k
views
What is known about the eigenvectors of the $2^n \times 2^n$ Hadamard matrix?
What is known about the eigenvectors of the $2^n \times 2^n$ Hadamard matrix defined recursively by $H_1=(1)$ and $$ H_N=\begin{pmatrix}H_{N/2} & H_{N/2} \\ H_{N/2} & -H_{N/2}\end{pmatrix}, $$ ...
- 1,324
11
votes
1
answer
2k
views
Eigenvalues of the complement of a graph
Let $A$ and $\widetilde A$ be the adjacency matrices of a graph $G$ and of its complement, respectively.
Is there any relation between the eigenvalues of $A + \widetilde A$ and the eigenvalues of $A$ ...
- 1,269
11
votes
1
answer
985
views
Exact eigenvalues of a specific tridiagonal matrix
I'm studying the following tri-diagonal matrix
$$
X = \begin{pmatrix}
0 & x_0 & 0 & 0 &\cdots & 0 & 0 & 0 \\\
x_0 & 0 & x_1 & 0 &\cdots & 0 & ...
- 161
10
votes
2
answers
615
views
Lower eigenvectors of nonnegative matrices with zero trace
Let $A$ be an $N\times N$ nonnegative matrix with all diagonal entries equal to zero and such that there is $n_0$ such that all entries of $A^{n_0}$ are strictly positive. Let $\lambda_1,\ldots, \...
- 1,897
9
votes
1
answer
3k
views
Frobenius-Perron eigenvalue and eigenvector of sum of two matrices
Suppose that I have two positive matrices, $A$, and $B$, and I know their Frobenius-Perron eigenvalues ($\lambda_A$, $\lambda_B$) and eigenvectors ($v_A$, $v_B$). I'm interested in what I can say ...
- 403
8
votes
1
answer
5k
views
Eigenvectors of Kronecker Product [closed]
Conjecture If $A$ and $B$ are two complex square matrices, then every eigenvector of $A\otimes B$ is of the form $x\otimes y$, where $x$ is an eigenvector of $A$ and $y$ is an eigenvector of $B$.
...
7
votes
1
answer
412
views
Sum of the absolute eigenvalues of A>=B
Kindly help me to prove/disprove the following statement.
Let $A$ be a symmetric matrix of order $n \times n$ with all the diagonal entry equal to $0$, and other non-diagonal entry equal to $k$ (...
7
votes
1
answer
6k
views
Eigenvectors as continuous functions of matrix - diagonal perturbations
The general question has been treated here, and the response was negative. My question is about more particular perturbations. The counterexamples given in the previous question have variations not ...
- 2,222
6
votes
0
answers
96
views
Finding the maximal component of a vector in sublinear time
Given a vector $u \in \Bbb R^n$, finding the value of the largest component of $u$ needs linear time in $n$. However, what if we additionally know that $u$ lies in some linear subspace $U \subset \Bbb ...
- 13.6k
5
votes
3
answers
2k
views
Proving that a certain non-symmetric matrix has an eigenvalue with positive real part
Suppose that
$X$ is the $n \times n$ matrix of all ones
$Y$ is an arbitrary $n \times n$ matrix with zeroes on the diagonal and all other entries equal to $0$ or $1$
$0 < \delta < 1$
Let $Z = ...
- 61
5
votes
2
answers
2k
views
rank-one perturbation of a matrix corresponding to a specific spectrum
Let $A$ be a real symmetric matrix whose spectrum is $\lambda_1,\lambda_2,\ldots,\lambda_n$.
Let $A'$ be the matrix obtained by adding a perturbation to $A$. The requirement is that only the second ...
- 111
5
votes
1
answer
8k
views
Eigenvalues and eigenvectors of tridiagonal matrices
What can I say about the eigenvalues and eigenvectors of the tridiagonal matrix $T$ given as
$T = \begin{pmatrix}
a_1 & b_1 \\
c_1 & a_2 & b_2 \\
& c_2 & \ddots & \ddots \\
&...
- 73
4
votes
1
answer
1k
views
Why are 1 and -1 eigenvalues of this matrix?
This is a subject I've been working on for a very long time now, but still did not manage to fully understand the interesting properties of this matrix $\mathbf{A}$.
First, let's define two matrices:
...
- 403
4
votes
1
answer
2k
views
Eigenvalues of a rank-one update of a symmetric matrix
I have a matrix $A$ $(n\times n)$ with eigenvalues $\lambda_i$, then I add another matrix to it as: $A+xx^\top$ where $x$ $(n\times 1)$ is a column vector.
and also $A=yy^\top$ with $y$ a $(n-1)$ rank ...
- 63
4
votes
2
answers
477
views
Non-asympototic version of Gelfand's formula
Let $A$ be a $n\times n$ matrix. Let $\|A\|$ be the spectral norm of $A$, and $\rho(A)$ be the spectral radius. I am wondering whether the following statement is true.
There exists universal ...
- 181
4
votes
1
answer
380
views
Rotatable matrix, its eigenvalues and eigenvectors
We say that a real matrix is rotatable iff after turning it clockwise on $90^{\circ}$ it doesn't change.
I'm interesting about eigenvalues and eigenvectors (belonging to non-zero eigenvalues) of such ...
4
votes
0
answers
447
views
How to find eigenvalues of following block matrices?
Is there a procedure to find the eigenvalues of A?
$$A=\begin{bmatrix}X & I &&&&&&&&& 0\\I & 0 & P &&&&&&&&\\& P^t ...
- 41
4
votes
0
answers
2k
views
What is the time complexity of the largest singular value and its vectors?
Full zero-error SVD on an $m \times n$ matrix $A$ would cost $O(\min(m^2n,mn^2))$. What is the time complexity if we need only the largest singular value and its corresponding vectors? I think it is $...
- 123
4
votes
0
answers
84
views
Matrices with almost constant coefficient have a simple eigenvalue
As a by-product of a general result for bounded operators of a Banach space, I have the following:
A matrix $L=(\ell_{ij})_{ij}$ that has almost constant coefficients in the sense that for some $c$,...
- 14.4k
3
votes
3
answers
357
views
Can a block matrix with at least 3 zero blocks of different size on the diagonal and 1's everywhere else have only integer eigenvalues?
Let $M=\begin{pmatrix}
\begin{array}{cccccccc}
0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\
0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\
1 & 1 & 0 & 0 & ...
3
votes
2
answers
394
views
Eigenvalues and eigenvectors of non-symmetrical tridiagonal matrix
The question is the following: given a matrix
$$A=\begin{pmatrix}
1& 2 & & & & \\
1& 0& 1 & & & \\
& 1& 0& 1 & &\\
& &...
- 145
3
votes
1
answer
264
views
When is this matrix singular?
Consider matrix $A$ with $(j,k)$′th entry $A_{j,k}=\sin(\omega_j t_k+\phi_j),\,\forall j,k\in\{1,2,...,n\}$, where $\omega_j,t_k,\phi_j\in \mathbf R$.
1) For $t_k=k$, what is the condition on $\...
- 2,251
3
votes
2
answers
1k
views
Eigenvectors and eigenvalues of Tridiagonal matrix with varying diagonal elements
is it possible to analytically evaluate the eigenvectors and the eigenvalues of a tridiagonal $n\times n$ matrix of the form :
\begin{pmatrix}
1 & b & 0 & ... & 0 \\\
b & 2 &...
- 31
3
votes
0
answers
220
views
Eigenvalues and eigenvectors of nonsymmetric complex tridiagonal matrix
I wonder if it is possible to find analytically all eigenvalues and eigenvectors of the following $2n \times 2n$ non-symmetric complex tridiagonal matrix
$$M = i \begin{pmatrix}
0 & a & 0 &...
2
votes
1
answer
1k
views
Diagonalizing a symmetric block matrix
Let us consider the matrix
$$ A = \begin{pmatrix} a & c+ib \\ c-ib& a \end{pmatrix},$$
then this matrix has eigenvalues $a\pm \sqrt{c^2+b^2}.$
Now, let us consider a block matrix
$$ A = \begin{...
- 29
2
votes
1
answer
474
views
When is there a solution to these coupled eigenvalue equations?
I am trying to find the fixed point of a dynamical system, which requires solving two coupled eigenvalue-like equations. These equations are, in general, overconstrained. I'd like to have a simple ...
- 403
2
votes
2
answers
564
views
Linear dynamical systems: interpretation of Frobenius eigenvector
Consider a positive linear dynamical system. $\frac{dx}{dt}=Ax$, where $A$ is a quasipositive/Metzler/essentially nonnegative matrix. By its properties, the vector $x$ will remain positive for all ...
- 143
2
votes
2
answers
269
views
Is my use of the eigendecomposition correct here?
I'm exploring different techniques to efficiently solve some matrix equations. My situation is that I have a matrix $\textbf{H} = \textbf{J}^T \textbf{J}$, where $\textbf{J}$ is a matrix with no ...
- 123
2
votes
1
answer
661
views
Leading eigenvector value problem as an optimisation problem for asymmetric matrices
As noted in 1806.05647, given a symmetric matrix $A$, the leading eigenvector value problem (LEVP)
$$Av = \lambda v,$$
where $A = A^T \in \mathbb{R}^{n \times n}$, $\lambda$ is the largest ...
- 21
2
votes
0
answers
86
views
Smallest eigenvalue of certain PD matrix decreases under sparse perturbation
Let $\omega_1<\dots<\omega_n\in\mathbb{R}$. Then, define $G\in\mathbb{C}^{n\times n}$ such that $G_{k\ell}=\frac{1}{1-i(\omega_\ell-\omega_k)}$. For example, if $n=3$ we obtain $$ G=\begin{...
- 83
2
votes
0
answers
148
views
Commutation relation and eigenvectors of infinite matrices [closed]
I'm given the Matrix $A$ and $A^T$:
$A =
\begin{bmatrix}
0 & 1 & 0 & 0 & \dots \\
0 & 0 & \sqrt{2} & 0 & \dots \\
0 & 0 & 0 & \sqrt{3} & \...
- 121
1
vote
3
answers
1k
views
Simple Spectrum of Jacobi matrices
I want to call a matrix a Jacobi matrix (cause there may be different notions of Jacobi matrices) if it is a tridiagonal matrix with positive off-diagonal entries. Now, I read that the spectrum of ...
- 21
1
vote
2
answers
416
views
Eigenvectors of a non-symmetric rank-one update of a symmetric matrix
I have a matrix $A$ $(n\times n)$ with eigenvalues $\lambda_i$, then I add another matrix to it as: $A+uv^\top$ where $u$ $(n\times 1)$ and $v$ $(n\times 1)$ are column vector. Also, $A=yy^\top$ with $...
- 63
1
vote
1
answer
136
views
Matrix transformation that always works?
Consider the matrix
$$A_2:= \begin{pmatrix} a & b_1 \\ b_2 & a\end{pmatrix}.$$
Let $\sigma_2 = \begin{pmatrix} 0 & i \\ -i & 0 \end{pmatrix}$, then
$$\sigma_2 A_2 \sigma_2 = \begin{...
1
vote
2
answers
449
views
Transforming matrix to off-diagonal form
I wonder if one can write the following matrix in the form $A = \begin{pmatrix} 0 & B \\ B^* & 0 \end{pmatrix}.$
The matrix I have is of the form
$$ C = \begin{pmatrix} 0 & a & b & ...
- 536
1
vote
1
answer
468
views
Trace minimization for generalized eigenvalue problem
In [1], it is shown in theorem 1.2 that for symmetric $n \times n$ matrices $A$, $B$, we have
$$
\min_{Y \in Y^*} \text{tr}(Y^TAY) =
\text{tr}(X^TAX) =
\sum_{i=1}^p \lambda_i,
$$
with
$$
\text{
$X^...
- 21
1
vote
0
answers
179
views
QR algorithm for eigenvalues and eigenvectors of large symmetric matrices
I am trying to write a QR algorithm in Python for eigenvectors and eigenvalues finding for large symmetric matrices,
My initial thought was to use Householder transformation with a Wilkinson shift ...
1
vote
0
answers
293
views
Eigenvalue decomposition of normalized adjacency matrix
Let $A$ be an adjacency matrix of undirected graph $G$, where $G$ is a connected graph. The normalized adjacency matrix is defined as $\hat{A}=D^{-1/2}AD^{-1/2}$, where $D$ is degree matrix of graph $...
- 11
1
vote
0
answers
86
views
Solution of a manipulated equation vs the maximum eigenvalue and eigenvector of a non-negative matrix
Lets assume we have the following equation:
$AU=\lambda U \Rightarrow\left[
\begin{array}{c|c|c}
0 &A_{12}&A_{13}\\
\hline
A_{21}& 0& A_{23}\\
\hline
A_{31}&A_{32}&0
\end{...
- 21
1
vote
0
answers
92
views
An inequality concerning the eigenvalues and eigenvectors of an SPD matrix
Let $Ax_i=\lambda_ix_i, \ (i=1,\cdots,n)$ be an eigensystem of the symmetric positive-definite diagonally-dominant matrix $A=\{a_{ij}\}$. Let
$$b_{jk}=\sum_{i=1}^{n}{\frac{(x_i(j)-x_i(k))^2}{\...
- 11
1
vote
0
answers
171
views
Eigenvalues of non-negative block matrices
$B$ is a non-negative irreducible block matrix as follows:
$$B=
\left[
\begin{array}{c|c|c}
0 &B_{12}&B_{13}\\
\hline
B_{21}& 0& B_{23}\\
\hline
B_{31}& B_{32}&0
\end{array}
\...
- 21
1
vote
0
answers
270
views
Eigenvalue of product of self adjoint compact operators
Suppose A is a self adjoint $m \times m$ real matrix with eigenpairs $\{e_j, \lambda_j\}$ such that $\lambda_j > \lambda_{j + 1}$. Let $B$ be another self adjoint real $m \times m$ matrix such that ...
- 157
1
vote
0
answers
136
views
when can I say that $UV^T$ is a permutation matrix? [closed]
suppose we have two p.s.d matrices A and B: so we can diagonalize them like this:
A= $UΛU^T$ and $B=VΣV^T$
1: on what condition for $A$ and $B$ I can say that $UV^T$ is a permutation matrix?
2: how ...
- 59
0
votes
1
answer
226
views
Faulty algorithm for simultaneous diagonalization?
I found a simple algorithm for simultaneous diagonalization of two commuting matrices (Nordgren - Simultaneous Diagonalization and SVD of Commuting Matrices), which seemed to be well-founded. For ...
- 103
0
votes
1
answer
106
views
Eigenvalue-related statements [closed]
(I understand this question might not be appropriate for this website, but it has been asked on MathStackexchange and did not receive any replies even with a bounty)
How can I prove that the ...
- 103
0
votes
0
answers
149
views
Diagonalizing a specific case of symmetric block matrix
Let's consider the following block matrix
$$ M = \begin{pmatrix}D&A^T\\A&-D\end{pmatrix},$$
where $A$ and $D$ are $n \times n$ matrices. The diagonal matrix $D$ is defined by $D_{kk} = k \...
- 1
0
votes
0
answers
166
views
Minimize a vector from a matrix operation
I want to minimize a certain vector that results from a matrix operation with some constraints and i don't exactly know how to tackle this problem.
Lets say we have
$$
(L+A)*s = v
$$
L is the ...
- 101
0
votes
0
answers
79
views
Eigendecomposition of $A=I+BDB^H$
Suppose that we have $$A = I_m + BDB^H$$ where matrix $A$ is $m \times m$, matrix $B$ is $m \times k$, $BB^H \neq I_m$ and $D$ is a $k \times k$ diagonal matrix. Can we obtain the eigendecomposition ...
- 13
-2
votes
1
answer
970
views
What can we say about the rank of the sum of a multiple of the identity matrix and a symmetric rank-$1$ matrix? [closed]
Suppose we have the following symmetric matrix.
$$A = \sigma^2 I + u u^T$$
What can we say about the eigendecomposition of $A$?
- 67