All Questions
202 questions
2
votes
1
answer
400
views
Eigenvalue perturbation under sparse perturbations
Let $A \in \{0,1\}^{n \times n}$ be an irreducible matrix whose entries are in $\{0,1\}$, and let $\lambda_1(A)$ be the eigenvalue with the largest magnitude. By Perron–Frobenius theorem, we know that ...
CommunityBot
- 1
0
votes
0
answers
46
views
max eigenvalue and schatten-1 norm of depolarizing channel acting on trace-0 Hermitian matrix
Denote $\mathcal{H}^n$ as the $n-$dimension Hermitian matrices. Depolarizing channel $\Delta_p:\mathcal{H}^2\to\mathcal{H}^2$ is defined as $\Delta_p(A)=p\text{ tr }(A)~I/2+(1-p)A$ where $A\in \...
- 91
0
votes
1
answer
309
views
Eigenvalues of $\operatorname{diag}({\bf v}) - {\bf v} {\bf v}^\top - \alpha({\bf v} - {\bf w})({\bf v} - {\bf w})^\top$
Given vectors ${\bf v}, {\bf w} \in [0,1]^n$ , where $n \in \mathbb{N} \setminus \{0\}$, and $\alpha > 0$, I would like to find the eigenvalues of the following matrix.
$$\operatorname{diag}({\bf v}...
CommunityBot
- 1
2
votes
1
answer
158
views
The relationship between a matrix and its coefficient matrix decomposed in Pauli matrix
For a dimension-$4$ Hermitian matrix $A$, denote pauli matrices $\{I,X,Y,Z\}$ as $\{\sigma_0,\sigma_1,\sigma_2,\sigma_3\}$ respectively. The pauli matrices form a basis of the matrix space if we take ...
- 256
20
votes
6
answers
42k
views
Eigenvalues of symmetric tridiagonal matrices
Suppose I have the symmetric tridiagonal matrix:
$$ \begin{pmatrix}
a & b_{1} & 0 & ... & 0 \\\
b_{1} & a & b_{2} & \ddots & \vdots \\\
0 & b_{2} & a & \...
- 21
1
vote
0
answers
63
views
Reference request for non-banded Toeplitz matrix
I want to know references that treat the property of eigenvalues and eigenstates of the non-banded Toeplitz matrix.
I mean for example, the Toeplitz matrix $A$ whose matrix element is given by $A_{ij}=...
- 11
0
votes
0
answers
32
views
Eliminating nullity for enhanced non-singularity
If we have an
$n\times n$ matrix $A$ with entries either $0$ or $1$, where all diagonal entries are $0$ and the rank is $k<n$, can we reach full rank by changing exactly $n-k$ zero off-diagonal ...
- 4,058
2
votes
1
answer
512
views
Submatrices of matrices in $\mathrm{SL}(4, \mathbb{Z})$ with all eigenvalues equal to $1$ [closed]
This is a follow-up question to my question from Math Stackexchange (Thank you Dietrich Burde and Michael Burr for the help).
Let $M\in \mathrm{SL}(4, \mathbb{Z})$ with all eigenvalues equal to $1$ (i....
CommunityBot
- 1
1
vote
1
answer
48
views
Iteration matrix representation with complex conjugate operator
I am studying the convergence of a particular class of radial power flows, whose goal is to obtain the voltage solution for a given electric grid, i.e., a complex vector $\mathbf{V}$ that gives the ...
- 63.9k
8
votes
1
answer
7k
views
Upper bound on largest eigenvalue of a real symmetric $n \times n$ matrix with all main diagonal entries positive, everywhere else nonpositive
Is there a good analytic upper bound on the largest eigenvalue of a real symmetric n*n matrix with all main diagonal entries strictly positive, all other entries <=0 with typically many of them ...
3
votes
1
answer
5k
views
Relation between the eigenvalues of a block matrix and the eigenvalues of its diagonal blocks
Consider the $(m+n) \times (m+n)$ block matrix
$$M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}$$
I need references where they are talking about the relation between the eigenvalues of $M$ ...
5
votes
2
answers
343
views
Maximal eigenvalue of a correlation matrix with some entries fixed as zeros
Let $A$ be real a positive semidefinite matrix of dimension $n$ and with $1$s on the diagonal. Those matrices are sometimes referred to as correlation matrices. From the positivity of the minors, we ...
2
votes
1
answer
133
views
Is the sum of the circulant matrix with a super upper triangular matrix diagonalizable?
By the circulant matrix $C$ in $M_n(\mathbb{R})$, we mean that
$$C=[e_n|e_1|\cdots|e_{n-1}]$$ where $e_1,\cdots,e_n$ are the standard basis vectors in $\mathbb{R}^n$. It is well-known that
$$C=\...
- 16.2k
0
votes
2
answers
588
views
Estimating the shift in the $\lambda_{\max}$ of a matrix under a diagonal perturbation
Given a matrix $A$ and a diagonal matrix $D$, how can we estimate $\lambda_{\max}(A+D) - \lambda_{\max}(A)$? Feel free to make other assumptions about the matrices that they are all symmetric and have ...
- 6,853
2
votes
4
answers
293
views
Find a square, stochastic matrix of odd size, not a permutation matrix, with an eigenvalue other than 1 on the unit circle
...or prove that none exists.
Note that such a matrix $M$ couldn't be primitive, so there would be at least one entry equal to zero in every power $M^k$ (Perron-Frobenius theory).
Preferably the ...
- 20.2k
6
votes
1
answer
4k
views
Minimum and maximum eigenvalue
I don't know if this is the right place to post this question, but I find it interesting and have not gotten an answer elsewhere. If it violates any rules, I will gladly delete it.
Let $\Lambda$ be ...
- 109k
1
vote
0
answers
223
views
Fastest algorithm for finding the closest semi-definite matrix?
Given a real-valued, symmetric matrix $A \in \mathbb{R}^{n \times n}$, I'm interested in finding the closest positive semi-definite matrix $X^*\in \mathbb{R}^{n \times n}$:
$$
X^* = \mathop{\text{...
5
votes
2
answers
721
views
Matrices with same eigenvalues
This question is a more precise version of this question.
Let's assume we have the matrix
$$\left(
\begin{array}{ccccc}
0 & a & 0 & 0 & 0 \\
f & 0 & b & 0 & 0 \\
0 &...
- 13.2k
3
votes
1
answer
309
views
Eigenvalues two-fold degenerate
Consider the matrix $$A:=\left(
\begin{array}{cccc}
0 & a & 0 & 0 \\
f & 0 & b & 0 \\
0 & e & 0 & c \\
0 & 0 & d & 0 \\
\end{array}
\right)$$
I ...
19
votes
1
answer
2k
views
Smallest eigenvalue of a tricky random matrix
While experimenting with positive-definite functions, I was led to the following:
Let $n$ be a positive integer, and let $x_1,\ldots,x_n$ be sampled from a zero-mean, unit variance gaussian. Consider ...
1
vote
0
answers
179
views
QR algorithm for eigenvalues and eigenvectors of large symmetric matrices
I am trying to write a QR algorithm in Python for eigenvectors and eigenvalues finding for large symmetric matrices,
My initial thought was to use Householder transformation with a Wilkinson shift ...
0
votes
0
answers
149
views
Diagonalizing a specific case of symmetric block matrix
Let's consider the following block matrix
$$ M = \begin{pmatrix}D&A^T\\A&-D\end{pmatrix},$$
where $A$ and $D$ are $n \times n$ matrices. The diagonal matrix $D$ is defined by $D_{kk} = k \...
1
vote
1
answer
136
views
Matrix transformation that always works?
Consider the matrix
$$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{...
- 128k
0
votes
1
answer
224
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 ...
- 128k
1
vote
0
answers
114
views
Higher dimensional Cauchy interlacing theorem
If $A$ is a Hermitian matrix and $A_j$ the principal minor with the $j$ row and column deleted and $\phi_A(x)$ the characteristic polynomial. The Cauchy interlacing iheorem states that the roots of $\...
- 63.9k
2
votes
1
answer
299
views
Eigenvalues of a specific matrix
I have a block matrix
$$M=\begin{bmatrix}
I_0& I_1& \cdots& I_1\\
I_2& I_0& \ddots& \vdots\\
\vdots& \ddots& \...
2
votes
1
answer
8k
views
Properties of eigenvalues of general nonnegative matrices
I am aware, that an answer to this question can be found via Perron-Frobenius theory or something very similar, but unfortunately I am far from being an expert in the field and I am unable to find the ...
-2
votes
1
answer
183
views
Property of positive semi-definite
Let $A$ is a positive semi-definite matrix like this:
$$ A = \begin{bmatrix}
1 & \alpha_{1,2} & \alpha_{1,3} & \alpha_{1,4}\\
\alpha_{1,2} & 1 & \alpha_{2,3} & \alpha_{2,4}\\
\...
- 785
4
votes
1
answer
187
views
largest eigenvalue of the difference between two quadratic forms
Let $U,V\in\mathbb{R}^{4\times n}$ such that $UU^T=VV^T=I$, and $A\in\mathbb{R}^{n\times n}$ be an Hermitian matrix.
Is it true that
$$\sqrt{\lambda_{\text{max}}\left(\left(UAU^T-VAV^T\right)^2\right)}...
- 52.3k
3
votes
1
answer
144
views
On the bounds of the sum of the squares of spectral variation of two real symmetric matrices
Suppose $A$ and $B$ are two symmetric real $\{0,1,-1\}$ matrices of order $n$ with diagonal elements as zeros (therefore the traces are zeros) and eigenvalues $\lambda_1\ge \lambda_2\ge \dotsb \ge \...
- 12.9k
1
vote
2
answers
446
views
Transforming matrix to off-diagonal form
I wonder if one can write the following matrix in the form $A = \begin{pmatrix} 0 & B \\ B^* & 0 \end{pmatrix}.$
The matrix I have is of the form
$$ C = \begin{pmatrix} 0 & a & b & ...
- 785
1
vote
0
answers
49
views
Bounds on Eigenvalues After Skew-Symmetric Perturbation
Consider two matrices $\mathbf{J} \in \mathbb{R}^{n \times n}$ and $\mathbf{L} = -\mathbf{L}^T \in \mathbb{R}^{n \times n}$. I am trying to upper bound the eigenvalues of their sum:
$$\mathbf{A} = \...
- 11
3
votes
2
answers
394
views
Eigenvalues and eigenvectors of non-symmetrical tridiagonal matrix
The question is the following: given a matrix
$$A=\begin{pmatrix}
1& 2 & & & & \\
1& 0& 1 & & & \\
& 1& 0& 1 & &\\
& &...
- 3,671
2
votes
1
answer
244
views
Expected minimal distance of eigenvalues
Let $A$ be an arbitrary symmetric matrix and $B$ be a random GUE matrix. I would like to know. Are there any results on the minimal eigenvalue distance between two distinct eigenvalues of $A+B$? I ...
CommunityBot
- 1
-1
votes
1
answer
155
views
Companion matrices must have geometric multiplicity one, linear recurrence sequence view [closed]
I posted this question on math stackexchange weeks ago, and it have not receive an answer yet after a bounty offer...
I've been recently playing around with the linear recurrence sequences. Consider ...
- 4,713
7
votes
0
answers
195
views
Hölder continuity of spectrum of matrices
Endow $\mathbb{C}^{d \times d}$ with the norm induced by the Euclidean norm on $\mathbb{C}^d$. It is well-known (to those who know it well, I guess) that the spectrum $\sigma(A)$ of a matrix $A \in \...
- 12.5k
1
vote
0
answers
163
views
An estimation of the largest eigenvalue of a submatrix of $\left(\cos(\frac{kl\pi}{4n})\right)_{k,l=1}^n$
Let us consider the following matrix $A=(a_{k,l})$ where
$$A=\left(\cos(\frac{kl\pi}{4n})\right)_{k,l=1}^n$$
Let us consider the submatrix $A_0$ of $A$ whose entries are those $a_{k,l}$ where $k\...
- 4,058
0
votes
0
answers
232
views
How to analyse the range of eigenvalues of a symmetric and indefinite matrix?
Let $G$ be a symmetric and indefinite matrix defined as follows
$$ G := S - \begin{pmatrix} I_n & I_n \\ I_n & I_n \end{pmatrix},$$
where $S$ is a symmetric positive definite matrix of size $...
4
votes
0
answers
989
views
Lower bound minimum eigenvalue of a positive semi-definite Hermitian matrix with bounded entries
Let $M \in \mathbb{C}^{n \times n}$ be a matrix with the following properties:
$M$ is Hermitian and positive semi-definite (all the eigenvalues are real and nonnegative).
The diagonal entries of $M$ ...
- 41
2
votes
1
answer
728
views
Maximum eigenvalue of Hadamard power of a positive semidefinite matrix
Let $K$ be a covariance matrix. It is positive semidefinite, its diagonal elements are all 1, and its off-diagonals are between -1 and 1. Let $K.^2$ be its element-wise power (Hadamard power). Can we ...
- 5,407
36
votes
2
answers
32k
views
Eigenvalues of the product of two symmetric matrices
This is mostly a reference request, as this must be well-known!
Let $A$ and $B$ be two real symmetric matrices, one of which is positive definite. Then it is easy to see that the product $AB$ (or $BA$...
- 101
2
votes
0
answers
69
views
Unimodular eigenvalue of a H-self-adjoint matrix (indefinite innerproduct)
Let $A,H \in \mathbb{C}^{n \times n}$ be such that $H$ is Hermitian and invertible and $A = H^{-1} A^* H$. In this case, $A$ is said to be $H$-self-adjoint. This is due to the fact that if $\langle \...
- 175
1
vote
1
answer
209
views
Eigenvalues invariant under 90° rotation
Consider $N \times N$ matrices
$$A = \begin{bmatrix}
0 & 0 & \cdots & 0 & 1 \\
1 & 0 & 0 & & 0 \\
\vdots & 1 & 0 & \...
- 3,446
21
votes
5
answers
2k
views
The middle eigenvalues of an undirected graph
Let $ \lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_{2n} $
be the collection of eigenvalues of an adjacency matrix of an undirected graph $G$ on $2n$ vertices. I am looking for any work or references ...
- 4,713
12
votes
1
answer
1k
views
Eigenvalues come in pairs
Consider the two matrices with some parameter $s \in \mathbb R$
$$A_1= \begin{pmatrix} s& -1 &0& 0 \\1&0 &0&0 \\ 0&0&1&0 \\0&0&0&1 \end{pmatrix}$$
and
$$...
- 188k
91
votes
5
answers
124k
views
Eigenvalues of matrix sums
Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their sum? What about the special case when the matrices are Hermitian and positive definite?
I am ...
3
votes
2
answers
853
views
Power of a matrix, largest eigenvalue in absolute value, and convergence acceleration
I want $S^k$, with $S=I-\Lambda^{-1}M$, to tend to zero quite fast as $k\rightarrow \infty$, as this is what drives the convergence in a fixed-point algorithm. Here $M=X^TX$ is a fixed $m\times m$ ...
- 621
2
votes
0
answers
121
views
Eigenvalues of two positive-definite Toeplitz matrices
Consider two positive-definite Toeplitz matrices $M_1$ and $M_2$ both with dimension $2^j \times 2^j$. Their matrix elements are:
$$M_1[x,y] = \frac{\text{sin}(\pi(x-y)/2^j)}{\pi(x-y)} \qquad M_2[x,y] ...
- 21
0
votes
1
answer
1k
views
How do eigenvalues change if we duplicate a row and column of a symmetric matrix?
Let $\bf A$ be a $n \times n$ symmetric positive semidefinite matrix whose first column is denoted by ${\bf a}_1$. We define a new matrix,
$$ {\bf B} = \begin{bmatrix} a_{11} & {\bf a}_1^T \\ {\bf ...
16
votes
3
answers
2k
views
Why is the set of Hermitian matrices with repeated eigenvalue of measure zero?
The Hermitian matrices form a real vector space where we have a Lebesgue measure. In the set of Hermitian matrices with Lebesgue measure, how does it follow that the set of Hermitian matrices with ...
- 12.9k