All Questions
Tagged with eigenvalues matrices
323 questions
2
votes
1
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398
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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
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0
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46
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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
4
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1
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Tight upper bound on a ratio involving symmetric PSD matrices and Kronecker products
Let $\mathbf{A}_i \in \mathbb{R}^{d \times d}$ ($i = 1, \dots, T$) be symmetric positive semidefinite (PSD) matrices. Define the quantity
$$
m = \frac{\lambda_{\max}\left(\sum_{i=1}^T \mathbf{A}_i^2\...
- 63.9k
4
votes
1
answer
2k
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Eigenvalues of a rank-one update of a symmetric matrix
I have a matrix $A$ $(n\times n)$ with eigenvalues $\lambda_i$, then I add another matrix to it as: $A+xx^\top$ where $x$ $(n\times 1)$ is a column vector.
and also $A=yy^\top$ with $y$ a $(n-1)$ rank ...
- 188k
3
votes
0
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250
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Generalized matrix determinant lemma for pseudo-determinant of symmetric matrix
The pseudo-determinant of a square matrix $A$ is the product of its nonzero eigenvalues. Consider the generalized matrix determinant lemma $$\det(A+UWV^\top) = \det A\det W\det(W^{-1} + V^\top A^{-1}U)...
- 141
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0
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57
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Class of covariance matrices invariant under permutations
I am reading a paper on covariance matrix estimation, and in this paper is introduced a class of covariance matrices:
\begin{equation}
U(q, c_0(p),M)=\{\Sigma: \sigma_{ii}\leq M,\quad \max_j\sum_{j=1}^...
- 1
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1
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309
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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.
$$\operatorname{diag}({\bf v}...
CommunityBot
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1
vote
1
answer
52
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Eigendecomposition of a Gram matrix with specified columns
I originally posted this question on Math SE, but I am starting to think it is more suitable to post it here. After waiting about two weeks, I didn't get any activity. The original question is linked ...
- 256
2
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0
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Limiting spectral distribution of a random matrix with specific structure
First, consider an $N \times N$ Hermitian random matrix $V$ from the Gaussian Unitary Ensemble (GUE). It is well known that the empirical spectral distribution of the GUE satisfies the semicircle law ...
- 3
3
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1
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Eigenvalues of certain matrices
We write $R(\theta)=\left(\begin{smallmatrix}\cos(2\pi\theta)&\sin(2\pi\theta)\\ -\sin(2\pi\theta)&\cos(2\pi\theta)\end{smallmatrix}\right)$ for any $\theta\in\mathbb R$.
Let $d,m,n,r$ be a ...
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2
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1
answer
158
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The relationship between a matrix and its coefficient matrix decomposed in Pauli matrix
For a dimension-$4$ Hermitian matrix $A$, denote pauli matrices $\{I,X,Y,Z\}$ as $\{\sigma_0,\sigma_1,\sigma_2,\sigma_3\}$ respectively. The pauli matrices form a basis of the matrix space if we take ...
- 256
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6
answers
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Eigenvalues of symmetric tridiagonal matrices
Suppose I have the symmetric tridiagonal matrix:
$$ \begin{pmatrix}
a & b_{1} & 0 & ... & 0 \\\
b_{1} & a & b_{2} & \ddots & \vdots \\\
0 & b_{2} & a & \...
- 21
5
votes
2
answers
392
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Maximal eigenvalue of a real symmetric Toeplitz matrix
The $n×n$ matrix $A_n$ is defined by the elements $a_{ij}=n−|i−j|$.
\begin{bmatrix}
n & n-1 & n-2 & \cdots & 1\\
n-1 & n & n-1 & \cdots & 2\\
n-2 & n-1 & n &...
- 3,671
2
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0
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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{...
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1
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0
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63
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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
12
votes
3
answers
737
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How to find Suleimanova's work on the Nonnegative Inverse Eigenvalue Problem?
Many papers cite the work of Suleimanova when studying inverse eigenvalue problems - in particular, the nonnegative inverse eigenvalue problem (NIEP). However, I cannot seem to find her work anywhere....
- 1,719
0
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0
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32
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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
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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
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148
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Is there a way to find the eigenvalues of a matrix using character table?
I am studying applications of representation theory. I want to know if there is a procedure to find the eigenvalues and eigenvectors of a matrix using the character table of the Group acting on its ...
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1
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1
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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 ...
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52
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8
answers
5k
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Is there a fast way to check if a matrix has any small eigenvalues?
I have hundreds of millions of symmetric 0/1-matrices of moderate size (say 20x20 to 30x30) which (obviously) have real eigenvalues.
I wish to extract from this list the tiny number of matrices that ...
- 25.7k
8
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1
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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 ...
0
votes
1
answer
104
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Change of the smallest positive eigenvalue after a rank-one update
Given natural numbers $n,r,R\in\mathbb{N}$ with $r,R\le n$, let $A\in\mathbb{R}^{n\times r}$ and $B\in \mathbb{R}^{n\times R}$ be two matrices with full column rank and let $c\in\mathbb{R}^n$.
Denote ...
3
votes
2
answers
2k
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Expected value of the largest singular value of a random matrix with entries in $N (0,1)$
Given a matrix $A \in \mathbb R^{n \times n}$ whose entries are i.i.d. $N(0,1)$, what is the expected value of its largest singular value? Equivalently, what is the expected value of the largest ...
- 2,442
2
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1
answer
2k
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Eigenvalues of directed Laplacian matrix $L$ and $DL$, where $D$ is a diagonal matrix with positive entries
I have a weighted Laplacian matrix $L$ of a strongly connected directed graph and a diagonal matrix $D$ with positive entries. Since the graph is directed, $L$ is non-symmetric real. Further, since ...
- 1,446
4
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1
answer
230
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Conditions for distinct nonzero eigenvalues in product DAD for symmetric matrix A with repeated nonzero eigenvalues and diagonal matrix D
Let $A\in\mathbb{R}^{n\times n}$ be an real symmetric matrix with eigenvalues $\lambda_1, \lambda_2, \cdots, \lambda_n$ with some of which be nonzero and repeated, i.e., there exist $\lambda_i \ne 0$ ...
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1
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1
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112
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square matrix depending on complex value: spectral radius continous? [closed]
Let $A(z)$ be a $n\times n$ square matrix depending on the complex value $z$ and $\lambda_z$ is its spectral radius.
Is $\lambda_z$ continous or is it possible that it can jump? Or maybe someone knows ...
- 6,357
3
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1
answer
5k
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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$ ...
0
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0
answers
114
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Expectation of the operator norm of projection of a random permutation matrix
Assuming I have a fixed dimension $p$ subspace of $\mathbb{R}^d$ orthogonal to $1^d$ and $VV^\top$ with $V \in \mathbb{R}^{d \times p}$ is the orthogonal projection to the subspace.
What bound can I ...
0
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0
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69
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Spectrum of Moore-Penrose pseudo-inverse multiplied by a constant
Consider a random rectangular matrix $X\in\mathbb{R}^{N\times P}$ where each entry is drawn from iid distribution with mean $m$ and variance $s^2$, and denote $X^+$ the Moore-Penrose pseudo-inverse.
...
0
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0
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79
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Approximate solution problem of rank-one modification matrix secular equation
In Golub's paper , page 327,the eigenvalues of a rank-one modification of a $n\times n$ symmetric matrix can be computed by findng the zeros of the secular equation
\begin{equation*}
w(\lambda_j)=...
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1
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2
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413
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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 $...
- 188k
3
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0
answers
145
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Eigenvalues of random matrices are measurable functions
I have read that if a random matrix is hermitian then its eigenvalues are continuous, hence also measurable.
If the random matrix is not hermitian, the eigenvalues are not continuous in some cases. ...
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5
votes
2
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343
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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
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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
8
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3
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8k
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Spectrum of an adjacency matrix
The adjacency matrix of a non-oriented connected graph is symmetric, hence its spectrum is real.
If the graph is bipartite, then the spectrum of its adjacency matrix is symmetric about 0. A few ...
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2
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588
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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
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4
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293
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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
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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
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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 $X^*\in \mathbb{R}^{n \times n}$:
$$
X^* = \mathop{\text{...
5
votes
2
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721
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Matrices with same eigenvalues
This question is a more precise version of this question.
Let's assume we have the matrix
$$\left(
\begin{array}{ccccc}
0 & a & 0 & 0 & 0 \\
f & 0 & b & 0 & 0 \\
0 &...
- 13.2k
3
votes
1
answer
309
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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
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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
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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
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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
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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
$$\sigma_2 A_2 \sigma_2 = \begin{...
- 128k
1
vote
0
answers
290
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Eigenvalue decomposition of normalized adjacency matrix
Let $A$ be an adjacency matrix of undirected graph $G$, where $G$ is a connected graph. The normalized adjacency matrix is defined as $\hat{A}=D^{-1/2}AD^{-1/2}$, where $D$ is degree matrix of graph $...
0
votes
1
answer
224
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Faulty algorithm for simultaneous diagonalization?
I found a simple algorithm for simultaneous diagonalization of two commuting matrices (Nordgren - Simultaneous Diagonalization and SVD of Commuting Matrices), which seemed to be well-founded. For ...
- 128k
1
vote
0
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114
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Higher dimensional Cauchy interlacing theorem
If $A$ is a Hermitian matrix and $A_j$ the principal minor with the $j$ row and column deleted and $\phi_A(x)$ the characteristic polynomial. The Cauchy interlacing iheorem states that the roots of $\...
- 63.9k
2
votes
1
answer
299
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Eigenvalues of a specific matrix
I have a block matrix
$$M=\begin{bmatrix}
I_0& I_1& \cdots& I_1\\
I_2& I_0& \ddots& \vdots\\
\vdots& \ddots& \...