All Questions
Tagged with matrices eigenvalues
323 questions
1
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39
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Characterisation of Coxeter matrices with all non-real eigenvalues having absolute value 1
Let $C$ be an invertible integer matrix. Then a matrix $M$ is called Coxeter matrix (following Sato in https://www.sciencedirect.com/science/article/pii/S0024379505001709?via%3Dihub ) when $M=-C^{-1} ...
- 26.5k
3
votes
0
answers
373
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Eigenvalues of block matrix
Given scalars $\alpha, \beta \in \mathbb{R}$, a symmetric positive definite matrix $A \in \mathbb{R}^{n\times n}$ and a flat matrix $B \in \mathbb{R}^{m\times n}$, where $m < n$, can I say ...
- 125
0
votes
0
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166
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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
2
votes
1
answer
263
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Limit law of eigenvalue of random matrix with mean different to 0
If $X$ denotes a $m \times n$ random matrix whose entries are independent identically distributed random variables with mean $\mu$ and $\sigma^2 < \infty$, let
$$Y = X X^T$$
with $X^T$ the ...
- 335
0
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0
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79
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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
4
votes
1
answer
220
views
Sum of divisors and unitary divisors as the eigenvalue and the spectral norm of some addition matrix?
Let $n$ be a natural number and $D_n$ be the set of divisors.
We can make this set to a ring by observing that each divisor $d$ has
$$0 \le v_p(d) \le v_p(n)$$
Hence we can add two divisors $d,e$ by ...
user6671
11
votes
1
answer
2k
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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
16
votes
2
answers
1k
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Spectral symmetry of a certain structured matrix
I have a matrix
$$ A= \begin{pmatrix} 0 & a & d & c\\ \bar a & 0 & b & d \\ \bar d & \bar b & 0 & a \\ \bar c & \bar d & \bar a & 0 \end{pmatrix} $$
As ...
- 536
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 $M$ ...
- 1,269
3
votes
2
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432
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Maximum eigenvalue of a covariance matrix of Brownian motion
$$ A := \begin{pmatrix}
1 & \frac{1}{2} & \frac{1}{3} & \cdots & \frac{1}{n}\\
\frac{1}{2} & \frac{1}{2} & \frac{1}{3} & \cdots & \frac{1}{n}\\
\frac{1}{3} & \frac{...
5
votes
1
answer
241
views
Trace inequality under consideration of definiteness
Let $G \in \mathbb{R}^{3 \times 3}$ a symmetric, but indefinite matrix and $U \in \mathbb{R}^{3\times 3}$ a symmetric and positive definite matrix. I would like to prove the inequality
$$ \text{Tr} \...
- 51
3
votes
1
answer
534
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Spectrum of the sum of two general matrices
Are there any restrictions on the possible spectrum of the sum of two arbitrary matrices with given spectra other than the trace identity?
In other words:
Let $\alpha, \beta, \gamma$ be $n$-tuples (...
- 2,934
0
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0
answers
54
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Is there a method to find a vector that optimizes a Rayleigh quotient over a subspace?
Let $M\in\mathbb{C}^{n\times n}$ be an arbitary Hermitian matrix and let $E$ be a subspace of $\mathbb{C}^n$.
Is there a method to find vectors $y,z\in E$ such that
$$\dfrac{y^*My}{y^*y}=\sup_{x\in E\\...
- 596
-2
votes
1
answer
352
views
Can we attain the maximum and minimum of a Rayleigh quotient over any subspace? [closed]
Let $M\in\mathbb{C}^{n\times n}$ be a Hermitian matrix and let $E$ be a subspace of $\mathbb{C}^n$.
$$\mbox{Are } \sup_{x\in E\\
x\neq0}\dfrac{x^*Mx}{x^*x}\mbox{ and }\inf_{x\in E\\
x\neq0}\dfrac{x^*...
- 596
1
vote
1
answer
847
views
Do there exist graphs whose adjacency matrix is positive semi-definite? [closed]
If so, could you provide examples and specify the conditions under which this occurs? Thank you in advance
4
votes
1
answer
431
views
Simple way to calculate the eigenvalues of a $2 \times 2 \times 2$ tensor
I am working with hypergraphs. The various matrices associated with hypergraphs are hypermatrix or tensors. I am interested in spectral aspects. In particular, I want to find all the eigenvalues ...
- 1,269
1
vote
0
answers
443
views
Eigenvalues of symmetric tridiagonal matrices with identical off diagonal elements
Is there a simple analytical solution to obtain eigenvalues (and eigenvectors) for this type of tridiagonal matrices ? ( Off diagonal elements are identical and the matrix is symmetric)
$$
\begin{...
- 111
1
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0
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126
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Fastest way to calculate the eigenvalues of a product of two Toeplitz matrices
I have the following problem:
I need to find the fastest way to calculate the eigenvalues of a matrix that is the product of two Toeplitz matrices. $B = A U$.
The first is a regular Toeplitz matrix $A$...
- 11
4
votes
1
answer
2k
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Relation of row sums to largest eigenvalue
I know that the largest eigenvalue of a graph is bounded between the minimal and maximal row sum of the matrix. If I have a $0-1$ symetric matrix (an adjacency matrix) and I know $k$ of the rows have ...
- 133
1
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1
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350
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A linear algebra question regarding the eigenvalues of the product of a diagonal matrix and a projection matrix
I need to prove a statement in my research. The statement seems to be fundamental linear algebra, and numerical studies in MATLAB supported this statement, but I wasn't able to prove it after a few ...
- 21
1
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1
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276
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Spectral decomposition of a $4\times4$ real nonsymmetric matrix with unknown elements
I'm trying to eigendecompose the following matrix $A$, i.e. to find $Q$ and $\Lambda$ such that
$$
A = \begin{bmatrix}
-\alpha & \alpha & -\gamma^{-1} & 0\\
\beta &...
- 207
2
votes
0
answers
106
views
Connections between eigenvalues of $B$ and $A+iB$
Consider two symmetric and real matrices $A,B\in\mathbb{R}^n$ and definie $A+iB$. Note that $A+iB$ is not hermitian in this case. There are many results based on Brendixson and Courant-Fischer, saying,...
- 21
1
vote
2
answers
2k
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Eigenvalues of tridiagonal symmetric matrix
Could you tell me please, are there any analytical methods how to find eigenvalues of matrix such this one?
$$
\begin{pmatrix}
a_1 & b_1 & 0 & 0 & 0 & \ldots & 0 \\
b_1 & ...
- 335
5
votes
3
answers
273
views
Significance of the length of the Perron eigenvector
Let $A$ be a positive square matrix. Perron-Frobenius theory says that there exist $\lambda,v$ with $Av=\lambda v$ and $\lambda$ equals the spectral radius of $A$, $\lambda$ is simple, and $v$ is ...
- 175
6
votes
1
answer
389
views
Maximum eigenvalue of a doubly stochastic matrix with deleted row and column
Consider an $n \times n$ irreducible and reversible (in the sense of a Markov chain) stochastic matrix $P$; assume that it has uniform stationary distribution (so, by reversibility, the matrix is ...
- 297
9
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0
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802
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Positive definiteness of matrix
This question is about the positive definiteness of a (non-random) matrix that is defined using random variables as follows:
We fix the vector $v=(1,1)$ (yet, it seems the final result does not ...
- 192
3
votes
1
answer
4k
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Eigenvalues of product of symmetric positive definite matrices
Let $T_1, \ldots, T_n$ by real symmetric positive definite matrices, with eigenvalues bounded below by $\mu > 0$.
Can I say
$$
\frac{x^T T_1 T_2 \ldots T_n x}{x^T x} \geq \mu^n
$$
If these matrices ...
- 41
13
votes
0
answers
809
views
Can one Gershgorin circle (only) contain all eigenvalues, when the other circles are not contained in it
In short, following a question from my students, I am trying to find a special case where all the eigenvalues of a matrix lie within only one circle, but not in the others, and the other circles are ...
- 673
-2
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3
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2k
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When is it possible to find the sum of all elements of inverse of a matrix?
Given sum of elements of each row of a positive definite square matrix $M$ of order $n$ all of whose entries are non-negative, when is it possible to find the sum of all elements of the matrix $M^{-1}$...
- 135
8
votes
1
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5k
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Eigenvectors of Kronecker Product [closed]
Conjecture If $A$ and $B$ are two complex square matrices, then every eigenvector of $A\otimes B$ is of the form $x\otimes y$, where $x$ is an eigenvector of $A$ and $y$ is an eigenvector of $B$.
...
1
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1
answer
311
views
Relation between the algebraic multiplicity of an eigenvalue and the subdiagonal elements of a symmetric tridiagonal matrix [closed]
Show that if $T$ is a symmetric tridiagonal matrix and an eigenvalue $\lambda$ has multiplicity $k$, then at least $k−1$ subdiagonal elements of $T$ are zero.
If we consider a submatrix $B$ that has ...
7
votes
0
answers
905
views
The Möbius function as eigenvalues
Let the $N$ by $N$ matrix $A$ be defined by the tetration:
$$\Large \text{If } \gcd(n, k)=1 \text{ then } A(n,k)= \underbrace{e^{e^{\cdot^{\cdot^{e^{\Re\left(\frac{1}{n^s}\right)}}}}}}_m \text{ else }...
- 1,183
1
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0
answers
132
views
Transformations preserving the number of distinct eigenvalues
Let $A\in\mathbb{R}^{n\times n}$ be an $n\times n$ symmetric, invertible matrix with nonnegative real entries, $\mathbf{1}$ be the all one $n$-dimensional vector, and $\mathrm{diag}(v)$, $v=[v_1,v_2,\...
- 2,712
2
votes
0
answers
146
views
Upper bound on some eigenvalue problem
Let $A_1,\ldots,A_m \in R^{n\times n}$ be symmetric and positive semidefinite, and suppose that their sum $A$ is positive definite. For some nonzero vector $u\in R^n$ with $u^TA_ju>0$ for all $j$, ...
- 3,873
1
vote
1
answer
711
views
Eigenvalues of the product of traceless unitary hermitian matrices [closed]
As a follow up of the question raised in Determinant involving traceless unitary hermitian matrices, I would like to pose a similar question.
If A and B are distinct traceless unitary hermitian ...
12
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2
answers
2k
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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
5
votes
1
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335
views
Projecting a symmetric matrix onto the space of linear operators with a particular eigenvalue
Specifically, I am interested in the case where one eigenvalue is exactly $0$. Given an $n \times n$ symmetric matrix, I would like to find the closest $n\times n$ symmetric matrix that has one ...
- 245
2
votes
0
answers
52
views
Large-scale projected minimum-eigenvalue computations
I am interested in efficient numerical procedures for solving large-scale instances of the following projected minimum-eigenvalue problem:
$$\mu := \min_{v \in \mbox{ker}(A)} \frac{v^T H v}{\lVert v \...
- 35
2
votes
1
answer
657
views
Leading eigenvector value problem as an optimisation problem for asymmetric matrices
As noted in 1806.05647, given a symmetric matrix $A$, the leading eigenvector value problem (LEVP)
$$Av = \lambda v,$$
where $A = A^T \in \mathbb{R}^{n \times n}$, $\lambda$ is the largest ...
- 21
0
votes
1
answer
154
views
Energy of a symmetric matrix with $0$, $1$ or $-1$ entries
I have a symmetric matrix with entries $0$, $1$ or $-1$ which appeared in my works in graph theory (the diagonal entries are all zero). I need a good upper bound for the energy of this matrix; i.e. "...
- 351
3
votes
1
answer
336
views
Eigenvalues of random graphs
At time $t=0$, let $G_n(V,E)$ be a graph with $n$ vertices and $m < n$ edges. Then there exists a unique symmetric adjacency matrix $A_n$ associated with $G_n(V,E)$, defined as follows: $a_{ij} = 1$...
- 101
2
votes
1
answer
1k
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Is it faster to compute eigenvalues or coefficients of characteristic polynomials?
Given $A \in \mathsf{M}_n(\mathbb{C})$ (no special structure) is it (generally) faster to compute its eigenvalues or the coefficients of its characteristic polynomial?
References/insights would be ...
- 1,719
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
2
votes
1
answer
372
views
The maximal eigenvalue of average of positive matrices
Let $A$ and $B$ be two square real positive (all entries are positive) matrices that differ only in the first row. Let $\lambda_A$ and $\lambda_B$ be the maximal real eigenvalues of $A$ and $B$, ...
- 745
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 ...
- 689
26
votes
1
answer
5k
views
Generalization of Cauchy's eigenvalue interlacing theorem?
Cauchy's Interlacing Theorem says that given an $n \times n$ symmetric matrix $A$, let $B$ be an $(n-1) \times (n-1)$ principal submatrix of it, then the eigenvalues of $A$ and those of $B$ interlace.
...
- 571
3
votes
1
answer
403
views
Eigenvalue-taking operator?
$\newcommand{Tr}{\operatorname{Tr}}$
Is there a continuous map $(p,t) \mapsto \lambda(p,t)$ which, given a path $p: [0,1] \to M(2,\mathbb R)$ and a $t \in \mathbb [0,1]$, gives back an eigenvalue of $...
- 4,943
1
vote
0
answers
152
views
Bound for Expectation of Singular Value
In my case, $X_{\boldsymbol{\delta}}\in\mathbb{R}^{d\times M}$ is a function of Rademacher variables $\boldsymbol{\delta}\in\{1,-1\}^M$ with $\delta_i$ independent uniform random variables taking
...
- 53
0
votes
0
answers
59
views
Dimension reduction
$A=({B}\otimes{I_{k}})C$ where $B$ is a $N$x$r$ matrix with rank $r$, and $C$ is a $rk$x$rk$ symmetric matrix
$M=DAE$ where $D$ is a $Nk$ x $Nk$ symmetric matrix and $E$ is a $rk$x$rk$ symmetric ...
- 31
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