All Questions
Tagged with matrices reference-request
190 questions
4
votes
1
answer
233
views
Generating of the matrix ring by two hermitian matices
Let $p$ be a prime and $q=p^n$. Let $\mathbb F_{q^2}$ be a field with $q^2$ elements and $\sigma$ its authomorhism of order two. A $m$ by $m$ matrix $A$ over $\mathbb F_{q^2}$ is hermitian if $A^\...
4
votes
1
answer
448
views
Is that series-transformation known in the context of divergent summation?
Note: I asked this question in math.stackexchange but did not receive an answer
Background:
In the context of divergent summation I'm analyzing the matrix of eulerian numbers for a regular matrix-...
- 5,342
4
votes
0
answers
59
views
Graph-class defined by matrix-like vertex-operations
Let $m$ be a positive integer. We define a (directed) graph on $m(m-1)$ vertices
$$V = \bigl\{(i,j) \mid i \ne j,\, i,j\in\{1,\dots,m\}\bigr\}$$
and edges as follows:
$(i,j) \in V$ is adjacent (...
- 577
4
votes
0
answers
1k
views
Reference for matrices with all eigenvalues 1 or -1
In a homological algebra problem I am in the situation that I have an invertible (over $\mathbb{Z}$) integer matrix $X$ and a permutation matrix $Y$ such that $N:=XY$ is a matrix with all eigenvalues ...
- 26.5k
4
votes
0
answers
98
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Rank of binary matrix related to the number of positive squarefree integers less than $n$
I posted this question at the Mathematics SE, but received no response there so I am posting it here.
The following fact is stated in the comments-section of sequence A013928 in the OEIS.
Let $C$ ...
- 1,719
4
votes
0
answers
148
views
A Toeplitz variant of the Hilbert matrix
It is well-known that the Hilbert matrix $H$, i.e., the symmetric Hankel matrix with entries
$$H_{m,n}=\frac{1}{m+n-1}, \quad m,n\in\mathbb{N},$$
determines a bounded operator on $\ell^{2}(\mathbb{N}...
- 2,188
4
votes
0
answers
244
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On the sum of the first row of the inverse of a certain symmetric Toeplitz matrix
(i) Consider a Toeplitz matrix $A_n = (a_{i, j})_{1 \le i, j \le n}$ of size $n$ defined as follows:
$$ a_{i, j} := |i-j|^{-1/2}, \text{ if } i \ne j; \ \ a_{i, j} := 2, \text{ if }i = j. $$
Let $...
- 195
4
votes
0
answers
188
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Distributions over permutation groups $\mathcal{S}_n$
Partly inspired by recent developments in enumeration of pattern avoiding permutations, which is known to be connected with Brownian excursions [Hoffman&Rizzolo]. The exciting milestone is the ...
- 8,071
4
votes
0
answers
96
views
Bessel in matrix?
Let $M_n$ be the matrix
$$M_n=\begin{pmatrix}
1&\binom{1}{1}\binom{1-1}{1-1} &0 &0\qquad \qquad \dots &0\\
1&\binom{2}{1}\binom{2-1}{1-1} &\binom{2}{2}\binom{2-1}{2-1} &0 \...
- 43.2k
3
votes
3
answers
1k
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Applications of rank factorization or full rank decomposition [closed]
I am teaching a course on linear algebra and came to this theorem: every $m \times n$ matrix $A$ with rank $r$ admits a factorization $A = CR$ where $C$ is an $m \times r$ matrix and $R$ is an $r \...
- 357
3
votes
1
answer
156
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How can I show $\{\mathbf{x}: \dim (\ker M_1(\mathbf{x}) \cap \ker M_2(\mathbf{x})) \geq C \}$ is an affine variety?
Let $M_1(\mathbf{x})$ and $M_2(\mathbf{x})$ be $m$ by $m$ matrices with each entry a homogeneous form in $\mathbb{C}[x_1, \ldots, x_n]$.
I would like to show that
$$
\{ \mathbf{x} \in \mathbb{A}^n_{\...
- 3,625
3
votes
1
answer
449
views
Explicit formula for the functional calculus of 2x2 matrices
Wikipedia gives the following explicit formula for the functional calculus of $2\times2$ matrices:
$$
f(A) = \frac{f(\lambda_+) + f(\lambda_-)}{2} I + \frac{\mathrm{tr}(A)/2 - \mathrm{adj}(A)}{\sqrt{\...
- 7,127
3
votes
1
answer
166
views
The spectral radius of a modified graph
Let $H$ be a graph and let $G=H \vee K_{1}$ be obtained by creating a new vertex and joining it to every vertex in $H$.
This situation has many different names: $G$ is called the cone or the ...
- 6,962
3
votes
3
answers
2k
views
'Sign matrices'-(-1,+1) square matrices
My question arises from a discussion on an answer given by Maurizio Monge here.I do not know if there is a known terminology for such matrices. By "sign matrices," I mean square matrices whose entries ...
- 2,855
3
votes
1
answer
343
views
Reference request: about “SNF” (Smith Normal Form)
I've read about some studies on the Paley I Construction. Among them I found the following notations ( See this page: https://documents.uow.edu.au/~jennie/matrices/32P02.html ).
$$SNF:1,2^a,4^{b},8^{b}...
- 41
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 $M$ ...
- 1,269
3
votes
1
answer
655
views
Upper bounds on the condition number of the eigenvector matrix
Let $A$ be an $n\times n$ real matrix with entries in a fixed interval $[a_\min,a_\max]$, with $a_\min$, $a_\max>0$.
Question: Are there any upper bounds on the condition number of the ...
- 2,712
3
votes
1
answer
347
views
The covariance matrix of quadratic form, without normal assumption
Assume $\mathbf{x}$ is a random vector with mean $\mathbf{\mu}$ and covariance matrix $\mathbf{\Sigma}$. Symmetric matrices $\mathbf{A}$ and $\mathbf{B}$ are given.
Without assuming normality, how to ...
- 51
3
votes
1
answer
270
views
What is the name of this measure of matrix "degenerateness"
Given a spanning set, consider the minimum number of vectors that you must remove in order to make it no longer span. What is this number called?
If the vectors are columns in a matrix $\Phi$, then ...
- 7,633
3
votes
1
answer
300
views
Deciding isometry of unimodular lattices by Gram matrices
Say I have two unimodular lattices $A$ and $B$, represented by their Gram matrices.
Question: Is there an algorithm to decide whether $A$ and $B$ are isometric, i.e. whether there exists a matrix $S \...
- 9,501
3
votes
1
answer
319
views
Matrix transformation that "rotates" a matrix by $45^\circ$
I have an $n \times n$ integer matrix $A$. I want to obtain an $m \times m$ matrix $B$, where $m \ge n$, such that the rows of $A$ are the diagonals of $B$ and the columns of $A$ are the anti-...
- 2,993
3
votes
1
answer
383
views
Matrices as dynamical systems
Matrices can be understood in different ways, e.g.
Linear systems of equations
(rich algebraic structure of) Linear mappings
Graphs
Evolution law of discrete-time Dynamical system
Well, 1. und 2. ...
- 5,327
3
votes
1
answer
203
views
Results on Boolean matrices
Matrices with entries in the finite field of two elements $\mathbb{F}_2$, and with the usual operations of matrix addition and multiplication, have been intensively studied, especially due to their ...
- 31
3
votes
1
answer
143
views
Reference request: Spectrum of intersection matrices
Let $P(A)$ be the set of all non-empty proper subsets of a finite set $A$. Let $M$ be a matrix indexed by the set in $P(A)$ whose $ij$ the entry is $1$ if the associated sets are disjoint and $0$ ...
- 1,269
3
votes
1
answer
621
views
Largest eigenvalue of a periodic Jacobi matrix
There is a vast literature on Jacobi matrices, I just don't know where to start looking. I'm interested in estimating the largest eigenvalue of the $n\times n$ periodic Jacobi matrix $D+P+P^{-1}$, ...
- 11.1k
3
votes
0
answers
147
views
Convolution integral and its matrix representation
My background is chemistry and I was exploring some one dimensional deconvolution problems i.e., resolution of two or more overlapping peaks. A lot of excellent work was done in the 1970-80s. However, ...
- 879
3
votes
0
answers
111
views
Infinite ordered products (reference request)
While writing arXiv:1510.05757v2, I found myself proving some basic facts about products of Banach algebra elements over an infinite totally ordered set. (Statements and proofs are in Appendix C.) The ...
- 2,284
3
votes
0
answers
148
views
Spectrum of symmetric Toeplitz matrix
A matrix is Toeplitz if it is constant on the diagonals parallel to the main diagonal.
I am looking for references on the spectrum of finite symmetric Toeplitz matrices over finite fields.
- 221
3
votes
0
answers
39
views
A non-singularity property for sets of real matrices
Let $M_N(\mathbb{R})$ be the ring of $N\times N$ real matrices. We say that a couple $(\mathcal{U},\mathcal{V})$, with $\mathcal{U},\mathcal{V}\subseteq M_N(\mathbb{R})$ is admissible if, for every $A\...
- 943
3
votes
0
answers
231
views
Singularity of symmetric block matrix with singular diagonal blocks
One can show that the following statement holds:
Given symmetric matrix $A \in \Re^{n \times n}$ and tall matrix $B \in \Re^{n \times p}$ with full column rank,
$$\begin{bmatrix}A & B \\ B^T &...
- 73
3
votes
0
answers
180
views
Automorphisms of infinite matrix algebra
This is a similar question to one that I posted in MSE a few days ago.
I recently came across this paper from Alahmedi, Alsulami, Jain and Zelmanov, which quoted the following result for $M_\infty(K)$...
- 195
3
votes
0
answers
359
views
Do we know what the impulse to "introduce" the Jordan canonical form was?
Mo-ers,
Do you know how it was that the study of the Jordan canonical form began?
There are certain things that may be said once one has thought about the matter: for instance, one can say that the ...
- 87
3
votes
0
answers
122
views
Algebra of block matrices with scalar diagonals
I am interested in block matrices $A$, that is $A\in M_{n\times n}(R)$ where $R=M_{s\times s}(k)$ and $k$ is a field, such that for every positive integer $m$ the matrix $A^m$ has only scalar blocks ...
- 3,041
3
votes
0
answers
65
views
How to show that a continuous family of symmetric matrices is uniformly positive?
My problem : I have a family of $4 \times 4$ symmetric matrices. More precisely consider an interger $d$, a real $\lambda> 0$ and define the family $S_{\lambda}$:
$ \{A(\lambda,x_1,x_2) ; (x_1,...
- 31
3
votes
0
answers
62
views
How likely is a matrix with exactly $n$ number $1$s per row to avoid a large wide empty submatrix?
Consider the finite collection $M(N,n)$ of all $N \times N$ matrices with exactly $n$ entries per row equal to $1$ and all other entries equal to zero $0$.
By an $a \times b$ submatrix of $M$ we ...
- 1,955
3
votes
0
answers
82
views
Maximum number of negative entries in a matrix with positive diagonal and given rank
Suppose $A \in \mathbb{R}^{n \times n}$ has positive entries on it main diagonal and $\mbox{rank}(A) =: d < n$. Then, what is the maximum number of many negative entries $A$ can contain?
If, in ...
- 31
3
votes
0
answers
162
views
Does the following characterization of subgroups of $GL_2(\mathbb{F}_p)$ generalise?
Let $p$ be a prime number. By a Cartan subgroup of $GL_n(\mathbb{F}_p)$ I mean an absolutely semisimple maximal abelian subgroup.
When $n=2$, it is well-known* that, for $G \subset GL_n(\mathbb{F}_p)$...
- 2,827
2
votes
1
answer
460
views
Maximum permuted row/column sum of a matrix
Given a real $n \times n$ matrix $A$ (feel free to assume its entries are non-negative, if it helps), what is known about the problem of computing the quantity
$$
\max_{\sigma \in S_n} \left\{\sum_{j=...
- 6,351
2
votes
1
answer
1k
views
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
2
votes
2
answers
535
views
Is anything known about the eigenspectrum of the regular representation of the permutation group?
I am looking for information like upper bounds on how many times any eigenvalue can occur or something like how many eigenvalues can be there in some given range. Is anything like this known?
The ...
- 1,893
2
votes
2
answers
3k
views
Statement of Lagrange's theorem on determinants(elementary question).
Apologies for this elementary question; but I was unable to find a reference otherwise.
Let $A, B, C$ be square matrices of the same dimension. Then,
$$\begin{vmatrix} A & C \\\ 0 & B \end{...
- 7,442
2
votes
1
answer
341
views
Symmetric orthogonal matrices with constant diagonal entries
$\newcommand{\al}{\alpha}
\newcommand{\be}{\beta}
\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\varepsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\...
- 128k
2
votes
3
answers
1k
views
On certain decomposition of unitary symmetric matrices
This is by any means elementary, but since I have asked this question on Stark Exchange but received no satisfactory answers I decide to post it here.
It is well known that a symmetric matrix over ...
- 355
2
votes
1
answer
810
views
On matrices that almost have the same eigenvalues
Let $A$ and $B$ be two $4\times 4$ matrices. Using Newton's identities, one can prove that if
$$\det(A) = \det(B)\quad \text{and}\quad \mathrm{tr}(A^i) = \mathrm{tr}(B^i)$$ for $i=1,2,3$, then $A$ and ...
- 2,154
2
votes
1
answer
153
views
What conditions on the rate matrix $Q$ ensure unique convergence in continuous-time Markov chains?
In the study of discrete-time Markov chains, the conditions under which all initial distributions converge to a unique stationary distribution are well-understood. Specifically, if the transition ...
- 807
2
votes
1
answer
460
views
dimensions of strata of Pfaffian varieties
Let $V$ a complex vector space of dimension $2n$. Let us consider $W=\wedge^2V$ and the Pfaffian variety $Pf\subset \mathbb{P}W$ that parametrize degenerate skew-symmetric matrices. $Pf$ is naturally ...
- 3,779
2
votes
1
answer
217
views
Diagonalising a symmetric matrix with polynomial entries
Suppose I have a symmetric $2$ by $2$ matrix $M$ whose $(i,j)$-th entry $F_{i,j}(\mathbf{x})$ belongs to $\mathbb{R}[x_1, \ldots, x_n]$ for each $i,j$. I know that for each $\mathbf{x} \in \mathbb{R}^...
- 3,625
2
votes
1
answer
311
views
How to find the set of vectors which are as nearly orthogonal as possible?
Let $\mathbf{A}=[a_{ij}]$ be a matrix of real numbers $a_{ij}>0$ for all $i\in\{1,\ldots,k\}$ and $j\in\{1,\ldots,n\}$ and let $v$ be a nonnegative real number. The coefficient of the matrix are ...
- 123
2
votes
1
answer
107
views
The ring of upper triangular $n$-by-$n$ matrices over a skew field is (left and right) Rickart
Let $T_n(R)$ be the ring of upper triangular $n$-by-$n$ matrices with entries in a (commutative or non-commutative) unital ring $R$. It happened to me to note that, if $R$ is a skew field, then $T_2(R)...
- 10.5k
2
votes
1
answer
741
views
rank of a linear combination of matrices
Let $A_1,..., A_s \in M_n(\mathbb{R})$ be symmetric matrices and suppose they are linearly independent over $\mathbb{R}$. This means that
$$
m = \min_{(c_1, ..., c_s) \in \mathbb{R}^s \backslash \{0\}}...
- 3,625