All Questions
Tagged with matrices reference-request
190 questions
8
votes
7
answers
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views
Source for roots of matrix polynomials?
A
matrix polynomial
is a polynomial whose variables are square $n \times n$ matrices,
let's say with entries in $\mathbb{C}$, and with coefficients in $\mathbb{C}$.
I am seeking a source of results on ...
- 151k
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
5
votes
2
answers
322
views
Stabilization of the pencil of skew symmetric matrices by the orthogonal group
During my researches I've come across the following question.
Let $A$ and $B$ be a couple of square $k\times k$ skew symmetric matrices on $\mathbb R$. Let us consider the (real) pencil generated by $...
user17697
27
votes
1
answer
1k
views
Properties to have matrices that commute in $\mathrm{GL}_n(\mathbb C)$
Let $G$ be a finite subgroup of $\mathrm{GL}_n(\mathbb C)$, $A,B \in G$ whose eigenvalues are thus in the unit circle.
Assume that the eigenvalues of $A$ are included in a circle arc of length $<\...
user49093
5
votes
2
answers
495
views
Existence of parametrizations of rational orthogonal matrices
I suppose that there are formulas which parametrize all the orthogonal matrices with rational coefficients. Does anyone know anything about it? And what are some publications that discuss this?
Thanks....
7
votes
2
answers
1k
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Dimension of the nilpotent centralizer of a nilpotent matrix
Fix a natural number $n$ and an algebraically closed field $k$. Let $\mathfrak{g}=\mathfrak{gl}_n(k)$. For any partition of $n$, $\lambda=(\lambda_1,\ldots,\lambda_r)$, let $A_{\lambda}$ be the $n\...
- 768
1
vote
1
answer
362
views
Odd subgroup of $\mathrm{GL}(n,\mathbb{Z})$
The group $\mathrm{GL}(n,\mathbb{Z})$ acts on $(\mathbb{Z}/2\mathbb{Z})^n$ by right multiplication (the same kind of things can be done with left action). I denote by $H\subset \mathrm{GL}(n,\mathbb{Z}...
- 7,722
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
4
votes
1
answer
630
views
Can sparse matrices satisfy the Null Space Property?
Definition A matrix $A \in \mathbb{C}^{m \times N}$ with $m < N$ satisfies the Null Space Property (NSP) of order $s$ if
$$\|x_S\|_1 < \|x_{\bar{S}}\|_1, \quad \forall x \in \ker A \setminus \{...
- 837
8
votes
2
answers
377
views
A family of skew-symmetric matrices corresponding to cycles in graphs
When investigating loops in Markov chains I ran into the following observation.
A cycle in a graph $G$ with $n$ vertices may be represented by a matrix $\Gamma \in \mathbb R^{n \times n}$ having the ...
- 985
3
votes
3
answers
1k
views
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
9
votes
1
answer
573
views
$SL_2(\mathbf{Z},8\mathbf{Z})$ differs from $E_2(\mathbf{Z},8\mathbf{Z})$. Has this result appeared in the literature?
I know a proof that the congruence subgroup $SL_2(\mathbf{Z},8\mathbf{Z})$ differs from its subgroup $E_2(\mathbf{Z},8\mathbf{Z})$, but can't find this fact in the literature. Does anyone know a ...
- 127
8
votes
1
answer
201
views
Growth of powers of non-negative integer matrices
In what I am currently doing, there naturally appears the following question: let $A$ be a square matrix with non-negative integer entries. Let $a_n$ be the sum of all entries of $A^n$.
Question: How ...
- 1,437
1
vote
1
answer
517
views
Books or references on multidimensional matrix operations [closed]
Have the 2D matrix operations been generalized to n-dimensional matrices?
Are there any books that define various operations on multidimensional matrix? I'd like to see operations such as ...
- 163
5
votes
1
answer
2k
views
Rank of a 0-1-matrix
Suppose $K$ is a field of characteristic $0$. Let $M \in K^{n \times m}$ be a matrix such that every entry of $M$ is either $0$ or $1$. About this matrix, I know further that each sum over a column ...
- 303
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
4
votes
2
answers
299
views
tracial triples
Say that a triple of real numbers $(a,b,c)$ is a realizable triple if there are matrices $A,B\in SL_2(\mathbb{R})$ such that $tr (A)=a$, $tr (B)=b$, and $tr (AB)=c$. Question: what is the shape of the ...
- 115
29
votes
6
answers
10k
views
how to find/define eigenvectors as a continuous function of matrix?
I asked this (with background) here
https://stats.stackexchange.com/questions/38494/principal-component-analysis-bootstrap-and-probability-of-eigenvalue-collision
but did not really get any answers. ...
- 2,600
1
vote
1
answer
416
views
Decomposition of Matrix to its sub-matrix with constant rank
When we study the structure of simple graphs with a lot of $1$ or $-1$ as its adjacency eigenvalues, the rank of its adjacency matrix is very important. The reason is, in these case, we can study the ...
- 4,784
1
vote
1
answer
198
views
Infinite Real Symmetric Toeplitz Matrix Reference
I am looking for a good starting point (book or articles) for studying Toeplitz matrices. Specifically as mentioned in the title, I am most interested in the case where they are of the form
$$A = \{\...
- 228
6
votes
0
answers
197
views
"Bell" or "Jabotinsky"-matrix - What's the canonical name (if any)?
I'm just reading J. Cigler's script for his talks "Konkrete Analysis" where I find the term "Jabotinsky-matrix" for that matrix, which I've (informally) been taught to call "Bell-matrix" (see at least ...
- 5,342
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 $BA$...
- 2,600
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^\...
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
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
18
votes
2
answers
5k
views
Nonvanishing of Jacobians implies global injectivity?
I am interested in obtaining injectivity of a $C^1$ map from the nonvanishing minors of its Jacobian matrix. Here is a brief history of the topic.
In 1953, Samuelson asked the following:
If the ...
- 981
1
vote
1
answer
254
views
references for families of conditionaly negative definite matrices
We say that a matrix $A\in M_n(\mathbb{C})$ is a conditionaly negative definite matrix if it is hermitian and if for all complex numbers $c_1,\ldots,c_n$ such that $c_1+\cdots +c_n=0$ we have
$$
\sum_{...
- 1,222
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
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
8
votes
4
answers
2k
views
Commuting matrices in GL(n,Z)
Suppose $M$ is a "hyperbolic" matrix in $GL(n,\mathbb Z)$, i.e., that its characteristic polynomial $p$ is irreducible over $\mathbb Z$ and has no roots of modulus 1.
Is there a closed description ...
- 2,125
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
6
votes
3
answers
3k
views
Defining Multiplication in Polynomials over Rings of Matrices
More explicitly, if $M_{2 \times 2}(\mathbb{R})[x]$ denotes the ring of polynomials over the ring of 2x2 matrices with real coefficients (with indeterminate x a 2 by 2 matrix with real coefficients), ...
- 370
6
votes
0
answers
998
views
Generalized Courant-Fischer theorem
Consider some quaternionic matrix $A$. A right eigvenvalue of $A$ is a quaternion $q$ such that $Ax=xq$ for some $x\in \mathbb{H}^n$. Similarly, a left eigenvalue of $A$ is quaternion $q$ such that $...
- 475
16
votes
4
answers
3k
views
How many minors I need to check to conclude all minors will vanish ?
Given a $m \times n$ matrix $n>m$, I was trying to check if all its $m \times m$ minor vanish.
I remember hearing that one really does not need to check all possible minors in order to conclude ...
- 1,795
4
votes
1
answer
1k
views
Integer vectors in the kernel of an integer matrix
Let $A$ be a non-zero symmetric $n \times n$-matrix with integer entries and suppose that $\det(A) =0$.
Question: How long is the shortest non-zero integer vector in the kernel of $A$?
Example: If ...
- 25.5k
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
16
votes
3
answers
791
views
Random products of projections: bounds on convergence rate?
The von Neumann-Halperin [vN,H] theorem shows that iterating a fixed product of projection operators converges to the projector onto the intersection subspace of the individual projectors. A good ...
- 181
38
votes
10
answers
18k
views
Fast matrix multiplication
Suppose we have two $n$ by $n$ matrices over particular ring. We want to multiply them as fast as possible. According to wikipedia there is an algorithm of Coppersmith and Winograd that can do it in $...
- 1,791
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)$$ for $i=1,2,3$, then $A$ and ...
- 2,154
2
votes
2
answers
3k
views
Statement of Lagrange's theorem on determinants(elementary question).
Apologies for this elementary question; but I was unable to find a reference otherwise.
Let $A, B, C$ be square matrices of the same dimension. Then,
$$\begin{vmatrix} A & C \\\ 0 & B \end{...
- 7,442