All Questions
Tagged with matrices reference-request
190 questions
7
votes
3
answers
1k
views
Determinant of correlation matrix of autoregressive model
I wonder if there is a paper that can point out how to compute the determinant of a $d \times d$ autoregressive correlation matrix of the form
$$R = \begin{pmatrix}
1 & r & \cdots & r^{d-...
- 719
4
votes
0
answers
188
views
Distributions over permutation groups $\mathcal{S}_n$
Partly inspired by recent developments in enumeration of pattern avoiding permutations, which is known to be connected with Brownian excursions [Hoffman&Rizzolo]. The exciting milestone is the ...
- 8,071
10
votes
1
answer
1k
views
Maximal order of elements in SL(n,q)
The maximal order of an element of $\mathrm{GL}(n,\mathbb{F}_q)$ is $q^n-1$, where the characteristic of $\mathbb{F}_q$ is odd $p$. See here for a nice proof that uses the Cayley-Hamilton Theorem.
...
- 8,529
4
votes
3
answers
667
views
Regularity for the roots of (characteristic) polynomials with given multiplicity
A classical result states that roots of a polynomial are continuous functions of its coefficients.
This is, for exemple, a direct consequence of Rouché's theorem.
Using the implicit function ...
- 2,135
1
vote
0
answers
47
views
Partial ordering of a matrix entries [closed]
I need this for experimentation in some work, so it is not without purpose.
Consider the in-spiraling and out-spiraling $4\times 4$ matrices
$$\begin{pmatrix} 1&2&3&4 \\ 12&13&14&...
- 43.2k
4
votes
0
answers
96
views
Bessel in matrix?
Let $M_n$ be the matrix
$$M_n=\begin{pmatrix}
1&\binom{1}{1}\binom{1-1}{1-1} &0 &0\qquad \qquad \dots &0\\
1&\binom{2}{1}\binom{2-1}{1-1} &\binom{2}{2}\binom{2-1}{2-1} &0 \...
- 43.2k
1
vote
1
answer
111
views
Decomposition of integral non-generate matrices [closed]
Let $GL_{\eta}(n,\mathbb{Z})=\left\{a\in
GL(n,\mathbb{R})\cap M^{n\times n}(\mathbb{Z})|det(a)=\eta\right\}$. Prove that there exists a
finite number of matrices $a_i$ in $GL_{\eta}(n,\mathbb{Z})$ ...
- 333
8
votes
2
answers
2k
views
Bounding the minimum singular value of a block triangular matrix
Question:
What is the sharpest known lower bound for the minimum singular value of the block triangular matrix
$$M:=\begin{bmatrix}
A & B \\ 0 & D
\end{bmatrix}$$
in terms of the properties ...
- 1,160
2
votes
1
answer
325
views
Determinant and inverse of a "stars and stripes" matrix
This is a variant of another MO question. Consider the matrix
$$M_n:=\begin{bmatrix}c_1& a & b&a& \ddots & a \\ b & c_2 & a& b&\ddots & b\\ a & b & c_3&...
- 13.4k
11
votes
3
answers
918
views
yet another determinant and inverse of a matrix
This problem is some variation of another MO question. Consider the matrix
$$M_n:=\begin{bmatrix}-c& a & a& \dots & a \\ b & c & a& \ddots & a\\ b & b & -c &...
- 43.2k
2
votes
0
answers
95
views
Reference request: Positive solution of positive system of linear equations
Let $A \in \mathbb{R}^{n\times n}$ be an invertible matrix with positive entries,
and $b \in \mathbb{R}^n$ a vector with positive entries.
When does $A^{-1}b$ have all positive entries?
I am looking ...
- 267
1
vote
2
answers
285
views
a follow up question on traces of matrices
In a recent MO post, pallab1234 ask for trace inequalities for which counterexample were given. I wish to probe in a different direction.
Suppose $A, B$ are $n\times n$ symmetric matrices (with ...
- 43.2k
5
votes
1
answer
199
views
Find the inverse of a more general matrix that is similar to the Hilbert matrix
In the last MO question , the following matrix is given:
$$M_{ij}=\left[\frac{1+(-1)^{i+j}}{i+j-1}\right]$$
and its inverse has been discussed.
Now the problem is further extended to a more general ...
- 153
10
votes
3
answers
829
views
Find the inverse of a matrix that is very similar to the Hilbert matrix
The standard Hilbert matrix $H$ is given by
$$H_{ij}=\frac{1}{i+j-1},$$
and it has an inverse given for example in this MO question.
Now I have encountered a matrix $M$ of similar form, namely,
$$...
- 153
17
votes
2
answers
1k
views
The GCD-matrix: generalizing a result of Smith?
Let $M$ be the $n\times n$ matrix, known as the GCD matrix, of entries $M_{ij}=\gcd(i,j)$. In the paper
H J S Smith, On the value of a certain arithmetical determinant, Proc. London Math. Soc. 7:208-...
- 43.2k
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
8
votes
1
answer
450
views
Do these matrices have a name?
I am looking for matrices $A\in \mathbb{R}^{m\times n}$ with $m>n$, $A^TA=\frac{m}{n}I$ and $diag(AA^T)=(1\ \dots\ 1)$ where $diag$ denotes the diagonal. Do such matrices have a name? An example ...
- 398
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
4
votes
1
answer
781
views
Determinant of a random row stochastic matrix
Does anyone know anything about the determinant of a random $n\times n$ row stochastic matrix? What I have in mind is that the rows are independently selected from the uniform distribution on the unit ...
- 23.2k
15
votes
2
answers
477
views
matrix inequality with orthogonal matrices
I would like to know if for $A,B\in SO(3)$ the inequality
$$
\|AB-BA\|_F\leq \|A-I\|_F\|B-I\|_F
$$
holds, where $\|\cdot\|_F$ denotes the Frobenius norm and $I$ the identity matrix. Using the identity
...
- 883
10
votes
1
answer
520
views
Homogeneous polynomials, mixed determinants, positive definiteness
Are there $n\times n$ real matrices $A_{1}, \ldots, A_{n}$ such that the $n$-homogeneous polynomial
$$
f(x_{1}, \ldots, x_{n}) = \det(x_{1} A_{1}+\cdots +x_{n} A_{n})
$$
never vanishes on $\...
- 3,946
4
votes
2
answers
133
views
$\|x\|_0$ approximation for very large matrices
Given a matrix $A \in F^{n \times m},$ $m\gg n,$ and a given $b \in F^n,$ with $F$ any (possibly finite) field, is there an algorithm that approximates the size of the minimal support solution for the ...
- 259
8
votes
2
answers
946
views
Best known bounds on (border) ranks of small matrix multiplication tensors?
The $(m,n,p)$-matrix multiplication tensor is a representation of the bilinear map $T\colon\mathbb{R}^{m\times n}\times\mathbb{R}^{n\times p}\rightarrow\mathbb{R}^{m\times p}$ given by $T(A,B)=AB$. ...
- 7,633
5
votes
2
answers
389
views
Pfaffian of several skew-linear transformations / matrices
Introduction: Let's assume we have a 2-form $\alpha=(1/2)\sum_{j,k=1}^n a_{jk}\ e_j\wedge e_k$, where $n=2m$, and $a_{jk}\in\mathbb C$. We know that $\alpha^{\wedge m}=\alpha\wedge\alpha\dots\wedge\...
- 53
10
votes
1
answer
537
views
Coefficient-wise powers of matrices. Reference wanted
Let $K$ be a commutative field and ${\rm M}_n (K)$ be the ring of $n\times n$ square matrices with coefficients in $K$ ($n\geqslant 1$ is an integer). For $k\geqslant 1$ and $A =(a_{ij})_{1\leqslant i,...
- 6,349
26
votes
1
answer
1k
views
Real square roots of symmetric matrices
In joint work with Andreas Fischle (TU Dresden, Germany) and Patrizio Neff (U Essen, Germany) we needed to use the following statement: If $S$ is a real $n\times n$ matrix with $S^2$ symmetric, then ...
- 5,186
13
votes
1
answer
1k
views
An inequality for the spectral radius of matrices used by J. Bochi
I am interested in the history of an inequality for the spectral radius of a $d\times d$ real or complex matrix, which occurs in Jairo Bochi's 2002 article Inequalities for numerical invariants of ...
- 6,206
1
vote
0
answers
105
views
Primitivity of $AA^\top$
Let $A\in\mathbb{R}^{n\times n}$ be a non-negative and irreducible matrix. Consider $B:=AA^\top$. It can be proved (I can post a proof if needed) that the following condition is necessary and ...
- 2,712
1
vote
0
answers
26
views
How can I filter the effects of a variable from a correlation matrix?
I have a correlation matrix (it contains 500 columns and 500 rows) and I would like to make an other correlation matrix in which one variable (and its influences) is filtered from the initial matrix. ...
- 11
5
votes
0
answers
179
views
When is a Hermitian matrix of the form $g^*g$ for some matrix $g$
I tried asking this question on Math StackExchange and didn't get any replies. I was hoping that maybe someone here could help, sorry for duplicating.
I'm trying to figure out some properties of ...
- 993
15
votes
4
answers
4k
views
Kernel of skew-symmetric matrix of rank $n-1$ with $n$ odd: is this a known result?
When $n$ is odd, the kernel of a skew-symmetric matrix $M$ of size $n\times n$ and rank $n-1$ is the span of $v$, where $v$ is a vector whose $i$-th component is the Pfaffian of the matrix obtained by ...
- 403
11
votes
1
answer
292
views
Have "sturdy squares" been studied before?
Over on PPCG I've just made a question that involves arranging the numbers 1 to 9 on a 3 by 3 grid such that every 2 by 2 subgrid has the same sum. I'm calling such 3 by 3 grids and their N by N ...
- 391
7
votes
0
answers
252
views
A conjecture of Lubotzky on ranks of subgroups of special linear groups over the integers
In a 1985 paper named "Dimension function for discrete groups" Lubotzky conjectured that:
For any integer $n \geq 3$ the group $\mathrm{SL}_n(\mathbb{Z})$
contains infinitely many finite index ...
- 11.3k
10
votes
0
answers
477
views
Name for an operation on matrices?
Given two matrices $A$ and $B$ of size $a \times n$ and $b \times m$ consider the following operation $A \dagger B$ whose result is an $a b^n \times n m$ matrix. $A \dagger B$ is a block matrix with $...
7
votes
1
answer
169
views
What is the maximal possible rank of a subgroup of a special linear group mod a prime?
Let $p$ be a prime number, and let $\mathbb{F}_p$ be the unique field of cardinality $p$.
What is $\max \{d(H) : H \leq \mathrm{SL}_3(\mathbb{F}_p)\}$?
Here we denote by $d(G)$ the smallest ...
- 11.3k
18
votes
0
answers
734
views
How boundedly generated is $SL_3(\mathbb{Z})$?
The group $G = \mathrm{SL}_3(\mathbb{Z})$ is known to be boundedly generated, that is, there exists some $m \in \mathbb{N}$, and $g_1, \dots, g_m \in G$ such that we have the following equality of ...
- 11.3k
1
vote
1
answer
250
views
characterize certain type of matrices
I am trying to characterize matrices with a certain property :
Define $U$ as an $n \times n$ matrix (over C or R; you can also assume
that it is unitary or orthogonal if it helps). Now take $n$
...
- 451
0
votes
1
answer
229
views
Reference request: Strong Connectivity and the Incidence Matrix
Question: What would be a good reference for characterizations of strong connectivity of a digraph in terms of its incidence matrix?
Details: Consider a digraph $(V, E)$ with vertex set
$$V = \{v_1,...
- 123
4
votes
1
answer
152
views
Enumerator Polynomials for Linear Anytime Codes
Let $C = \{c \in \mathbb{F}^n_2 : Hc=0\}$ be a binary linear code where $H \in \mathbb{F}^{k \times n}_2$ is a block lower-triangular matrix of full rank called the parity-check matrix of $C$. Clearly ...
- 313
8
votes
0
answers
2k
views
Possible values of eigenvalues of Hadamard product of Hermitian matrices
One of the most important (and very well-known) result in the study of the spectrum of Hermitian matrices is Horn's conjecture (or theorem?), which provides a complete answer to the following problem:
...
- 891
6
votes
1
answer
192
views
Monte-Carlo computation of the Smith normal form
Quite some time ago I saw an article where a Monte-Carlo algorithm for computing the Smith normal form of an integer matrix was described. In this article the following problem was posed:
Suppose $P, ...
7
votes
2
answers
251
views
What methods do we have to understand the spectrum of matrices with restricted entries?
Consider questions of the form (or the "most probable value of" version of these questions rather than the "largest possible"),
What is the largest possible spectral radius of a $...
- 1,893
0
votes
1
answer
130
views
Reference for measures of commutativity needed
I'm looking for an appropriate measure to quantify the extent to which two matrices commute. In other words, if A and B are two n×n Hermitian matrices, and [A,B]=C.
I'd like a function μ:Cn×n→[0,∞) ...
- 209
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
0
answers
276
views
An (open?) problem about a sequence of nested principal sub-matrices and their determinants
Problem: Let $A$ be a $n \times n$ integer matrix, $\det(A) = \pm 1$. Under which conditions there exist a nested sequence of principal submatrices of size $n$ such that they all have determinant $\pm ...
- 59
5
votes
0
answers
254
views
A weak Perron-Frobenius property for sets of positive matrices
A popular form of the Perron-Frobenius theorem states the following result: if $A$ is a $d \times d$ real matrix all of whose entries are positive, then the spectral radius of $A$ is a simple ...
- 6,206
8
votes
3
answers
2k
views
Transforming a binary matrix into triangular form using permutation matrices
I am interested in the complexity of the following problem:
Given an $m\times n$ binary matrix $M$, can we permute its rows/columns to obtain a triangular matrix?
I am also interested in ...
- 1,271
2
votes
1
answer
336
views
Bringing a (Least Squares Problem) Matrix into Block Upper-triangular Shape via Matrix-reordering
I have the problem of solving very large and very sparse least squares problems and, a bit dissatisfied with the run-times of the full-fledged QR-algorithm, I would like to bring the instances into ...
- 13.2k
13
votes
2
answers
678
views
Closure of the orbits of the $SL(2,\mathbb{Z})$-action on $\mathbb{R}^2$
I'm coming with a very basic question for which I can't find an answer. Please forgive me if I didn't search efficiently enough.
What can the closure of an orbit of an element $X$ of $\mathbb{R}^2$ ...
- 2,696
27
votes
3
answers
13k
views
What is known about the distribution of eigenvectors of random matrices?
Let $A$ be a real asymmetric $n \times n$ matrix with i.i.d. random, zero-mean elements. What results, if any, are there for the eigenvectors of $A$? In particular:
How are individual eigenvectors ...
- 433