All Questions
Tagged with matrices reference-request
190 questions
7
votes
3
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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
7
votes
1
answer
223
views
Result attribution for eigenvalues of a matrix of Pascal-type
A few years ago, I wanted to cite a result in a paper, for which I could not find a reference. I ended up not using the full strength of it, and the part that I needed could be easily proved. Still, I'...
- 1,190
7
votes
1
answer
169
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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
7
votes
2
answers
251
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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
7
votes
0
answers
195
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Hölder continuity of spectrum of matrices
Endow $\mathbb{C}^{d \times d}$ with the norm induced by the Euclidean norm on $\mathbb{C}^d$. It is well-known (to those who know it well, I guess) that the spectrum $\sigma(A)$ of a matrix $A \in \...
- 12.5k
7
votes
0
answers
133
views
Removing rows to reduce the rank
What is the smallest number of rows one can delete from a matrix to reduce its rank (by $1$)? Is there any standard name / notation for this characteristic? Has it been studied?
I am in fact ...
- 23k
7
votes
0
answers
197
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A special eigenvalue problem
For my research I need to solve a generalised eigenvalue problem
$Ax=\lambda B x$, where $A$, $B$ are general matrices, and selectively find only eigen-pairs $\lambda, x$ such that $\lambda\in \mathbb{...
- 492
7
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0
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252
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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
6
votes
3
answers
1k
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Product of the entries of a matrix
Given a $n \times n$ matrix $A = (a_{ij})$, I was wondering if there was any theory or research interest relevant to the term
$$ \prod_{i,j} a_{ij}$$
the product of all the entries of the matrix.
- 401
6
votes
3
answers
3k
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Defining Multiplication in Polynomials over Rings of Matrices
More explicitly, if $M_{2 \times 2}(\mathbb{R})[x]$ denotes the ring of polynomials over the ring of 2x2 matrices with real coefficients (with indeterminate x a 2 by 2 matrix with real coefficients), ...
- 370
6
votes
1
answer
239
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Attempts to define a matrix exponential over (as much as possible) general fields
Given a $n \times n$ matrix $A$ over the complex numbers, the exponential of $A$ is defined as
$$\exp(A) := \sum_{k = 1}^\infty \frac1{k!} A^k , \qquad \tag{$\star$}\label{468645_star}$$
where ...
- 361
6
votes
1
answer
423
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Difference between parallel transport and ambient projection
Consider a $d$-dimensional complete embedded Riemannian submanifold $(M,g)$ of a Euclidean space $\mathbb{R}^D$ (The major examples we consider are sphere and Stiefel manifold). Assume the sectional ...
- 125
6
votes
1
answer
192
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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, ...
6
votes
0
answers
111
views
Factorization to sparse matrices
$\newcommand{\lrank}{\operatorname{lrank}}$
$\newcommand{\rank}{\operatorname{rank}}$
Given a matrix $A$, we can define its Hamming weight, $w(A)$, as the number of non-zero elements in it.
Now, given ...
- 3,319
6
votes
0
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392
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Divisibility properties of minors of matrices
Let $A$ be an $m\times n$ matrix with integer entries. Let $d_i(A)$ be the greatest common divisor of all $i\times i$ minors of $A$, and define $d_0(A)=1$. Whenever $i\leq j$, one has that $d_i(A)$ ...
- 191
6
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0
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197
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"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
6
votes
0
answers
998
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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
5
votes
2
answers
495
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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....
5
votes
1
answer
474
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An inequality for certain positive-semidefinite matrices
Suppose that $G=(G_{ij})$ is a positive-semidefinite symmetric matrix with the diagonal entries all equal $1$ and all off-diagonal entries $\le0$. Does it then necessarily follow that
$$\sum_{i,j}(G^5)...
- 128k
5
votes
1
answer
319
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Is there a name for this type of matrix?
For my thesis in neural networks, I was trying to find a way to generalize a Sobel operator. I quickly thought of this:
$$
\begin{bmatrix}
a&b&c\\
d&0&-d\\
-c&-b&-a
\end{...
- 153
5
votes
2
answers
249
views
Eigenvalue density of a symmetric tridiagonal matrix
Let $A_n\in\mathbb{R}^{n\times n}$ be defined as
$$
A_n=\begin{bmatrix} a & b & 0 & \cdots & \cdots & 0 & 0\\ b & a & b & \cdots & \cdots & 0 & 0\\ 0 &...
- 2,712
5
votes
1
answer
175
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Growth of the word norm for elementary matrices in $\rm SL_3 (\mathbb{Z})$
This is a reference request, since the answer is probably well known, but I could not find it.
Given a finitely generated group $\Gamma$ with a generating set $S$, define the word norm $l = l_S : \...
- 1,163
5
votes
1
answer
1k
views
Spin groups in terms of matrices and/or linear operators
Thus far, the books and articles I have read dealing with spin groups $\mathbf{Spin}(n)$ and $\mathbf{Spin}(p,q)$ consider them only in terms of either Clifford algebras or topologically as the double ...
- 349
5
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1
answer
473
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higher order analogues of sylvester's law of inertia?
Sylvester's law of inertia (here I quote wikipedia)
If A is the symmetric matrix that defines the quadratic form, and S is any invertible matrix such that D = SAS^{T} is diagonal, then the number ...
- 305
5
votes
1
answer
103
views
Interpolation between two matrices so that $L^p$ norm is controlled
Assume to have two square matrices $A$ and $B$ acting on $\mathbb{R}^n$ such that all their entries are in the interval $[0,1]$ and such that $||Ax||_1 = ||x||_1$ and $||Bx||_\infty \leq ||x||_\infty$....
- 697
5
votes
1
answer
514
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Do matrices with only elements along the main and anti-diagonals have a name?
To expand upon the title, I am wondering if there is a specific name for square matrices of the form: $$M = \begin{bmatrix} a_{11} & 0 & \cdots & 0 & \cdots & & 0 & b_{1n} ...
- 169
5
votes
2
answers
322
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Stabilization of the pencil of skew symmetric matrices by the orthogonal group
During my researches I've come across the following question.
Let $A$ and $B$ be a couple of square $k\times k$ skew symmetric matrices on $\mathbb R$. Let us consider the (real) pencil generated by $...
user17697
5
votes
1
answer
141
views
On the half-skew-centrosymmetric Hadamard matrices
Definition 1: A Hadamard matrix is an $n\times n$ matrix $H$ whose entries are either $1$ or $-1$ and whose rows are mutually orthogonal.
Definition 2: A matrix $A$ is half-skew-centrosymmetric if ...
- 696
5
votes
1
answer
315
views
Rank-constrained least-squares solution of the Sylvester matrix equation
For the Sylvester matrix equation $AX+XB=C$, I want to find the least-squares solution $X$ via
$$\begin{array}{ll} \text{minimize} & \| AX + XB - C \|_{\text{F}}^2\\ \text{subject to} & \mbox{...
- 191
5
votes
1
answer
2k
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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
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
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
5
votes
0
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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
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
4
votes
2
answers
828
views
English translation of “A multidimensional generalization of the Wronskian”
I am trying to generalize an algorithm for the construction of a certain linear combinations of functions $\boldsymbol{f}:x\in\mathbb{R}\mapsto \boldsymbol{f}(x)\in\mathbb{R}^n$ that utilizes ...
- 13.2k
4
votes
2
answers
1k
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Reference request: Oldest linear algebra books with exercises?
Inspired by the recent success of my "soft question" here, I also have to ask, what are some of the oldest linear algebra books out there with exercises? I'm fine with or without solutions, either way....
4
votes
3
answers
369
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Determinant in terms of certain $2\times 2$ minors
Let $A$ be an $n\times n$ matrix with entries $a_{i,j}$. Define an $(n-1)\times(n-1)$ matrix $B$ with entries $b_{i,j}=a_{1,1}a_{i+1,j+1}-a_{1,j+1}a_{i+1,1}$. Then $\det(B)=a_{1,1}^{n-2}\det(A)$.
I ...
- 191
4
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1
answer
170
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About $CW(512,16^2)$
Definitions: A weighing matrix $W = W(n,k)$ with weight $k$ is a square matrix of order $n$ and entries $w_{ij}$ in $\{0, \pm 1\}$ such that $WW^T=kI$,
where $I$ is the identity matrix. A circulant ...
- 696
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
4
votes
2
answers
133
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$\|x\|_0$ 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
4
votes
1
answer
372
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Eigenvalues of random matrix conditional on positive definiteness
Consider the Gaussian Orthogonal Ensemble, considered as a probability measure $\mu$ on the space of real symmetric matrices. Let $\mu|PD$ denote this measure conditioned on the event that the matrix ...
- 505
4
votes
1
answer
414
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A Handbook of Matrix Factorizations
I am looking for a good collection of facts regarding the various types of matrix factorizations, something like a "Handbook of Matrix Factorizations" or a very-thorough review paper. I am hoping for ...
- 143
4
votes
1
answer
781
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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
4
votes
1
answer
1k
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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
4
votes
1
answer
132
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Reference request for Bessel function of the second kind with matrix argument
As the title says, I would like to know if anyone could provide a reference which provides the definition and properties of the Bessel function of the second kind with matrix argument. If possible, I ...
- 49
4
votes
1
answer
152
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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
4
votes
2
answers
299
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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
4
votes
1
answer
230
views
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 $\lambda_i \ne 0$ ...
- 187
4
votes
1
answer
172
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Smith normal form and affine buildings
In Smith Normal Form of powers of a matrix someone has commented saying that one can reformulate many questions about Smith normal forms in the language of affine buildings. I wanted to know of a ...
- 41
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