All Questions
Tagged with linear-algebra co.combinatorics
513 questions
9
votes
3
answers
409
views
Determinant of a block matrix with many $-1$'s
For an array $(n_1,...,n_k)$ of non-negative integers and non-zero reals $a_1,...,a_k$, define a block matrix $M$ of size $n=n_1+\cdots+n_k$ as follows:
The main diagonal has blocks of sizes $n_i$ and ...
- 13.4k
4
votes
2
answers
665
views
The maximal size of intersection of two sets
Let $S=\{1,2,\cdots,2n\}$, and $S_i \subseteq S(i=1,2,\cdots,n+1)$ be $n+1$ subsets, each of which contains half of the $2n$ elements, namely $|S_i|=n$. Consider the following expression:
$$M=\max_{1\...
- 601
3
votes
0
answers
298
views
Matrix Inequality: Traces of $n$th powers
Let $A, B$ be matrices over $\mathbb{C}$ of the same dimensions (not necessarily square). With $'$ denoting conjugate-transpose, and tr the trace, show for $n\in\mathbb{N}$ that
$ 2\,\mathrm{Re}\, \...
- 131
6
votes
2
answers
283
views
Nonlinear boolean functions
Let $\mathbb{F}_2=\{0,1\}$ be the field with two elements. I wonder if there is any known algorithm/construction that, given any $n\geq 1$, returns a boolean function $f:\mathbb{F}^n_2\rightarrow \...
- 333
2
votes
2
answers
128
views
Spectral decomposition of a combinatorial matrix/Random walks on $s$-sets
$\newcommand{\Z}{\mathbb{Z}}
\newcommand{\J}{\mathcal{J}}
\newcommand{\la}{\lambda}
\newcommand{\1}{\mathbf{1}}
\newcommand{\R}{\mathbb{R}}$
Take any $n\in[3;\infty]$. Here and in what follows, $[k;\...
- 128k
8
votes
2
answers
323
views
Matrix rescaling increases lowest eigenvalue?
Consider the set $\mathbf{N}:=\left\{1,2,....,N \right\}$ and let $$\mathbf M:=\left\{ M_i; M_i \subset \mathbf N \text{ such that } \left\lvert M_i \right\rvert=2 \text{ or }\left\lvert M_i \right\...
- 225
6
votes
1
answer
278
views
Is this bound uniform in $N$?
I encountered this small combinatorial problem and do not quite know how to solve it:
Consider a set $\mathbf N:=\left\{1,2,....,N \right\}.$ This set has $\binom{N}{2}$ many subsets of cardinality $...
- 225
2
votes
1
answer
83
views
On submatrices: size bound
Let $M$ be a generic $2n\times 2n$ matrix and fix $k\leq n$.
Suppose $\mathcal{F}$ is a family of submatrices under the conditions that $A\in\mathcal{F}$ provided
(a) $A$ is a $k\times k$ ...
- 43.2k
9
votes
1
answer
349
views
A binomial determinant formula: a new variant
In a previous MO question, the OP asks a proof for $\det_{1\leq i,j\leq n}\left(\binom{i}{2j}+\binom{-i}{2j}\right)=1$. Subsequently, Gjergji Zaimi generalized the problem to
$$\det_{1\le i,j\le n}\...
- 43.2k
5
votes
1
answer
198
views
A nowhere-zero point in a linear mapping conjecture
I found a very interesting problem in the Open Problem Garden, which I am surprised is not as well-known as I would think it would be:
Prove that If $p>3$ is prime and $A$ is an invertible $n \...
- 2,649
3
votes
0
answers
182
views
Motivation/intution behind using linear algebra in these combinatorics problems
What is the motivation behind using linear algebra in these three problems ?
A pair $(m,n)$ is called nice if there is a directed graph with (self edge are allowed, but multiple edge are not allowed) ...
- 353
4
votes
2
answers
275
views
Hyperrectangle that contains most of cube's interior (except its vertices)
Let $n>0$, and let $p,q\in (0,1)$ such that $p<q$.
Is there a hyperrectangle $H$ that satisfies the following:
$\forall i\in{1,\dots,n}:\\ H\supset \prod_{j=1,\dots,n}
\begin{cases}
[p,q], &...
- 265
4
votes
0
answers
98
views
Ref. request: Enumerating elements of Bruhat cells
Given a field $F$ and a natural number $n$, let $B$ be the group of lower triangular, invertible $n \times n$ matrices over $F$. Then
$$GL_n(F) = \biguplus_{\pi \in S_n} B \pi B,$$
where we embed the ...
- 809
4
votes
1
answer
296
views
Is there a fast algorithm to test positivity of all principal minors of non-symmetric matrix?
I have a matrix $A \in \mathbb{R}^{n \times n}$ with positive eigenvalues. In the symmetric case, Sylvester's criterion implies that all the principal minors are positive. In the non-symmetric case, ...
- 188
2
votes
0
answers
86
views
A system of homogeneous linear equations
This is the "real-life" (but slightly more technical) version of a question I have asked recently.
For a prime $p>10$, let $\mathcal L_X$, $\mathcal L_Y$, and $\mathcal L_Z$ denote the pencils of ...
- 23k
15
votes
2
answers
862
views
What are the periodic Dyck paths?
I changed the thread completely so that everything is now elementary linear algebra.
A Dyck path of length $n$ is a list of positive integers $[c_1,c_2,...,c_n]$ with $c_i -1 \leq c_{i+1}$ for all $i$...
- 26.5k
3
votes
1
answer
164
views
A system of linear equations related to the geometry of a finite plane
Let $\mathcal L$ denote the set of all lines in $\mathbb F_p^2$ parallel to one of the lines
$$ X:=\{(x,0)\colon x\in\mathbb F_p \},
\ Y:=\{(0,y)\colon y\in\mathbb F_p \},
\ Z:=\{(z,z)\...
- 23k
2
votes
0
answers
41
views
Efficient $H$ representation of matrices with distinct cyclic shift permuted entries
Given points $v_1,\dots,v_n\in\mathbb Z^n$ in codimesion $1$ hyperplane $x_1+\dots+x_n=t$ with $0\leq x_{i}$ and a cyclic shift permutation $\sigma$ where
$v_1,\dots,v_n$ when written as columns of ...
- 13.9k
4
votes
0
answers
94
views
Totally Unimodular matrix edited from ordinary matrix
Given a matrix $M\in\{0,1\}^{m\times n}$ is there an algorithm to tell if we can convert some of $1$s to $-1$s and make $M$ Totally Unimodular and output such a Totally Unimodular in polynomial in $mn$...
- 13.9k
1
vote
1
answer
255
views
Is this totally unimodular family?
Is it possible to prove this matrix family only contains totally unimodular matrices?
The matrix has dimensions $\frac{3n(n-1)}2$ rows and $n+\frac{n(n-1)}2$ columns.
To every pair $(i,i')$ with $1\...
- 13.9k
17
votes
2
answers
1k
views
Constructive proof of a rational version of Perron-Frobenius?
In the following, we work with vectors and matrices whose entries are rational numbers. Inequalities between such vectors are understood to be coordinatewise: e.g., two vectors $a = \left(a_1,a_2,\...
- 33.8k
2
votes
0
answers
197
views
Full-rank factorization property of integer-valued matrices
$\newcommand{\al}{\alpha}
\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\varepsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\...
- 128k
5
votes
1
answer
109
views
Improved estimates of $n$ quantities via $n$ measurements
$\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\epsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\si}{\sigma}
\newcommand{\Si}{\...
- 128k
3
votes
0
answers
116
views
Trace of Symmetric matrices in fixed rank
I am solving some problem related to symmetric matrices over a finite field $\mathbb{F}_q$ and I am stuck at the following problem:
For every $a\in\mathbb{F}_q $, let $S_a(t,m)$ be the set of all $m\...
- 179
13
votes
1
answer
625
views
A difficult determinant
(EDIT: I have removed the denominators I had in a previous version as they were superfluous)
The $N\times N$ determinant
$$D(a,\vec{b})=\det\left((2N+a+b_j-i-j)!\right)$$
has the nice form
$$D(a,\...
- 2,552
3
votes
0
answers
119
views
Forcing $A_1x,\dotsc,A_Kx$ to lie in a proper subspace
(This is a re-worked version of a question I asked several days ago.)
Let $J$ be the all-one matrix with $m$ rows and $n$ columns, and suppose that $J=A_1+\dotsb+A_K$ is a decomposition of $J$ into a ...
- 23k
2
votes
0
answers
90
views
Representable integer matrices
Let $C, R \in \mathbb{Z}^n$. If there is an $n \times n$-matrix $M$ with all entries being integers such that the sum of the entries of column $k$ equals $C(k)$, and the sum of the entries of row $k$ ...
- 48.9k
9
votes
3
answers
356
views
Spectrum of orthogonality graph (2)
The orthogonality graph, $\Omega(n)$, has vertex set the set of $\pm 1$ vectors of length $n$, with orthogonal vectors being adjacent.
I am only interested when $4|n$, since otherwise $\Omega(n)$ is ...
- 421
5
votes
0
answers
397
views
spectrum of orthogonality graphs
The orthogonality graph $\Omega(n)$ with $2^n$ vertices is the graph with vertex set $\{-1,+1\}^n$, with two vertices being adjacent if and only if they are orthogonal (as vectors in the standard ...
- 421
8
votes
1
answer
604
views
Number of zeros of the derivatives of a polynomial
What is the maximum total number of zeroes a univariate polynomial $f\in\mathbb{C}[z]$ of degree $d$, together with all of its derivatives, can have at $k$ given points of $\mathbb{C}$?
I am ...
- 2,179
6
votes
1
answer
257
views
Families of subsets whose characteristic vectors are spanning sets
Let $X$ be a finite set and $\mathbb CX$ be a vector space with basis $X$. If $Y\subseteq X$ is a subset, then by the characteristic vector of $Y$ I mean $\sum_{y\in Y}y$.
My question is:
...
- 38.6k
2
votes
1
answer
924
views
Number of Symmetric matrices of fix rank over finite fields
This might be a question that shouldn't be asked here. But I need some help.
I want to count the number of $n\times n$ symmetric matrices over the finite field $\mathbb{F}_q$ and rank $r$. I found the ...
- 179
0
votes
0
answers
181
views
Number of Symmetric matrices
Let $S_m(q)$ denote the space of all $m\times m$ symmetric matrices over the finite field $\mathbb{F}_q$ of size $q$. What is the number of matrices $A=(a_{ij})\in S_m(q)$ of rank at most $3$ and $a_{...
- 179
2
votes
1
answer
302
views
For which finite projective planes can the incidence structure be written as a circulant matrix?
It is well known that the projective plane of order $2$ can be represented by the circulant matrix $M_2:=circ(x,x,1,x,1,1,1)= \begin{pmatrix}
x&x&1&x&1&1&1\\
1&x&x&...
- 13.4k
13
votes
1
answer
468
views
Near-linear mappings from $\mathbb F_p$ to $\mathbb R$
$\newcommand{\F}{{\mathbb F}}$
$\newcommand{\R}{{\mathbb R}}$
$\renewcommand{\phi}{\varphi}$
Let $p\ge 5$ be a prime.
If the functions $\phi_1,\phi_2,\phi_3\colon\F_p\to\R$ satisfy $\phi_1(x)+\...
- 23k
5
votes
1
answer
394
views
Disjoint union of affine subspaces contains a larger affine subspace
I'd like to say that a large structured subset of the $n$-dimensional Boolean cube $\{0,1\}^n$ contains a non-trivial affine subspace. To be more specific, I want to prove/disprove that for some ...
- 211
2
votes
0
answers
122
views
Number of distinct rows and columns in a matrix with bounded number of entries
How many distinct rows and columns a real square matrix can have (at least in symmetric case) such that rank of matrix is $r$ and entries:
are from $\{-b,-b+1,\dots,0,\dots,b-1,b\}$?
are from $\{-b,-...
- 13.9k
3
votes
0
answers
91
views
Number of vectors such that the projection is decomposable
Let $V$ be a vector space of dimension $n\geq 6$ over the finite field $\mathbb{F}_q$. Let $\omega\in\bigwedge^{n-3}V$ be a nonzero element. Define the annihilator subspace of $\omega$ by $V_\omega=\{...
- 179
0
votes
1
answer
258
views
Rank of the connected components
Let $G$ be a simple graph with adjacency matrix $A(G)$. Let $v$ be a cut-vertex of $G$. Let $G_1, G_2,\dots, G_k$ be the connected components of the induced graph $G-v$ ( the subgraph resulting after ...
- 640
0
votes
1
answer
138
views
On sum of matrices
Suppose we have a matrix $M\in\Bbb Q^{n\times n}$ with no $0$ elements and we write as sum of two matrices $M_1$ and $M_2$ on following constraint.
$M_{1,ij}=M_{ij}$ and $M_{2,ij}=0$ or $M_{1,ij}=0$ ...
- 13.9k
1
vote
0
answers
51
views
Relation between nullity of components to its parent graph
Let $G$ be an undirected graph and the corresponding adjacency matrix be $A$. Let $v$ be a cut-vertex of $G$. Let $G_1, G_2,\dots, G_k$ are the connected components of the induced graph $G-v$ ( the ...
- 640
6
votes
0
answers
218
views
Reconstruct orthogonal from an orthostochastic matrix
Given an $n \times n$ orthostochastic matrix $\mathbf{A}$, i.e., there exists an orthogonal matrix $\mathbf{O}$ with $A_{ij} = O_{ij}^2$ for all $1\leq i,j \leq n$. What is the fastest way to find $\...
- 909
0
votes
1
answer
390
views
An upper bound for skew symmetric rank 2 matrices
Earlier, I had asked a similar question but that was not the correct problem where I got stuck. After a few quick answer, I realized that and I apologize for that.
Let $B_m$ be the space of all skew-...
- 179
2
votes
2
answers
505
views
Maximal number of intersecting subspaces of a finite dimensional vector space
For a given $k,n$ such that $0<k \leq n/2$, is there a number $N_{n,k}$, such that if one has $N$ different $k$-dimensional subspaces $V_1, V_2,...,V_N$ in $\mathbb{R}^n$ satisfying:
1) $\bigcap_{...
- 1,101
8
votes
2
answers
824
views
Matroids of rank two
I am interested in matroids of rank two and would like to understand how interesting/big this class of matroids is.
I know that the 2-uniform matroid on (k+2) elements is not representable over any ...
- 613
7
votes
0
answers
296
views
Counting 0-1 $n\times n$ matrices with a given rank r
What is the number $N$ of $n \times n$ $0$-$1$ matrices with rank $k$?
I read this sequence is
"OEIS A064230 Triangle $T(n,k)$ = number of rational (0,1) matrices of rank $k$ ($n\ge 0$, $0\le k\le ...
- 1,677
2
votes
0
answers
113
views
Complexity of tensor decomposition vector over $\Bbb F_q$ or $\Bbb Z$
Suppose we have a matrix $$T\in\Bbb K^{n^k\times m}$$ and a target vector $v\in\Bbb F_q^m$ where $m<n^k$ and $1<k$ holds.
We need to find $k$ vectors $u_1,\dots,u_k\in\Bbb K^n$ such that $$v=...
- 13.9k
1
vote
0
answers
55
views
Maximum number of matrices satisfying given rank conditions
Assume that we have $2k$ matrices $S_1,\ldots,S_k$ and $\Phi_1,\ldots,\Phi_k$ over some finite field $F$ such that
(i) $S_i\in F^{l/2\times l}$ and $\dim S_i=l/2$ for any $i\in\{1,\ldots,k\}$;
(ii)...
- 147
2
votes
0
answers
363
views
Permanent of a matrix
Let $n \geq 2$ $a,b$ complex numbers (or in some other ring if you wish).
What is the permanent of the matrix
$$M(a,b,n)=
\begin{bmatrix}
a & a & a & ... & a & a \\
a &...
- 26.5k
7
votes
1
answer
404
views
Counting with tensor products
Suppose I've got vectors $v = (1,-1)$ and $w = (1,1)$ and any $m \in \mathbb{N}$. Let $a = v \otimes v \otimes w^{\otimes m}$ and let $\tilde{a}$ be the sum over all $\binom{m}{2}$ unique vectors ...
- 238