All Questions
Tagged with co.combinatorics matrices
256 questions
35
votes
0
answers
1k
views
Orthogonal vectors with entries from $\{-1,0,1\}$
Let $\mathbf{1}$ be the all-ones vector, and suppose $\mathbf{1}, \mathbf{v_1}, \mathbf{v_2}, \ldots, \mathbf{v_{n-1}} \in \{-1,0,1\}^n$ are mutually orthogonal non-zero vectors. Does it follow that $...
- 6,351
34
votes
1
answer
789
views
Which graphs on $n$ vertices have the largest determinant?
This is a question that seems like it should have been studied before, but for some reason I cannot find much at all about it, and so I am asking for any pointers / references etc.
The determinant of ...
- 12.7k
30
votes
1
answer
1k
views
Sum over 0-1 matrices
I stumbled across the following formula when working on a research problem in theoretical computer science. I am looking for a simple proof of it, or any idea which might prove useful.
I checked its ...
- 303
28
votes
3
answers
2k
views
Is every positive integer the permanent of some 0-1 matrix?
In the course of discussing another MO question we realized that we did not know the answer to a more basic question, namely:
Is it true that for every positive integer $k$ there exists a balanced ...
- 82.7k
28
votes
4
answers
2k
views
Matrices: characterizing pairs $(AB, BA)$
Let $A$ be an $m\times n$-matrix and $B$ an $n \times m$-matrix over the same field. Consider the matrices $C=AB$ and $D=BA$. It is probably well known (and not difficult to show) that the only ...
- 6,919
26
votes
1
answer
5k
views
Generalization of Cauchy's eigenvalue interlacing theorem?
Cauchy's Interlacing Theorem says that given an $n \times n$ symmetric matrix $A$, let $B$ be an $(n-1) \times (n-1)$ principal submatrix of it, then the eigenvalues of $A$ and those of $B$ interlace.
...
- 571
23
votes
3
answers
2k
views
On determinants formed by binomial coefficients
Let $q$ be a number. Let us consider the $q^2-1$-th line of the Pascal triangle (i.e. numbers ${{q^2-1} \choose i}$, $i=0,1,...q^2-1$). We have $q^2$ numbers.
Let us form naively a $q \times q$ ...
- 305
21
votes
2
answers
2k
views
Lifting matrices mod 2 to integers.
The following question was motivated by my research.
Consider a $n\times n$ matrix whose elements are $0$'s or $1$'s such that the determinant is odd. The question is: is it possible to assign signs ...
- 4,736
21
votes
0
answers
2k
views
The Fourier Transform of taking Eigenvalues
The purpose of this question is to ask about the Fourier transform of the map which associate to an $n$ by $n$ matrix its $n$ eigenvalues, or some function of the $n$ eigenvalues. The main motivation ...
- 24.7k
21
votes
0
answers
904
views
Cauchy matrices with elementary symmetric polynomials
$\newcommand{\vx}{\mathbf{x}}$
Let $e_k(\vx)$ denote the elementary symmetric polynomial, defined for $k=0,1,\ldots,n$ over a vector $\vx=(x_1,\ldots,x_n)$ by
\begin{equation*}
e_k(\vx) := \sum_{1 \...
- 28.6k
20
votes
2
answers
1k
views
Euler numbers and permanent of matrices
Motivated by Question 402249 of Zhi-Wei Sun, I consider the permanent of matrices
$$e(n)=\mathrm{per}\left[\operatorname{sgn} \left(\tan\pi\frac{j+k}n \right)\right]_{1\le j,k\le n-1},$$
where $n$ is ...
- 884
19
votes
4
answers
719
views
The rank of a perturbed triangular matrix
$\DeclareMathOperator{\rk}{rk}$
The question below is implicit in this MO post, but I believe it deserves to be asked explicitly, particularly now that I have some more numerical evidence.
Suppose ...
- 23k
18
votes
3
answers
8k
views
Number of invertible {0,1} real matrices?
This question is inspired from here, where it was asked what possible determinants an $n \times n$ matrix with entries in {0,1} can have over $\mathbb{R}$.
My question is: how many such matrices ...
- 32.1k
18
votes
3
answers
1k
views
Orthogonal matrices with small entries
Is it true that for any $n$, there exists a $n \times n$ real orthogonal matrix with all coefficients bounded (in absolute value) by $C/\sqrt{n}$, $C$ being an absolute constant ?
Some remarks :
If ...
- 4,653
18
votes
3
answers
1k
views
looking for proof or partial proof of determinant conjecture
Math people:
I am looking for a proof of a conjecture I made. I need to give two definitions. For distinct real numbers $x_1, x_2, \ldots, x_k$, define $\sigma(x_1, x_2, \ldots, x_k) =1$ if $(x_1, ...
- 533
18
votes
3
answers
6k
views
Number of unique determinants for an NxN (0,1)-matrix
I'm interested in bounds for the number of unique determinants of NxN (0,1)-matrices. Obviously some of these matrices will be singular and therefore will trivially have zero determinant. While it ...
- 359
18
votes
2
answers
488
views
Encoding primes via ranks of sign matrices
(Reposted from math.SE)
Recently I came across a very simply defined family of matrices: for $n \in \mathbb{N}$, set $A_n := (a_{ij})_{0 \le i, j \le n-1}$, where
$$\displaystyle a_{ij} := (-1)^{\big\...
- 281
17
votes
4
answers
2k
views
Possible values of the determinant for matrices with elements $\{1, 0, -1\}$
For matrices with elements $\{-1, 1\}$ it is known from here that the possible absolute values of determinants of $n \times n$, $n \leq 6$ matrices with entries $\{-1, 1\}$ are as follows:
...
- 539
17
votes
2
answers
911
views
Find the determinant of a matrix given the determinant of all $p\times p$ sub-matrices?
Is it possible to find the determinant of an $n\times n$- matrix, only given the determinant of all $p\times p$ sub-matrices in it? Here $p\leq n$ is fixed. This is obviously true if $p=1,n$. But what ...
- 293
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
17
votes
0
answers
402
views
Number of $F_p$-matrices ac=ca, bd = db , ad - da = cb - bc is polynomial in p ? ("Manin matrix variety" - normal ? Cohen–Macaulay ? )
Consider four $n\times n$ matrices $a,b,c,d$ over finite field $F_q$ (or $F_p$ for simplicity), such that they satisfy three equations: $ac=ca,bd=db, ad-da=cb-bc $. Thus an affine algebraic manifold ...
- 24.9k
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
16
votes
2
answers
504
views
The number of 0-1 normal matrices
Let $A\in\{0,1\}^{n\times n}$ be a $n\times n $ matrix with entries in the discrete set $\{0,1\}$.
My question: What is the number of matrices in $\{0,1\}^{n\times n}$ that are normal, that is, ...
- 2,712
15
votes
7
answers
6k
views
Binary matrices with constant row and column sums
My question is about $m \times n$ binary matrices (aka $\{0,1\}$-matrices), whose rows all sum to the same value, and whose columns all sum to the same value (but these two values may be different).
...
- 878
15
votes
1
answer
1k
views
Free subgroups of $\mathrm{GL}(2,\mathbb{Z})$
Is there a bound $B$ such that every 2-generator subgroup
$G = \langle a, b \rangle \le {\rm GL}(2,\mathbb{Z})$
whose generators do not satisfy a relation of length $\leq B$ is free?
If it exists, ...
- 19.6k
15
votes
2
answers
851
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
15
votes
0
answers
446
views
The rank of a "triangle-free" matrix
This is a version of the question I asked recently, but the assumptions got now strengthened substantially.
Suppose that $A=(a_{ij})_{1\le i,j\le n}$ is a square matrix with all elements in $\{0,\...
- 23k
14
votes
5
answers
989
views
Eigenvalues of a matrix with entries involving combinatorics
Let $F(n, l, i, j)$ be the cardinality of the set
\begin{eqnarray*}
\{(k_1, \cdots, k_n)\in\mathbb{Z}^{\oplus n}|0\leq k_r\leq l-1\text{ for }1\leq r\leq n\text{, }k_1+\cdots+k_n=lj-i\}.
\end{eqnarray*...
- 383
14
votes
2
answers
873
views
"sinc'n determinant"
The function $\text{sinc}(x)=\frac{\sin x}x$ permeates mathematics and physics in several aspects, and it carries multiple presentations/formulations. My interest is to inject yet another one of such.
...
- 43.2k
14
votes
1
answer
295
views
Product of a Laver table and a Hadamard matrix has mostly 0 rows
I recently noticed (while playing around) that the product of a Laver matrix with a Hadamard matrix gives a very sparse matrix. In particular, all but logarithmically few rows are all zero. The ...
- 1,203
14
votes
1
answer
865
views
Enumeration of $0-1$ matrices with determinant $1$
Has the number $f(n)$ of $n \times n$, $0{-}1$ matrices whose determinant is $+1$
been enumerated?
E.g., for $n{=}2$, there are $f(2)=3$ such matrices:
$$
\left(
\begin{array}{cc}
1 & 0 \\
0 &...
- 151k
13
votes
2
answers
697
views
in search of a transformation between determinants
Motivated by this MO question. Consider the two matrices $A_n$ and $B_n$ with entries $\binom{2j}i$ and $\binom{n+1}{2j-i}$, respectively; for $1\leq i, \,j\leq n$.
I can show $\det A_n=\det B_n=2^{\...
- 43.2k
13
votes
1
answer
2k
views
Number of idempotent $n\times n$ matrices over $\mathbb{Z}/m\mathbb{Z}$?
Is there any known formula for the number of idempotent $n\times n$ matrices over $\mathbb{Z}_m:=\mathbb{Z}/m\mathbb{Z}$ ?
The number of idempotent matrices over a finite field is well-known and ...
user31416
13
votes
3
answers
746
views
Is there a row vector $x$ with integer entries such that no entry of $xM$ is $0 \text{ (mod }p\text{)}$?
Let $p$ be a prime and let $M$ be an $n \times m$ matrix with integer entries such that $M\vec{v} \not\equiv \vec{0} \text{ (mod }p\text{)}$ for any column vector $\vec{v} \neq \vec{0}$ whose entries ...
- 305
13
votes
2
answers
913
views
Almost Hadamard matrices
As well-known, a Hadamard matrix is a square matrix with all coefficients $\pm 1$
and pairwise orthogonal rows or columns. Such matrices exist conjecturally
in every dimension divisible by $4$. Call ...
- 17.9k
13
votes
0
answers
188
views
Why is $ULU=NU$ (a refinement of $|N|=q^{n^2-n}$)?
Let $G=GL_n(\mathbb{F}_q)$, $U$, $L$, $N$ the subsets of upper-triangular unipotent, lower-triangular unipotent, all unipotent matrices respectively. Then $ULU=NU$ means that for any $g\in G$ the ...
- 3,772
13
votes
0
answers
1k
views
Pointwise (Hadamard) matrix product and the rank
$\DeclareMathOperator{\rk}{rk}$
Suppose that $A$ is a square matrix of order $n$. If, for any polynomials $P$ and $Q$ with $\deg P+\deg Q\le 2$, we have
$$ P(A)\circ Q(A^t) = P(1)Q(1)\, I_n \tag{$\...
- 23k
13
votes
0
answers
713
views
Regular languages of matrices and their generating functions
My question is somewhat related to this question.
Let us fix natural numbers $k$ and $C$. Let $A$ be an automaton whose alphabet consists of $k\times k$ matrices with integer coefficients of ...
- 3,897
12
votes
6
answers
692
views
Invertibility of a certain matrix indexed by the Hamming cube
For reasons which the margin of this page is too small to hold, I have been reading parts of a recent paper by O. Selim
On submeasures on Boolean algebras, arXiv 1212.6822v3
and in Section 7 the ...
- 25.8k
12
votes
1
answer
290
views
Largest subset of $GL_n(p)$ in which pairwise subtraction is also in $GL_n(p)$
Suppose $X\subset \mathrm{GL}_n(p)$ is a set of invertible matrices such that for every $A,B\in X$ then also $A-B\in \mathrm{GL}_n(p)\cup \{0\}$. (If anyone knows a name for such sets I would be ...
- 407
12
votes
1
answer
553
views
A matrix identity related to Catalan numbers
Let $$C_n=\frac{1}{2n+1}\binom{2n+1}{n}$$ be a Catalan number. It is well-known that $$(\sum_{n\ge{0}}C_n x^n)^k=\sum_{n\ge{0}}C(n,k)x^n$$ with $$C(n,k)=\frac{k}{2n+k}\binom{2n+k}{n}.$$
It is also ...
- 5,622
12
votes
1
answer
267
views
Total positivity of $q$-Pascal matrix?
A matrix of real numbers is called totally positive if all its minors are non-negative. A well-known example is the Pascal matrix $(\binom{i}{j})$.
Is it true that the minors of the $q$-Pascal matrix ...
- 5,622
12
votes
0
answers
321
views
Combinatorial proof of invertibility of a symmetric matrix associated to the ring of matrices over a finite field
Let $F$ be a finite field of $q$ elements with characteristic $p$. Let $M_n(F)$ be the ring of $n\times n$ matrices over $F$. We define a $q^{n^2}\times q^{n^2}$ symmetric matrix $L$ over the ...
- 38.6k
11
votes
4
answers
5k
views
Maximum determinant of $\{0,1\}$-valued $n\times n$-matrices
What's the maximum determinant of $\{0,1\}$ matrices in $M(n,\mathbb{R})$?
If there's no exact formula what are the nearest upper and lower bounds do you know?
- 111
11
votes
2
answers
388
views
Sum-regular $\{0,1\}$-matrices
Let $n\in\mathbb{N}$ be a positive integer. We say that an $n\times n$-matrix $A$ with all entries in $\{0,1\}$ is $k$-regular for some $k\in \{0,\ldots,n\}$ if the sum of every row and the sum of ...
- 48.9k
11
votes
2
answers
367
views
Kernel of a matrix and the Catalan numbers
Let $B_n$ denote the Boolean lattice of a set with $n \geq 2$ elements and $C_n$ the matrix with entries $c_{i,j}=1$ if $i \leq j$ and $c_{i,j}=0$ else, where $i,j\in B_n$.
Let $M_n:=C_n+C_n^T$ (this ...
- 26.5k
11
votes
3
answers
591
views
Non-singular matrix with restricted entries
Given a set $S$ of integers with $1 \not\in S$, let us consider the set $\mathcal{M}$ of all the symmetric matrices $M$, such that:
All the diagonal entries of $M$ are equal to $1$.
All the off-...
- 571
11
votes
1
answer
636
views
A simple proof for a theorem of Szekeres and Turán
Szekeres and Turán found in 1937 a formula for the sum of the squares and the sum of the fourth powers of determinants of all $n$ by $n$ matrices with $\pm 1$ entries. (The sum of squares case follows ...
- 24.7k
11
votes
1
answer
467
views
Correspondence between matrix multiplication and a graph operation of Lovász
In his book "Large networks and graph limits", Lovász describes a multiplication operation (he calls it concatenation) on "bi-labeled graphs". An $(m,n)$ bi-labeled graph is a ...
- 2,339
11
votes
0
answers
253
views
Poset of nonvanishing minors of a matrix
This question was posed on MSE here three days ago, but hasn't gotten any answers or suggestions. I hope it's okay to ask it on MO, but if I should wait a little longer, please just let me know.
Say $...
- 6,237