All Questions
Tagged with linear-algebra co.combinatorics
177 questions with no upvoted or accepted answers
1
vote
0
answers
413
views
Combinatorial Interpretation of an Extension of Gaussian Polynomials
It is well-known that the Gaussian polynomial (or Gaussian coefficient, q-binomial coefficient) $\binom{n}{k}_q$ counts the number of $k$-dimensional subspaces of an $n$-dimensional vector space over $...
- 443
1
vote
0
answers
229
views
Counting equivalence classes in the transitive closure of two equivalence relations
Let $X$ be a finite set, and let $P_i$ and $Q_j$ be two partitions of $X$:
$$\bigsqcup_i P_i = \bigsqcup_j Q_j = X.$$
The finest partition which is nevertheless coarser than both $P$ and $Q$ is ...
- 5,898
1
vote
0
answers
165
views
Affine space partition of a general set
Olof Heden, in his work "A survey of the different types of vector space
partitions", discusses various results regarding the following qustion - given a vector space $V$ over a finite field, how can ...
- 119
0
votes
0
answers
121
views
Closed form of coefficients of a finite field polynomial
I want to find a valid polynomial for a finite field $\mathbb{Z}_p[x]_{f(x)}$ with $d=deg(f(x))$. For this definition to hold, it can be deduced that $p$ must be prime and the polynomial $f(x)$ ...
- 47
0
votes
0
answers
163
views
Generalization of polynomial coefficients
I'm dealing with a hard combinatorial problem where for every positive integer value of a variable $n$ I have to calculate a list of numbers, specifically $n^2$, that depend on $n$ and its list index ...
- 47
0
votes
0
answers
133
views
On nilpotent singular $\mathbb F_2^{n\times n}$ matrices
Let $M$ be a $0/1$ matrix over $\mathbb F_2^{n\times n}$ with determinant $0$.
The set of such singular matrices form a semigroup.
The set of nilpotent matrices of size $n\times n$ form a semigroup.
...
- 13.9k
0
votes
0
answers
145
views
Bound on solutions of $Ax \ge b$
Let $A \in \mathbb{Z}^{m \times n}, b \in \mathbb{Z}^{m \times 1}$.
One can show that if there is a solution of $Ax \ge b, x \in \mathbb{R}^n$ then there is one such that $\|x\|_{\infty} \le c (\|A\|_{...
- 439
0
votes
0
answers
144
views
Given a binary parity check matrix, find a parity check matrix for the same code such that no row has weight greater than $k$
A parity check matrix for a binary linear code is a matrix $H$ (with linearly independent rows) such that the (right) kernel of $H$ is the codespace. Clearly we can replace any row $r_i$ by $r_i + \...
0
votes
0
answers
99
views
Invertible matrices with bounded nonnegative coefficients
I am teaching a class in linear algebra and I asked myself the following question: what is the chance to get an invertible matrix if I write a random one? My impulsive answer is "very likely"...
- 2,152
0
votes
0
answers
56
views
Determining the total number of nonzero expansion terms in a (0,1)-matrix
Let $A=(a_{ij})_{n\times n}$ be a $(0,1)$-matrix such that it contains equal number of $1$s in each row and column. Is there any general method to count the total number of the nonzero terms $\prod_{i=...
- 133
0
votes
0
answers
54
views
Rank decomposition of matrices over $\mathbb F_2$
Given an integer matrix $M\in\mathbb Z^{n\times n}$ of real rank $k$ what is the minimum and maximum number of rank $1$ matrices $B_1$ to $B_t$ we require so that $M\equiv\sum_{i=1}^tB_i\bmod 2$?
If $...
- 13.9k
0
votes
0
answers
299
views
Question on rank of matrices over $\mathbb F_2$
$A$ is a square matrix in $\mathbb F_2^{n\times n}$ of rank $k\leq n-1$.
$B$ is a square matrix in $\mathbb F_2^{n\times n}$ of rank $n$.
$T$ is a square matrix in $\mathbb F_2^{n\times n}$ of rank $1$...
- 13.9k
0
votes
0
answers
127
views
Maximizing a vector after a series of matrix multiplications
Problem Statement
Let's say we have a set of $n\times n$ matrices $X=\{M_1,\ldots,M_r\}$ and weights of these matrices $\{w_1,\ldots,w_r\}$ along with a set of "initial vectors" $\{v_1,\...
- 509
0
votes
0
answers
30
views
Signs of difference matrices (sum of submatrices)
Given matrix $A \in \mathbb{R}^{m \times n}$, are there any results related to its difference array
$$A^* \triangleq \left[sign(a_{i,j} + a_{r, s} - a_{r, j} - a_{i, s})\right]_{i<r, j<s}?$$
Or ...
- 75
0
votes
0
answers
45
views
On full rank submatrices of a construction
Take two matrices $T_1$ and $T_2$ in $\mathbb Z^{n\times n}$ with entries uniformly in $[-b,b]\cap\mathbb Z$ at some $b>0$. The matrices will be of rank $n$ each with probability at least $1-\frac1{...
- 1,826
0
votes
0
answers
81
views
Constructing set with maximal independent subset
What is the minimal $m$ such that there exists a set $A = \{a_1,...a_n\}$ of vectors : $a_i \in \{0,1\}^m$ ($n$ is given) such that every subset of vectors of size $k$ is independent, but only with ...
- 51
0
votes
0
answers
84
views
A permutation statistic and determinantal identity
I'm trying to read this paper, Total positivity, Grassmannians and networks by Postnikov (https://arxiv.org/abs/math/0609764) and I'm stuck on Lemma 5.1, which is essentially an identity about maximal ...
- 1
0
votes
0
answers
67
views
Singular values and the chromatic number
What relation, if any, is there between the singular values of the adjacency matrix ( or possibly incidence matrix) of a simple graph and its chromatic number. Typically, do we have Hoffmann type, or ...
- 2,089
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
0
votes
0
answers
185
views
Sum of unit vectors always has a binary span after constrained permutations
Conjecture:
Let $e_1 = (1,0,\ldots,0), \ldots , e_{m_1+m_2} = (0,\ldots,0,1)$ be the unit vectors of the standard basis $E$ of $\mathbb{R}^{m_1+m_2}$.
An enumeration $ E \cup -E = \{f_1, \ldots, ...
- 121
0
votes
0
answers
89
views
Decomposing matrices to lower ranks
Given real matrix $M\in\{0,1\}^{n\times n}$ of rank $r$. How many $M_i\in\{0,1\}^{n\times n}$ of rank $s$ does one need to write $M'=\sum_{i=1}^ta_iM_i$ for some $a_i\in\Bbb R$ where maximum absolute ...
- 13.9k
0
votes
0
answers
131
views
The largest size of a boolean subgraph (a hypercube) of a given graph
Let $G(\mathbb{F}_2^n)$ denote the graph that represents the lattice of all subspaces of $\mathbb{F}_2^n$ (also called a Hasse diagram). I am interested in knowing if there exists a large hypercube ...
- 313
0
votes
0
answers
266
views
Finding the effective maximum number of subspaces in a finite dimensional vector space
Hi mathoverflow community, may be some one may give me a hint on the following problem before I spend much time on brute force search.
For $q$ a prime number and $n=6$, let $\mathbb {F}_{q}^{n}$ be ...
-1
votes
1
answer
825
views
How to calculate determinants of such types?
Consider next determinant that we want to expand around $h=1$
\begin{eqnarray}
Z_q \ = \ h^{N N_f} \ \ \left ( \prod_{n=1}^{N} \ \sum_{l_n=0}^{N_f -q} \ h^{2l_n+q} \ \binom{N_f}{l_n} \right ) \ \...
-1
votes
1
answer
65
views
A follow-up question in a proof in a paper on complete multipartite graphs
A follow-up question from the following article/paper:
"Proof of a conjecture on distance energy change of complete multipartite graph due to edge deletion"
by Shaowei Sun and Kinkar Chandra ...
- 199
-1
votes
1
answer
200
views
Collecting terms of a linear expression with nested sums and combinatorics in coefficients
I need to collect the $\Pr(\cdot)$ terms of the following expression:
$\sum_{m=3}^{n}\frac{g_{m}\left( \cdot \right) }{\left( \sqrt{\theta \left(
1-\theta \right) }\right) ^{m}}\left[ \sum_{j=2}^{m-1}...
-1
votes
1
answer
383
views
On the permanent dominance conjecture for symmetric group
The Lieb's permanent dominance conjecture states that the expression $$\frac{d_{\chi}^HA}{\chi(e)}\le per(A)$$ holds for all positive semidefinite matrices $A$, where $d_{\chi}^HA=\sum_\limits{\sigma\...
- 2,089