Tagged Questions
5
votes
1answer
57 views
Is there a name for this given type of matrix?
Given a finite set of symbols, say $\Omega=\{1,\ldots,n\}$, is there a name for an $n\times m$ matrix $A$ such that every column of $A$ contains each elements of $\Omega$?
(The motivation for this ...
8
votes
3answers
157 views
Find a ternary $4\times 39$ matrix satisfying the conditions below
Can you find a matrix $A_{4\times39}$ with elements from $\{-1,0,1\}$ so that
No column is all zero.
All columns are different.
No column is $-1$ times another column.
Each row consists of $13$ of ...
3
votes
1answer
53 views
Counting 0-1 matrices up to symmetry
I'm interested in counting the number of n×n 0-1 matrices with a given number of 1s up to rotation and reflection. What is the best way to do this if n is not too small?
For example, consider ...
0
votes
0answers
15 views
How will an orthogonal array look for 3 levels and 3 factors?
I understand that an orthogonal array with 3 factors (parameters) will have 3 columns and if there are 3 levels, then it means each parameter can have 3 values.
However, when using a selector from ...
4
votes
1answer
65 views
Find a lower bound
Let $M$ be an $N\times N$ symmetric real matrix, and let $J$ be a permutation of the integers from 1 to $N$, with the following properties:
$J:\{1,...,N\}\rightarrow\{1,...,N\}$ is one-to-one.
$J$ ...
1
vote
1answer
161 views
How to deduce the psition mapping of entries of a matrix?
I would be thankful if any peer shed light on me.
Assume that the mapping of a set is unknown. By knowing n number of E element sets and the transformed sets with positioned elements, How can I ...
1
vote
2answers
42 views
Sum of the selected elements of matrix is $255$
A $5\times 10$ matrix is given:
$$\begin{pmatrix}
1 & 6 & 11 & 16 & 21 & 26 & 31 & 36 & 41 & 46\\
2 & 7 & 12 & 17 & 22 & 27 & 32 & 37 ...
0
votes
1answer
64 views
Prove $MM^t=A+kI$ for matrices associated to graphs
How can I prove that $MM^t=A+kI$ for incidence matrix $M$ and adjacency matrix $A$ of a $k$-regular graph with $n$ vertices?
It is easy to see that $MM^t$ is an $n\times n$-matrix (like $A$), ...
2
votes
1answer
28 views
Can an alternating sign matrix that is not a permutation matrix be non-singular?
An alternating sign matrix is a $n\times n$ matrix with entries in the set $\{-1,0,1\}$ such that for each row and column, the non-zero entries alternate between $1$ and $-1$, starting and ending with ...
5
votes
0answers
39 views
How many Matrices exist with increasing row and increasing column condition?
Given $N$, I would like to know the number of matrix constructed from $1$ to $N$ which satisfies the following condition:
1. The each row entries should be in increasing order.
2. The each column ...
1
vote
1answer
48 views
$m\times n$ matrix with an even number of 1s in each row and column
So I want to find the number of ways to fill an $m\times n$ matrix with only 0s and 1s such that each row and column has an even number of 1s. I'm pretty stumped here. I've set up m+n equations ...
1
vote
1answer
89 views
Finding all possible solutions to an $n \times n$ matrix, given row and column sums.
What I am ultimately attempting to do is find the solution that maximizes a given equation so I need to find all possible solutions so I can check them. I need all possible solutions to an $n \times ...
1
vote
1answer
37 views
Counting degrees of freedom
Why does a tensor with 3 indices in $n$ dimensions, such that if you swap any two of the indices gives the same value, has degree of freedom equal to $$n+2 \choose 3$$? I would have thought that it's ...
7
votes
1answer
110 views
Probability that a $3\times 3$ matrix with entries in $\{0,1,2,3\}$ is invertible.
Let $A$ be a $3\times 3$ matrix, and each of its entries takes value from $\{0, 1, 2, 3\}$ with probability $1/4$ for each value. What is the probability that A is invertible?
I have tried to list ...
0
votes
2answers
31 views
Invertible subblocks
I'm interested in invertible matrices that are built out of invertible sub-blocks. For example, four sub-blocks from $GL_n(F)$ (i.e. the group of $n \times n$ invertible matrices over a field $F$) ...
3
votes
5answers
104 views
Translating matrix fibonacci into c++ (how can we determine if a number is fibonacci?)
Is it possible to determine if a number is a fibonacci number in less than N time (where N is the Nth fibonacci number) using the matrix method? I'm trying to exclude external libraries like cmath or ...
1
vote
1answer
47 views
The lower bound on the number of numbers needed to fill a matrix in a special way.
Let's take any natural number $n>0$. Let $k$ be the smallest natural number greater than $n/2.$ Now let $A$ be any $n$-element set, and let $M$ be a $k\times k$ matrix over $A$. Suppose that for ...
0
votes
0answers
53 views
4
votes
3answers
172 views
What is the next number of this sequence?
Consider the sequence $ (a_{n})_{n \in \mathbb{N}} $ of positive integers whose first few entries are
$ 2 ~~ 6 ~~ 20 ~~ 70 ~~ 252 ~~ \ldots $
Now, consider the infinite matrix
...
6
votes
1answer
294 views
When does the adjacency or incidence matrix of a graph have consecutive ones property?
Given a graph, what are some sufficient (and necessary) conditions to tell if its adjacency matrix has the consecutive ones property?
Similar question for its incidence matrix?
Note that a ...
1
vote
1answer
110 views
interesting matrix
Let be $a(k,m),k,m\geq 0$ an infinite matrix then the set
$$T_k=\{(a(k,0),a(k,1),...,a(k,i),...),(a(k,0),a(k+1,1),...,a(k+i,i),...)\}$$is called angle of matrix
$a(k,0)$ is edge of $T_k$
...
0
votes
0answers
45 views
Abbreviations in Combinatorial Graph/Matrix theory
I'm getting started with research in combinatorics. I have come across a reference that uses a great deal of abbreviations. I was able to figure most of them out but there are a few that I can find.
...
0
votes
1answer
43 views
Counting commuting Pauli Strings of a certain weight
Let a $n-$length pauli string represent any tensor product of finitely many pauli matrices, Ex:
$X\otimes Z\otimes \mathbb{I}\otimes Y\otimes \mathbb{I}\otimes X\otimes\cdots\otimes Z$ where the ...
0
votes
2answers
47 views
Would cartesian product be the best approach for this
Not sure on how to migrate a question yet but over on SO someone said I might get better results here. Also please retag as I'm not allowed to create new and might not know the best tagging.
Link to ...
3
votes
0answers
158 views
Closed-form expression for sum of Vandermonde matrix elements
Given the Vandermonde matrix:
$$\begin{pmatrix}1^0 & 1^1 & 1^2 & ... & 1^n \\
2^0 & 2^1 & 2^2 & ... & 2^n \\
\vdots & \vdots & \vdots & \ddots & ...
2
votes
1answer
58 views
Commutants for matrices
Let ${\mathbb K}={\mathbb R}$ or $\mathbb C$. Let $V$ be a vector space over $\mathbb K$ and fix a basis $\cal B$ of $V$. We say that a family of vectors of $V$ is nice (relatively to $\cal B$) if
if ...
2
votes
0answers
117 views
The number of $(0,1)$ matrices with even row- and column sums.
Inspired by this question and some related ones here, I'd like to find the number of ways, $S(k, r, c)$, to place $k$ ones and $rc-k$ zeros in an $r\times c$ array so that the row sums and column sums ...
41
votes
3answers
736 views
Alice and Bob matrix problem.
Alice and Bob play the following game with an $n*n$ matrix, where $n$ is odd.
Alice fills in one of the entries of the matrix with a real number. then Bob fills one. Then Alice and so on so forth ...
2
votes
1answer
110 views
Number of zero entries in symmetric (0-1)-matrix with full diagonal
Let $S$ be an $n\times n$ symmetric matrix whose diagonal consists only of $1$s and whose other entries are either $0$ or $1$ .
If the determinant and rank of $S$ are known, what can be said about ...
4
votes
1answer
150 views
Permute the values in each row in a matrix such that the columns sum to the same amount.
The general problem
Given a matrix, I would like to permute the order of values in each row, so that all the columns of the matrix sums to the same value.
A simple example
For example, given:
...
3
votes
1answer
120 views
“Sorting” matrix entries by swapping rows and columns
Suppose one has an $n \times n$ matrix whose entries are filled with the numbers $1,2,\ldots,n^2$, but not in that order.
Q1:
What is the largest number of row or column swaps ever needed to end up ...
8
votes
1answer
206 views
The first column of the $n$th power for a triangular matrix
I have found a interesting thing but I cannot prove it. Given $k_i$ are positive for any $i\geq1$, and we have $M+1$ by $M+1$ matrix $A$, which is
$$
A=\left[\begin{array}{ccccc}
0\\
k_{1} & 0\\
...
2
votes
2answers
79 views
Counting the number of matrices which cause collision
Let $m,n \in \mathbb{N}$, and $q$ be a prime number.
Let $\mathbf{A} \in \mathbb{Z}^{m \times n}_q$ be a matrix. In the following, assume that all additions and multiplications are performed modulo ...
2
votes
1answer
173 views
Proving determinant product rule combinatorially
One of definitions of the determinant is:
$\det ({\mathbf C})
=\sum_{\lambda \in S_n} ({\operatorname {sgn} ({\lambda}) \prod_{k=1}^n C_{k \lambda ({k})}})$
I want to prove from this that
...
10
votes
1answer
635 views
4 by 4 Matrix Puzzle
I was solving the puzzle for the Company interview exam. I found this puzzle, I cannot come up with the solution. How to solve it and what is the correct answer?
Determine the number of $4\times ...
4
votes
0answers
133 views
What is the exponential generating function of the inverse matrix of an integer triangle?
Let ${Z}$ denote an integer triangle (like Pascal's), ${Z}_{n,k}$ for $0\leq k \leq n$ and let $f$ be an exponential generating function for polynomials $p_{n}(x)$ with $[x^k]p_{n}(x)= Z_{n,k}$.
...
0
votes
1answer
48 views
number of possible matrix entries swaps
Let us say that there is a $n \times n$ matrix with entries defined/given as some natural number. (duplicates are fine.)
What would be the number of possible matrices that result from ...
0
votes
0answers
46 views
cryptographyVs matrices
I need a good example(s) and justification to the following. If any one can help, I am so glad to them. I understand the original computation. However, I am little confused in writing coding and ...
3
votes
2answers
71 views
Counting matrices over $\mathbb{Z}/2\mathbb{Z}$ with conditions on rows and columns
I want to solve the following seemingly combinatorial problem, but I don't know where to start.
How many matrices in $\mathrm{Mat}_{M,N}(\mathbb{Z}_2)$ are there such that the sum of entries in each ...
1
vote
0answers
164 views
Counting question on permutation matrices with rotation and imprinting
Please read question of distinct permutation matrices with rotation at first, then new counting questions are below:
For a distinct $N\times N$ zero-symmetry permutation matrix, we could rotate it 3 ...
0
votes
0answers
103 views
Matrix & Partition & Natural Number & Pattern
I would like to know if someone know, how is called a matrix M*N, where m represents the row index in the matrix and the sum of the N columns at this row. Meaning that each row represents a possible ...
3
votes
0answers
129 views
How to invert this function on matrices which involves the permanent?
I'm interested in understanding whether a particular natural function on matrices, closely related to the permanent of a matrix, is invertible, and whether its inverse admits a simple closed form. The ...
3
votes
1answer
105 views
Existence of a binary symmetric matrix from row sums
Give a sequence of column/row sums, is it possible to determine (other than by brute force) whether there exists a binary, symmetric matrix with the same column/row sums?
I did find this article, ...
6
votes
1answer
117 views
Constructing Magic Squares over $\mathbb{Z}$ from Magic Squares over $\mathbb{Z}_m$
A magic square over $\mathbb{Z}$ is an n x n matrix whose entries are $\{1, \ldots, n^2\}$, with the sum of every row and column identical (in particular, my magic squares are all normal, but the sum ...
2
votes
1answer
60 views
Generalization of a Hadamard matrix
As a generalization of a Hadamard matrix, a c-matrix is a square matrix with one entry 0 and all other entries -1 or 1 in each row. With each row being pairwise orthogonal. Show that the transpose of ...
0
votes
1answer
78 views
Hadamard block matrix proof
$A$ is a Hadamard matrix of side $n$, and $H$ is a Hadamard matrix of side $m+n$, where $H=\pmatrix{A& B \\\ C& D}$ for some matrices $B,C,D$. Prove that $m \geq n$.
A Hadamard matrix is a ...
6
votes
2answers
174 views
Checkerboard coloring problem
Consider a $n\times n$ checkerboard. Each cell can be colored white or black. $n$ is even. How many configurations are there so that each row and each column have an odd number of white cells?
1
vote
1answer
404 views
Probability of full rank of a random matrix.
Suppose, $G$ is a $k \times n$ binary matrix with $\operatorname{rank}(G) = k$. The first $k$ columns of $G$ are linearly independent and the next $n-k$ columns are linear combinations of the first ...
2
votes
1answer
93 views
Hadamard Matrix
Prove that if $H$ is a (normalized) Hadamard matrix, then so is the matrix $\pmatrix{ H& H\\\ H& -H}$.
I have been working on this and I know this statement is true. My book just simply says ...
3
votes
0answers
59 views
Amount of matrices produced by mirroring
Given an $n \times n$ matrix whose entries are pairwise distinct, how many different matrices can you generate by:
exchanging columns $i$ and $n-i+1$
exchanging rows $i$ and $n-i+1$
mirroring the ...
