0
votes
1answer
37 views

Number of ways two matrices can be multiplied?

Given the dimensions of two matrices what are the different ways they can be multiplied? Example A[2][2] and B[2][2] then answer is 2. Let the dimensions of first matrix be n x m and second be m x p
4
votes
1answer
55 views

Order of group $GL_{2}\left( \mathbb{F}_{p}\right) $

I'm having a hard time counting. I need to count the number of elements for the multiplicative group of invertible $2\times 2$ matrices $GL_{2}\left( \mathbb{F}_{p}\right) $ with elements from the ...
1
vote
2answers
46 views

How to show the identity relating to Matrix

Suppose that $$ A=\begin{bmatrix}a_{11}&a_{21}\\a_{21}&a_{22}\end{bmatrix}, \ \ B=\begin{bmatrix}d&-1\\1&0\end{bmatrix}. $$ and $$A=B^N$$ Show that $$a_{11}=\sum_{i=0}^{[N/2]}(-1)^i ...
0
votes
1answer
42 views

2D boolean matrix number of unique combinations without mirrored/rotated ones

Given a $n \times n$ boolean matrix, it's well known that number of all possible combinations of 0s and 1s in that matrix would be $2^{n^2}$, as there are $n^2$ places which could take exactly 2 ...
0
votes
1answer
39 views

What is the minimum number of sign patterns in $\frac n2$ of columns (or rows) of Hadamard matrices?

Given a Hadamard matrix of size $n$, I want to know what is the minimum number of unique sign patterns in any $\frac n2$ columns (or rows). I count a sign pattern and its negation to be the same. My ...
2
votes
0answers
31 views

Counting symmetric binary matrices with constant line-sum

I'm trying to count, as the title suggests, symmetric matrices with entries $0$ or $1$ and constant line-sum $k$. ($0 \leq k \leq n$). If you start listing the number of these on a table you'll get a ...
0
votes
0answers
37 views

show by using leibniz formula

There are given $ r, s,n \in\mathbb N$ and $r+s=n$. It also given $A \in M_{r,K} $, $B \in M_{r\times s,K} $ and $C \in M_{s,K} $. Let $M$ be the matrix $\begin{bmatrix}A & B\\0 & ...
1
vote
1answer
48 views

Meaning of the characteristic polynomial of a matroid

From wikipedia The characteristic polynomial of a matroid $M$ (which is sometimes called the chromatic polynomial,[29] although it does not count colorings), is defined to be $$ p_M(\lambda) ...
0
votes
1answer
55 views

How many nonnegative integer matrices of size $N$ have all row and column sums equal to $D$?

Given the positive integer $N$ and $D$, generate all the non-negative integer matrices which satisfy matrix dimension is $N\times N$; sum of each row elements equals to $D$ sum of each column ...
4
votes
1answer
65 views

Rearrange columns and rows of a matrix such that it can be split in half

I am not a mathematician, so please excuse me if this question turns out to be trivial. I need this at work, but I could not figure out how to solve this efficiently, though it looks like it might be ...
2
votes
0answers
198 views

How to distribute 5-digit numbers in 5x5 matrices

I have 98000 5-digit numbers, from 00001 to 98000. I need to distribute these 98000 numbers in 14000 5x5 matrices. A matrix cell must contain only a digit from 0 to 9. Each matrix must receive 7 ...
2
votes
2answers
47 views

Sum of $\mathbf{x}\mathbf{x}'\mathbf{x}\mathbf{x}'$ when $\mathbf{x}$ is a binary vector of length $n$

Let's say $\mathbf{x}$ is an $n\times 1$ column vector where each entry can be either 1 or -1. There can be $2^n$ possibilities for this vector. Let $\mathbf{x}'$ be a transpose of $\mathbf{x}$. I ...
0
votes
1answer
31 views

Transform a symmetric matrix to a vector and multiplicate

Assume that we have a matrix symmetric matrix $A = \mathbb{R}^{nxn}$. It es enough to look only at the lower triangle. What I want to do is to transform the matrix into a vector: \begin{pmatrix} 1 ...
0
votes
0answers
42 views

Number of combinations in a matrix

Given the size of a matrix is $N \times N$, how many unique matrices are there given the following restrictions: Matrix entries can only contain numbers $\left[0,b\right]$ A valid matrix cannot have ...
1
vote
1answer
24 views

Determinants of matrices with constrained entries

Let $A \in GL(n,\mathbb{Z})$, written $A = (a_{i,j})$. Define the Height of $A$ to be $\max_{i,j} |a_{i,j}|$. The Laplace expansion of $\det(A)$ clearly implies that if $A$ has height $N \in ...
3
votes
0answers
78 views

A problem on 0-1 matrices.

Given a 0-1 matrix $A$, is there an efficient way to find all 0-1 vectors $x$ such that $Ax = v$ where the entries of $v$ belong to a set $\{a,b\} \subseteq \mathbb{Z}$ of size $2$? Note that $v$ is ...
0
votes
0answers
24 views

A p-Sylow-subgroup of the group $GL(n, \mathbb{F}_p)$ for prime $p$ and $n\geq 2$

I would like some help to handle the following matter: I have to find a p-Sylow-subgroup of the group $GL(n, \mathbb{F}_p)$ for prime $p$ and $n\geq 2$. My own tries I guess I have to know how ...
-1
votes
1answer
55 views

chosing between matrix theory and combinatroics

I have to take one more math course to finish my math minor , i am a computer science major and i want to know which course will benefit me more matrix theory or combinatorics and which takes more ...
0
votes
0answers
28 views

The lower bound of Cheeger Inequality as the degree goes to infinity

Consider an undirected graph $G(V,E)$ with adjacency matrix $A$ and define the graph Laplacian as \begin{equation} L = D - A \end{equation} where $D$ is a diagonal matrix such that $D(i,i) = d_i$. ...
0
votes
1answer
34 views

Construct matrix of ones and zeros based on sequences

We are given ($a_1,a_2,....,a_m$) and ($b_1, b_2,....,b_n$) sequences with non-negative integers. Decide whether it's possible and if it is construct a matrix $\Re^{m x n}$ of ones("1") and ...
2
votes
1answer
59 views

Proving that if $n\times n$ Hadamard matrix exists, then 4 divides $n$

Im looking for an explanation of the following: a standard way to prove that, if there exists Hadamard matrix of dimension $n > 2$, then $4|n$, is to suppose that without loss of generality every ...
2
votes
1answer
42 views

How do you handle this kind of probability?

What is the probability of selecting a singular matrix from $\Bbb{R}^{3\times 3}$? I calculated it to be zero based on their being approximately $9$ degrees of freedom to choose entries of $A$ such ...
1
vote
1answer
71 views

counting Number of matrices

We have a $2 \times 2$ matrix. We are given the trace of the matrix as $N$. Also, all elements of the matrix are greater than or equal to $1$. And, the determinant of matrix is $\geq 1$. QUESTIONS: ...
2
votes
2answers
119 views

Powers of permutation matrices.

Let $P$ be a permutation matrix obtained by the identity matrix by switching 2 rows $n$ times, (with no two rows switched more than one time). How to show that $$P^{\ n+1} = I$$? Is it true that, ...
0
votes
2answers
61 views

How many skew symmetric matrices are possible?

I just heard the term skew symmetric matrix and upon discovering what it was, I thought to myself, "Jeez, there could only be so many of those." I'm not good with the whole permutation thing and this ...
1
vote
1answer
83 views

On Adjacency Matrix of a Graph with a Cut Vertex and a Bridge

Let $G$ be a graph. If $v_i$ (resp. $v_iv_j$) is a cut vertex (resp. a bridge) of $G$, what can you say about its adjacency matrix $A(G)$?
2
votes
1answer
179 views

Number of binary n x m matrices, with at most k consecutive number of 1 in each column

I am trying to compute the number of $n x m$ binary matrices with at most $k$ consecutive values of $1$ in each column. I've figured out that I it will be enough to find the vectors with $1$ column ...
1
vote
0answers
61 views

Prove that the minimum of row sums of a nonnegative symmetric matrix is preserved

Let $A$ be an $n\times n$ adjacency (nonnegative, irreducible and symmetric) matrix with zeros on the diagonal. Denote $i$-th row sum of $A^k$ as $r^{(k)}_i$, where $k\geq1$. I want to prove that if ...
1
vote
1answer
25 views

Information content of an unlabelled matrix

I'm trying to get an idea of the amount of information that is "stored" in an "unlabelled" matrix. I assume that the vector $(x,y,z)$ contains more information than the set $\{x,y,z\}$. But purposely ...
7
votes
1answer
291 views

Number of binary $M\times N$ matrices with even row sums, even col sums and $K$ ones, $K$ even

A combinatorial problem arising with certain checksums: When sending messages, the user data are protected by adding a parity bit for bit positions $1\dots8$ and a parity bit for each byte. So, the ...
1
vote
4answers
165 views

How to tell if two matrices are equal up to a permutation

Given two real rectangular matrices A, B how can I tell if they are equal up to a permutation of their rows/column without trying all possible permutations? (This is closely related to the question I ...
0
votes
0answers
56 views

Summing the product of combinations of matrix elements

I have a situation where I have an $NxN$ matrix $A$ where each element $a_{i,j}\in\mathbb{R}_{\leq 0}$. I would like to consider the set of all collections of elements such that each collection of $N$ ...
6
votes
3answers
344 views

Number of matrices with no repeated columns or rows

If you consider all $10$ by $15$ matrices with entries that are either $0$ or $1$, there are ${2^{15} \choose 10}$ with no repeated rows (up to row permutation) and ${2^{10} \choose 15}$ with no ...
1
vote
2answers
40 views

combinatorics - permutations question, possibly with pigeon hole

Let $A \in Mat_n(\mathbb R)$ such that $\forall i,j: a_{ij}\geq 0$ We are given: $$\forall j: \sum_{i=1}^n a_{ij}=\sum_{i=1}^n a_{ji}=1$$ show there's a permutation $\pi \in S_n$ such that $$\forall ...
0
votes
0answers
36 views

Can one find arbitrarily large subsets of a vector space of dimension n, such that any subset of n vectors is a basis?

I thought of this mildly interesting question earlier this evening: Given a vector space $V$ of dimension $n$, for what values of $m > n$ is it possible to create a set $S$ of $m$ vectors such ...
0
votes
1answer
73 views

number of ways to fill a 2D grid

We have a 2D grid with n rows and m columns, we can fill it with numbers between 1 and k (both inclusive). Only condition is that for each r such that 1<=r<=k ,no two rows must have exactly the ...
3
votes
3answers
111 views

Find three $10\times10$ orthogonal Latin squares.

Does anyone know if there is a mathematical "trick" in finding mutually orthogonal Latin squares? Or is it basically trial and error?
0
votes
1answer
48 views

Inverse of particular matrix

I have a $ n \times n $ matrix with the following form: $$ \begin{pmatrix} a^n+b^n & C_{n,1} \; a b^{n-1} & \cdots & C_{n,n-1} \; a^{n-1} b \\ C_{n,n-1} \; a^{n-1} b & a^n + b^n & ...
1
vote
3answers
116 views

Generalization of permutation matrix

For integers $n$ and $k$, I am interested in $n\times n$ matrices with exactly $k$ non-zero entries in each row and each column. The case $k=1$ corresponds to (generalized) permutation matrices. In ...
3
votes
4answers
193 views

Coefficient of $x^n$ in the series

How will we find the coefficient of $x^n$ in the following series: $$(1+x+2x^2+3x^3+...)^n$$ Please suggest if there is some formula or if it can be computed using the computer in $\log n$ time. I ...
5
votes
1answer
126 views

Question about matrices whose row and column sums are zero

I am interested in $n \times n$ matrices over some field $K$ all whose rows and all whose columns sum to zero. First question: do these matrices have a name? Pending an answer I will call these ...
10
votes
1answer
141 views

How many arrays with crossed cells, order of rows/columns irrelevant

I've been struggling with this simple problem for months though as I am a newbie to… well, maths, there's high chance someone more educated than myself may get it right! Let's consider an array or a ...
18
votes
1answer
182 views

How many pairs of nilpotent, commuting matrices are there in $M_n(\mathbb{F}_q)$?

As a follow-up to this question, I've been doing some work counting pairs of commuting, nilpotent, $n\times n$ matrices over $\mathbb{F}_q$. So far, I believe that for $n=2$, there are $q^3+q^2-q$ ...
10
votes
1answer
121 views

How many matrices in $M_n(\mathbb{F}_q)$ are nilpotent?

I have strong computational evidence to think that the answer is $q^{n(n-1)}$, although a proof eludes me. Any ideas?
1
vote
3answers
180 views

What is the *correct* (matrix) square-root of $A_2=\begin{bmatrix} 0&-1 \\ 1& 2 \end{bmatrix} $?

In studying the problem of some trivial(?) generalization of the NSW-numbers [ OEIS,wikipedia ] (see my other related question) there came up one detail where I think I have the correct answer but ...
1
vote
1answer
164 views

Uniqueness of doubly stochastic matrix descomposition

this is my first question in the site. Thanks in advance for all answers. It is well known that each bistochastic matrix can be represented as a convex combination of permutation matrices. I am ...
4
votes
1answer
104 views

Periodic (-1,0,1) matrices of two types

similar question: http://mathoverflow.net/questions/9547/how-to-construct-matrices-with-periodicity Definition: a (-1,0,1) matrix is a matrix with entries either -1, 0 or 1. I am trying to understand ...
2
votes
1answer
69 views

Using matrix theory to solve this problem

I'm sorry that I couldn't find a better title for this. I was wondering if my solution is valid for the following problem, or if I've made some mistake. Problem: Let $N=\{a_1, \dots, a_n\}$ be a ...
3
votes
2answers
208 views

Characterizing sums of permutation matrices

Given an $n$ by $n$ matrix $A$ whose rows and columns sum to $m \in \mathbb N$ and entries are nonnegative integers, does there exist a permutation matrix $P$ such that $A - P$ has only nonnegative ...
2
votes
1answer
92 views

How many Jordan normal forms are there when the characteristic polynomial is $(\lambda+4)^5(\lambda-2)^2$?

Let $A\in M_7(\mathbb{C})$ be a matrix in with the characteristic polynomial $p(A)=(\lambda+4)^5(\lambda-2)^2$. I need to find all Jordan normal forms for this. I think that i can use that the ...