2
votes
0answers
44 views

What do you call the following operations on a symmetric matrix?

Suppose we have a symmetric matrix of the following form, where the diagonal is always zero: \begin{array}{cccc} 0 & 1 & 1 & 0\\ 1 & 0 & 1 & 1\\ 1 & 1 & 0 & 0\\ 0 ...
1
vote
1answer
26 views

permutations with a given condition!

What will be the number of permutations of n different things, taken r at a time,when p particular things is to be always included in each arrangement? I know the answer to this question but could not ...
3
votes
1answer
45 views

If matrix $A$ is invertible, then there is a permutation of its rows leaving no-zeros on the diagonal

I need to prove this statement: "If $A$ invertible, then exist a permutation of its rows leaving no-zeros on the diagonal" and I tried using the definitos of invertible matrices and $LU$ ...
1
vote
1answer
33 views

Permutations expressed as product of transpositions

There is a theorem that states that all permutations can be expressed as a product of transpositions. I have a couple of questions about this theorem: Does the product which is equal to the ...
0
votes
1answer
34 views

Prove that $sgn(\sigma_1 \circ \sigma_2) = sgn(\sigma_1)sgn(\sigma_2)$

Lete $n\in \mathbb{N}$. Show that the transformation $$sgn: S_n \rightarrow \{\pm 1\}$$ (where $S_n$ is the set of all permutations of the integers in the set $\{1,...,n\}$),given by $\sigma \mapsto ...
2
votes
2answers
42 views

Proof that $\det(A)=\det(A^T)$ using permutations.

I'm reading a proof for the identity $\det(A) = \det(A^T)$ and I'm trying to udnerstand why the following rows are equivalent: $$\eqalign{ & \det ({A^T}) = \sum\limits_{\pi \in {S_n}} ...
0
votes
0answers
24 views

Fixed points and permutations.

Let $\psi ,\varphi \in {S_n}$ two permutations. Let $M$ a matrix such that $a_{i,j}=1$ iff $i=\sigma(j)$ where $\sigma \in S_n$ ($0$, otherwise) I already showed that $tr(M) = \left| {\left\{ {k \in ...
3
votes
0answers
24 views

(Counting problem) more challenging Modular N algebraic eqs - for combinatorics-permutation experts

Experts in algebra please help - Part II after Part I: we would like to know the number of solutions for this set of six of modular N algebraic equations: $$ x_1 y_2 = x_2 y_1 \pmod N \qquad (1) \\ ...
0
votes
1answer
82 views

List all the permutations of {1,2,3,4}. Which are even, and which are odd?

The answer is: There are 24 permutations. The 12 even permutations are: id , (1 2 3 4) , (1 3 2 4) , (1 4 2 3) , (1 2 3) , (1 2 4) , (1 3 2) , (1 3 4) , (1 4 2) , (1 4 3) , (2 3 4) , (2 4 3). The ...
2
votes
2answers
101 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, ...
1
vote
4answers
143 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 ...
2
votes
1answer
39 views

How to convert a permutation to permutation polynomial?

Let Fq be the finite field with q elements, where q is a prime power. A permutation on Fq is a bijection from Fq to itself. Let Fq[x] be the ring of polynomials in a single indeterminate x over Fq. A ...
4
votes
0answers
93 views

Wikipedia's Cayley Table and Pictures for 3 by 3 Permutation Matrices

Are there any explanations or clarifications of the pictures at https://en.wikipedia.org/wiki/Permutation_matrix#Permutation_of_rows_and_columns? ...
1
vote
1answer
49 views

A canonical form for this equivalence relation on matrices

This question is inspired by http://cs.stackexchange.com/q/19250/755. Define the equivalence relation $\sim$ as follows: If $M,N$ are two $8\times 8$ (or $n\times n$ if you prefer generality) $(0,1)$ ...
1
vote
1answer
26 views

transformation of DFT matrix

$\mathbf{F}$ is a unitary DFT matrix where the $(m,n)$-th entry of $\mathbf{F}$ is given by $\frac{1}{\sqrt{M}}e^{-\imath2\pi(m-1)(n-1)/M}$. Note that $\imath=\sqrt{-1}$. Let $\mathbf{A}$ be a matrix ...
5
votes
1answer
56 views

possible determinants of permutations

this is taken from Gilbert Strang's Linear Algebra book: What are all the possible $4\times4$ determinants of $I + P_{even}$? (P - permutation matrix) I seem to be stuck on this question except for ...
-1
votes
1answer
37 views

permutation of the elements of a matrix with respect to sign?

Let matrix $\mathcal A$ with a $(m\times n)$. Every element $a_{i,j}$ of the matrix is either a positive or negative integer, or zero. Question: How many distinct matrices could be generated with ...
3
votes
2answers
124 views

Set of permutation matrices

I'm stuck in this problem. Prove the set $P$ of $n×n$ permutation matrices spans a subspace of dimension $(n−1)^2+1$
0
votes
1answer
52 views

Inequality related to Inner Product and Permutation Matrix

I think I've seen this from one of my undergraduate linear algebra textbook, but I cannot find it now. Is this true? If it is, how can we prove it? Given column vectors $\mathbf ...
0
votes
1answer
82 views

Circular permutation of 20 person and 1 host.

20 were invited for a party, find the no ways in which 20 persons and the host can be seated around a circular table such that two particular person be seated on either side of host. Solution : ...
1
vote
1answer
68 views

Whether solutions of a particular matrix equation are only the permutation matrices

Let $V \in R^{d \times d} $ be a real orthogonal matrix. Denote by $V \circ V$ the Hadamard product (elementwise product). I wish to show that if $V \left(V \circ V \right)^T V$ is "close" to $V ...
0
votes
1answer
50 views

decomposition of m-cycle in m-1 transpositions

I am searching for a proof. Every m-cycle $\sigma = (x_1 x_2 ... x_m)$ can be expressed as an composition of m-1 transpositions. I found many formulas, for example: $\sigma = (x_1 x_2)(x_2 x_3) ... ...
2
votes
1answer
65 views

Are these two permutation matrices equivalent?

Definition: The permutation matrix $P_{ij}$ is the identity matrix with rows $i$ and $j$ reversed. When left-multiplied with another matrix, it exchanges rows $i$ and $j$. Am I right in thinking that ...
0
votes
2answers
199 views

A step in finding the determinant of transpose of a matrix

The following question involves the permutation group, which I am horrible at handling. Any help will be greatly appreciated. Let $A$ be an $n \times n$ matrix with entries $(a_{ij})_{i = 1,2 \cdots, ...
-2
votes
1answer
59 views

What will be total number of solutions of $n_1a+n_2b+n_3c=n$?

What will be total number of solutions of $n_1a+n_2b+n_3c=n$? Here, $n_1,n_2,n_3,n$ are constants already provided in the question and $a,b,c$ are variables. What we have to do is find out the total ...
2
votes
1answer
38 views

Projectivities $\pi:\mathbb KP^1\rightarrow\mathbb KP^1$

I am a little bit confused conerning the following example of projectivities $\pi:\mathbb KP^1\rightarrow\mathbb KP^1$. On the affine part $\mathbb K\subseteq \mathbb KP^1$ they are exactly the ...
5
votes
1answer
92 views

Is it possible to reverse this sequence of permutations?

Let $ S = (a_1, a_2, ..., a_N) $ be a finite (arbitrarily long) sequence of elements, and let $p_1, p_2, ..., p_n $ be the first $n$ prime numbers, with $n \ge 3$. We apply a sequence of ...
2
votes
1answer
67 views

Expectation Involving Permutation Matrices

Let $L$ be $n\times n$ bidiagonal matrix such that its diagonal is all $1$ and its subdiagonal is all $-1$ (and zero elsewhere). Let $D$ be any diagonal matrix and $x,y$ be any $n$-dimensional column ...
4
votes
2answers
288 views

Nearest matrix in doubly stochastic matrix set

Suppose $\mathcal{D}_N$ denote an $N\times N$ doubly stochastic matrix, given any element $M\in \mathcal{D}_N$ , the singular value decomposition for $M$ is $$ M=USV'$$ where $U$ and $V$ are two ...
1
vote
1answer
191 views

Any permutation in symmetric group n can be rewritten as a composition of transpositions

I just want to show that a permutation can be written as a composition of transpositions. I cannot use cycles.
2
votes
0answers
31 views

order of elements in a partition using Maple

I determined this whole partition but I just want to have the finer the partition for example: I have this ...
1
vote
2answers
154 views

compositions of permutations

For compositions of permutations on a set $X = \{1,2,3\}$, my lecture notes say that the composition $\phi_2 \phi_1$ is the permutation $\phi_1$ followed by the permutation $\phi_2$. So consider the ...
2
votes
3answers
304 views

Permutation as Linear Transformation

Could you help me please to move forward with the problem. I'm trying to show that a function $\varphi_{\sigma }: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ $\varphi_{\sigma }(x_{1}, x_{2}, ... ...
8
votes
2answers
705 views

Span of Permutation Matrices

The set $P$ of $n \times n$ permutation matrices spans a subspace of dimension $(n-1)^2+1$ within, say, the $n \times n$ complex matrices. Is there another description of this space? In particular, ...
3
votes
2answers
371 views

Inversion Problem

I have a problem that says: Find the number of inversions in each of the following permutations of S = {1,2,3,4,5}: (a) 52134 (b) 45213 (c) 42135 In the text it doesn't do that great of a ...
11
votes
2answers
463 views

How many $n\times m$ binary matrices are there, up to row and column permutations?

I'm interested in the number of binary matrices of a given size that are distinct with regard to row and column permutations. If $\sim$ is the equivalence relation on $n\times m$ binary matrices such ...