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Questions tagged [skew-mapping]

For questions related to skew-symmetric (or antisymmetric or antimetric) matrix.

A skew-symmetric matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition:
$A \text{ skew symmetric }\iff A^\text T =-A$

In terms of the entries of the matrix, if $a_{ij}$ denotes the entry in the $i$-th row and $j$-th column:
$A \text{ skew symmetric }\iff a_{ji}=-a_{ij}$.

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The normal form of a skew symmetric matrix

In Werner Greub's book Linear Algebra, 4th ed. on p. 230, he gives this proof of the normal form for a skew transformation on a finite-dimensional real inner product space. (Note Greub's convention for the matrix of a transformation is the transpose…
blargoner
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Determine wether a skew-symmetric tensor is decomposable or not.

For $x_1,x_2,x_3,x_4$ elements of a vector space $V$, I would like to find out wheter the skew-symmetric tensor $11x_1 \wedge x_2 + 10x_1 \wedge x_3 + 17x_1 \wedge x_4 − 13x_2 \wedge x_3 − 10x_2 \wedge x_4 + 11 x_3 \wedge x_4$ of $\bigwedge^2V$ is…
Rhaena
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ergodicity of irrational rotation of product

I need some help with the following. Suppose $(X,\mathcal{B},\mu,R)$ is an ergodic measure preserving dynamical system. Consider the torus $Y=\mathbb{R}/\mathbb{Z}\times\mathbb{R}/\mathbb{Z}$ and the map $S:Y\to Y$ given by $Sy=y+c$ mod $1$ where…
Mr Martingale
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Showing measure is invariant/ergodic for a skew product.

Let $(X,\mathcal{B},\mu,T)$ be a measure preserving dynamical system for which the probability measure $\mu$ is ergodic. Suppose we have a compact group $G$ with Haar measure $\nu$ and a measurable function $h:X\to G$. Consider the skew product…
Mr Martingale
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Skew-symmetric dimension

I'm asked what is the smallest possible dimension of a vector space with a non-degenerative skew-symmetric bi-linear form. I realized it could not be size one because then $\delta_{ij}=-\delta_{ij}$, since, taking the skew-symmetric bi-linear form…
arturo velasquez
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How to prove that every real skew-symmetric matrix is congruent to a block diagonal matrix by using bilinear forms?

For any real $n \times n$ skew-symmetric matrix $A,$ show there exists an orthogonal matrix $P$ such that $$PAP^T = \begin{pmatrix}{} \Lambda_1 & \\ & \Lambda_2 & \\ & & \ddots & \\ & & & \Lambda_\frac{n}{2} \end{pmatrix},$$ where $\Lambda_i =…
Abdul Miah
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What is the geometric interpretation of the action of a degenerate real skew-symmetric matrix?

As discussed in What is the geometrical action of a skew-symmetric matrix on an arbitrary vector?, an arbitrary real skew-symmetric matrix can be brought into the block diagonal form $$\begin{bmatrix} \begin{matrix}0 & \lambda_1\\ -\lambda_1 &…
tparker
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Projecting 8D dataset on onto skew coordinates

I have an 8 dimensional dataset (as an Nx8 matrix), and I am hypothesising that much of the dataset can be described simply by linear addition of two known non-orthogonal vectors (i.e. two 1x8 vectors). As a result I want to project the data onto…
AlexLipp
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Where does this proof of ergodicity fail?

I am a bit confused and was hoping the folk of MSE would help (and point out where the error is made). Suppose $(X,\mathcal{B},\mu,R)$ and $(Y,\mathcal{C},\nu,S)$ are ergodic measure preserving dynamical systems. Let $m=\mu\otimes \nu$ and define…
Mr Martingale
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Can any linear map, $\phi$, can be expressed a as $\phi = S + N - N^T,$ where $S$ is a self-adjoint and $N$ is a nilpotent map

Inspired from this and this , I claim that Any linear map, $\phi$ on a real vector space $E$, can be written as a $$\phi = S + N - N^T,$$ where $S$ is a self-adjoint and $N$ is a nilpotent map ($N^T$ denotes the adjoint of $N$, and its matrix…
Our
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Prove that the characteristic polynmial of a skew mapping satisfies $c(-\lambda) = (-1)^n c(\lambda)$

In the book of Linear Algebra by Greub at page 233, it is asked that Prove that the characteristic polynomial of a skew mapping satisfies the equation $$c(-\lambda) = (-1)^n c(\lambda).$$ I have tried showing this directly from the definition of…
Our
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Prove that the third partial derivative over the space is a formally skew-adjoint operator in $\mathbb{R}$

I'm trying to solve the following question: Prove that $\mathcal{J} = \frac{\partial^3}{\partial z^3}$ is formally skew-adjoint on the space $C^ \infty(I, \mathbb{R}) $ of smooth real-valued functions on $I=[a,b]\subset\mathbb{R}$. I'm trying to…
olenscki
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Closed form for a multilinear skew-symmetric map applied in sums?

Suppose $\varepsilon$ is a multilinear skew-symmetric function of its arguments with vales in $\mathbb R$. Is it true that: $$\varepsilon(a_1+\theta_1, \ldots, a_p+\theta_p)=\varepsilon(a_1, \ldots, a_p)+\\ +(-1)^p\sum_{i=1}^{p-1}\sum_{1\leq…
PtF
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Distance between varying skew lines

Let $s \equiv$ $x=t, y=\left(\frac{t}{3}\right)+2$ and $z=\left(\frac{bt}{3}\right)+1$ $$$$ $r \equiv x=2bu-4b+1, y=u, z=2u-4$ Question: determine if there exists any value(s) of $b$ s.t. $r$ and $s$ are skew. If such value(s) exist then find the…
B2K
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how is the rank of a skew map is even ?

In the book of Linear Algebra by Greub, at page 230, it is claimed that More general, it will now be shown that the rank of a skew transformation is always even. Since every skew mapping is normal (see sec. S.5) the image space is the…
Our
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