PlanetMath (more info)
 Math for the people, by the people. Sponsor PlanetMath
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
Owner confidence rating: Very high Entry average rating: No information on entry rating
strain transformation (Definition)

Let $E$ be a Euclidean plane. Fix a line $\ell$ in $E$ and a real number $r\ne 0$ . Take any point $p\in E$ . Drop a line $m_p$ from $p$ perpendicular to $\ell$ . Denote $d(p,\ell)$ the distance from $p$ to $\ell$ . Then there is a unique point $p'$ on $m_p$ such that $$d(p',\ell)=r\cdot d(p,\ell).$$ The function $s_r:E\to E$ such that $s_r(p)=p'$ is called a strain transformation, or simply a strain.

One can visualize a strain stretches a geometric figure if $|r|>1$ and compresses it if $|r|<1$ . If $r=1$ , then $s_r$ is the identity function, the only time when a strain is a rigid motion. For example, let $\ell$ be the $x$ -axis and $C$ be a circle in the upper half plane of the $x$ -$y$ plane. Then the following diagrams show how a strain transforms $C$ :

unit=1.5cm
\begin{pspicture} % latex2html id marker 91 (-4,-2)(5,3) \psline(-4,0)(4,0) \rpu... ...(C)$} \rput(1,-2){$s_{\frac{1}{2}}(C)$} \rput(3,-2){$s_{-1}(C)$} \end{pspicture}

Again, if $\ell$ is the $x$ -axis, then $s_r$ is the function that sends $(x,y)$ to $(x,ry)$ . Representing the ordered pairs as column vectors and $s_r$ as a matrix , we have

$s_r \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & r \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x \\ ry \end{pmatrix} .$

Nevertheless, a strain, as a (non-singular) linear transformation, takes lines to lines, and parallel lines to parallel lines.

In general, given any finite dimensional vector space $V$ over a field $k$ , a strain $s_r$ is a non-singular diagonalizable linear transformation on $V$ such that $s_r$ leaves a subspace $W$ of codimension $1$ fixed. $0\ne r\in k$ is called the strain coefficient.

Remark. By choosing an appropriate base for $V$ of dimension $n$ , $s_r$ can be represented as a diagonal matrix whose diagonals are $1$ in at least $n-1$ cells and $r$ in at most one cell.

It is easy to see that every non-singular diagonalizable linear transformation on $V$ can be written as a product of $n$ strains, where $n=\operatorname{dim}(V)$ .




"strain transformation" is owned by CWoo.
(view preamble | get metadata)

View style:

Other names:  strain
Also defines:  strain coefficient
Log in to rate this entry.
(view current ratings)

Cross-references: product, easy to see, cells, diagonals, diagonal matrix, dimension, base, codimension, subspace, diagonalizable, field, vector space, finite dimensional, parallel lines, linear transformation, non-singular, matrix, column vectors, ordered pairs, Transforms, diagrams, plane, upper half plane, circle, rigid motion, identity function, function, distance, perpendicular, point, real number, line, fix, Euclidean plane
There are 6 references to this entry.

This is version 2 of strain transformation, born on 2007-07-27, modified 2007-07-27.
Object id is 9804, canonical name is StrainTransformation.
Accessed 1433 times total.

Classification:
AMS MSC15A04 (Linear and multilinear algebra; matrix theory :: Linear transformations, semilinear transformations)

Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)