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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

$\displaystyle d(p',\ell)=r\cdot d(p,\ell).$
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 $ \vert r\vert>1$ and compresses it if $ \vert r\vert<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... ...ix}\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.
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Other names:  strain
Also defines:  strain coefficient
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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 738 times total.

Classification:
AMS MSC: 15A04 (Linear and multilinear algebra; matrix theory :: Linear transformations, semilinear transformations)
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