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strain transformation
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(Definition)
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Let be a Euclidean plane. Fix a line in and a real number . Take any point . Drop a line from perpendicular to . Denote the distance from to . Then there is a unique point on such that
The function
such that is called a strain transformation, or simply a strain.
One can visualize a strain stretches a geometric figure if and compresses it if . If , then is the identity function, the only time when a strain is a rigid motion. For example, let be the -axis and be a circle in the upper half plane of the - plane. Then the following diagrams show how a strain transforms :
unit=1.5cm

Again, if is the -axis, then is the function that sends to . Representing the ordered pairs as column vectors and as a matrix , we have
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 over a field , a strain is a non-singular diagonalizable linear transformation on such that leaves a subspace of codimension fixed.
is called the strain coefficient.
Remark. By choosing an appropriate base for of dimension , can be represented as a diagonal matrix whose diagonals are in at least cells and in at most one cell.
It is easy to see that every non-singular diagonalizable linear transformation on can be written as a product of strains, where
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"strain transformation" is owned by CWoo.
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(view preamble)
| 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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Pending Errata and Addenda
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