Matrix groups and Symmetry
Linear algebra
See this video and these notes.
We use some facts from linear algebra:
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A map is linear if for all and all .
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An matrix yields a linear map from to via matrix multiplication .
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Given a linear map , we can associate to it a matrix with entries by computing where and are the - and - dimensional column vectors with a one in position (resp. ) and zeros elsewhere.
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If and are linear maps with associated matrices and , then the matrix associated to the composition is the matrix product .
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If is bijective then its inverse is also linear and the associated matrix is the inverse matrix . Conversely if the associated matrix is invertible then is bijective.
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An matrix is invertible if and only if its determinant is non-zero.
Matrix Groups
See this video and these notes
The general and special linear groups
Definition: The general linear group of rank , written , is the group of bijective linear transformations with group operation given by composition of maps. Equivalently:
- is the group of invertible matrices with matrix multiplication
- is the group of matrices with nonzero determinant under matrix multiplication.
Definition: The special linear group is the subgroup of of linear maps whose associated matrices have determinant one.
The orthogonal group
Definition: The Euclidean inner product on is the dot product If this is also written .
The inner product is symmetric, bilinear, satisfies for all , and if and only if .
Also is the euclidean length of the vector and is the distance between and .
Definition: A matrix is orthogonal if for all vectors .
Proposition: is orthogonal if and only if or, equivalently, if the rows (or columns) of form an orthonormal set.
Definition: The orthogonal group is the subgroup of consisting of orthogonal matrices. The special orthogonal group is the subgroup of consisting of matrices with determinant .
Extra topic (orientation)
See this video and these notes
The euclidean group
See this video and these notes
Definition: The euclidean group is the group of distance preserving maps .
The Euclidean group can be described as the subgroup of of matrices with the following block form:
where the block is an orthogonal matrix.
If is an element of , and is a point in , then the transformation of by is given by matrix multiplication after extending by adding a :
Alternatively, .
Plane symmetries
See this video and these notes
By the definition is the group of distance preserving maps of the plane to itself (the group of isometries).
Proposition: The group is abelian and consists of matrices of the form where corresponding to rotations through an angle where
The group includes as well as matrices of the form of determinant .
Elements of are rotations, reflections, translations, or “glide reflections.”
Definition: Two regions in the plane are congruent if there is an element of the Euclidean group carrying one bijectively to the other.
Proposition: The only finite subgroups of are isomorphic to or for some
Lattices
A lattice in the plane is the set of integer linear combinations of two independent vectors , :
Wallpaper and crystals
See these three videos:
and these notes
See also the interactive visualization at the bottom of this web page.
Definition: A wallpaper group is a subgroup of that preserves a lattice; in other words, there is a lattice such that for all .
Proposition: Up to isomorphism there are 17 wallpaper groups. Each such group yields a characteristic “wallpaper” made up of repeating patterns organized in a lattice shape.
This result is proved in the paper The 17 plane symmetry groups, by R. L. E. Schwarzenberger, The Mathematical Gazette, vol. 58, number 404, June 1974, pp. 123-131. The techniques used are covered in this course, so the proof, though somewhat long, is accessible.
See this page for examples of the 17 groups and their associated patterns.
Definition: A crystallographic group is a subgroup of that preserves a lattice. There are different crystallographic groups corresponding to different crystal structures. (Sometimes this number is given as depending on how you define “different.”)
Every possible crystal structure occurs for some mineral. See this page for a discussion.
You can see all possible plane symmetries here.