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Matrix groups and Symmetry

Linear algebra

We use some facts from linear algebra:

  • A map $T:\mathbb{R}^{n}\to\mathbb{R}^{m}$ is linear if $f(ax+by)=af(x)+bf(y)$ for all $x,y\in\mathbb{R}^{n}$ and all $a,b\in\mathbb{R}$.

  • An $n\times m$ matrix $A$ yields a linear map from $\mathbb{R}^{n}$ to $\mathbb{R}^{m}$ via matrix multiplication $x\mapsto Ax$.

  • Given a linear map $T:\mathbb{R}^{n}\to\mathbb{R}^{m}$, we can associate to it a matrix $A$ with entries $(a_{ij})$ by computing \(T(\mathbf{e}_j)= \sum_{i=1}^{m} a_{ij}\mathbf{f}_{i}\) where \(\mathbf{e}_{j}\) and \(\mathbf{f}_{i}\) are the $n$- and $m$- dimensional column vectors with a one in position $j$ (resp. $i$) and zeros elsewhere.

  • If $T:\mathbb{R}^{n}\to\mathbb{R}^{m}$ and $V:\mathbb{R}^{m}\to\mathbb{R}^p$ are linear maps with associated matrices $A$ and $B$, then the matrix associated to the composition $TV$ is the matrix product $AB$.

  • If $T:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is bijective then its inverse is also linear and the associated matrix is the inverse matrix $A^{-1}$. Conversely if the associated matrix is invertible then $T$ is bijective.

  • An $n\times n$ matrix is invertible if and only if its determinant is non-zero.

The general and special linear groups

Definition: The general linear group of rank $n$, written $\mathrm{GL}_{n}(\mathbb{R})$, is the group of bijective linear transformations $T:\mathbb{R}^{n}\to\mathbb{R}^{n}$ with group operation given by composition of maps. Equivalently:

  • $\mathrm{GL}_{n}(\mathbb{R})$ is the group of invertible $n\times n$ matrices with matrix multiplication
  • $\mathrm{GL}_{n}(\mathbb{R})$ is the group of $n\times n$ matrices with nonzero determinant under matrix multiplication.

Definition: The special linear group \(\mathrm{SL}_{n}(\mathbb{R})\) is the subgroup of \(GL_{n}(\mathbb{R})\) of linear maps whose associated matrices have determinant one.

The orthogonal group

Definition: The Euclidean inner product on $\mathbb{R}^{n}$ is the dot product \((\sum_{i=1}^{n} a_{i}\mathbf{e}_{i})\cdot (\sum_{i=1}^{n} b_{i}\mathbf{e}_{i})=\sum_{i=1}^{n} a_{i}b_{i}.\) If $x,y\in\mathbb{R}^{n}$ this is also written $\langle x,y\rangle$.

The inner product is symmetric, bilinear, satisfies $x\cdot x\ge 0$ for all $x$, and $x\cdot x=0$ if and only if $x=0$.

Also $x\cdot x=|x|^2$ is the euclidean length of the vector $x$ and $(x-y)\cdot (x-y)=|(x-y)|^2$ is the distance between $x$ and $y$.

Definition: A matrix $A\in\mathrm{GL}_{2}(\mathbb{R})$ is orthogonal if $|Ax|^2=|x|^2$ for all vectors $x$.

Proposition: $A$ is orthogonal if and only if $AA^{t}=Id$ or, equivalently, if the rows (or columns) of $A$ form an orthonormal set.

Definition: The orthogonal group $O_{n}(\mathbb{R})$ is the subgroup of \(\mathrm{GL}_{n}(\mathbb{R})\) consisting of orthogonal matrices. The special orthogonal group \(SO_{n}(\mathbb{R})\) is the subgroup of \(O_{n}(\mathbb{R})\) consisting of matrices with determinant $1$.

The euclidean group

Definition: The euclidean group \(E_{n}(\mathbb{R})\) is the group of distance preserving maps \(f:\mathbb{R}^{n}\to\mathbb{R}^{n}\).

The Euclidean group can be described as the subgroup of $\mathrm{GL}_{n+1}(\mathbb{R})$ of matrices with the following block form:

\(\left(\begin{matrix} A & \Bigg\vert & \begin{matrix} x_1 \\ \vdots \\ x_n \end{matrix}\\ \hline \begin{matrix} 0 &\cdots & 0 \\ \end{matrix} &\vert &1\\ \end{matrix}\right)\)

where the $n\times n$ block $A$ is an orthogonal matrix.

If $M$ is an element of $E_n(\mathbb{R})$, and $y$ is a point in $\mathbb{R}^{n}$, then the transformation of $y$ by $M$ is given by matrix multiplication after extending $y$ by adding a $1$:

\[My = \begin{pmatrix} A & \Bigg\vert & \begin{matrix} x_1 \\ \vdots \\ x_n \end{matrix}\\ \hline \begin{matrix} 0 &\cdots & 0 \\ \end{matrix} &\Bigg\vert &1\\ \end{pmatrix}\begin{pmatrix} y_1 \\ \vdots \\ y_n \\ 1\end{pmatrix}\]

Alternatively, $(A,x)y=Ay+x$.

Plane symmetries

By the definition $E(2)$ is the group of distance preserving maps of the plane to itself (the group of isometries).

Proposition: The group $SO(2)$ is abelian and consists of matrices of the form \(\left( \begin{matrix} a & b \\ -b & a \\ \end{matrix}\right)\) where $a^2+b^2=1$ corresponding to rotations through an angle $\theta$ where \(\mathrm{cis}(\theta) = a+bi.\)

The group $O(2)$ includes $SO_{2}$ as well as matrices of the form \(\left( \begin{matrix} b & a \\ a & -b \\ \end{matrix}\right)\) of determinant $-1$.

Elements of $E(2)$ are rotations, reflections, translations, or “glide reflections.”

Definition: Two regions in the plane are congruent if there is an element of the Euclidean group $E(2)$ carrying one bijectively to the other.

Proposition: The only finite subgroups of $E(2)$ are isomorphic to \(\mathbb{Z}_{n}\) or \(\mathbb{D}_{n}\) for some $n\ge 1.$

Lattices

A lattice $L$ in the plane is the set of integer linear combinations of two independent vectors $v_1$, $v_2$: \(L =\{av_1+bv_2:a,b\in\mathbb{Z}\}.\)

Wallpaper and crystals

Definition: A wallpaper group $G$ is a subgroup of $E(2)$ that preserves a lattice; in other words, there is a lattice $L$ such that $gL=L$ for all $g\in G$.

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.

See this page for examples of the 17 groups and their associated patterns.

Definition: A crystallographic group $G$ is a subgroup of $E(3)$ that preserves a lattice. There are $219$ different crystallographic groups corresponding to $219$ different crystal structures. (Sometimes this number is given as $230$ depending on how you define “different.”)

Every possible crystal structure occurs for some mineral. See this page for a discussion.