Orthogonal group

Orthogonal group - a group of all linear transformations $n$ ${\ displaystyle n}$ $n$ -dimensional vector space $V$ ${\ displaystyle V}$ $V$ over the field $k$ ${\ displaystyle k}$ $k$ preserving a fixed non-degenerate quadratic form $Q$ ${\ displaystyle Q}$ $Q$ on $V$ ${\ displaystyle V}$ $V$ (i.e. such linear transformations $\varphi$ ${\ displaystyle \ varphi}$ $\ varphi$ , what $Q(\varphi (v))=Q(v)$ ${\ displaystyle Q (\ varphi (v)) = Q (v)}$ $Q (\ varphi (v)) = Q (v)$ for anyone $v\in V$ ${\ displaystyle v \ in V}$ $v \ in V$ )

Content

Symbols and related definitions

Elements of an orthogonal group are called orthogonal (with respect to $Q$ ${\ displaystyle Q}$ ) transformations $V$ ${\ displaystyle V}$ as well as automorphisms of the form $Q$ ${\ displaystyle Q}$ (more precisely, by automorphisms of space $V$ ${\ displaystyle V}$ regarding form $Q$ ${\ displaystyle Q}$ )
Designated $O_{n}$ ${\ displaystyle O_ {n}}$ , $O_{n}(k)$ ${\ displaystyle O_ {n} (k)}$ , $O_{n}(Q)$ ${\ displaystyle O_ {n} (Q)}$ and so on. When the quadratic form is not indicated explicitly, it means the form specified by the sum of the squares of the coordinates, that is, expressed by the identity matrix .
Above the field of real numbers, an orthogonal group of an indefinite form with a signature ( $l$ ${\ displaystyle l}$ pluses $m$ ${\ displaystyle m}$ cons) where $n=l+m$ ${\ displaystyle n = l + m}$ is denoted by $O(l,m)$ ${\ displaystyle O (l, m)}$ see e.g. O (1,3) .

Properties

If the characteristic of the main field is more than two , then with $Q$ ${\ displaystyle Q}$ connected non-degenerate symmetric bilinear form $F$ ${\ displaystyle F}$ on $V$ ${\ displaystyle V}$ defined by the formula
$F(u,\;v)=Q(u+v)-Q(u)-Q(v).$ ${\ displaystyle F (u, \; v) = Q (u + v) -Q (u) -Q (v).}$

Then the orthogonal group consists precisely of those linear transformations of the space

V

{\ displaystyle V}

that save

F

{\ displaystyle F}

, and is denoted by

O_{n}(k,\;F)

{\ displaystyle O_ {n} (k, \; F)}

or (when it is clear about which field

k

{\ displaystyle k}

and form

F

{\ displaystyle F}

in question) just through

O_{n}

{\ displaystyle O_ {n}}

.

If a $B$ ${\ displaystyle B}$ - shape matrix $F$ ${\ displaystyle F}$ in a certain basis of space $V$ ${\ displaystyle V}$ , then the orthogonal group can be identified with the group of all such matrices $A$ ${\ displaystyle A}$ with coefficients in $k$ ${\ displaystyle k}$ , what
$A^{T}BA=B.$ ${\ displaystyle A ^ {T} BA = B.}$
In particular, if the basis is such that $Q$ ${\ displaystyle Q}$ is the sum of the squares of the coordinates (i.e., the matrix $B$ ${\ displaystyle B}$ unit), then such matrices $A$ ${\ displaystyle A}$ called orthogonal .
Over the field of real numbers , group $O$ $n$ $($ $R$ $,$ $V$ $)$ {\ displaystyle O_ {n} ({\ mathbb {R}}, \; V)} compact if and only if the form $Q$ {\ displaystyle Q} sign-defined .
- In this case, any element of $O_{n}({\mathbb {R} })$ ${\ displaystyle O_ {n} ({\ mathbb {R}})$ , for a suitable basias, is represented as a block-diagonal matrix
  ${\begin{bmatrix}{\begin{matrix}R_{1}&&\\&\ddots &\\&&R_{k}\end{matrix}}&0\\0&{\begin{matrix}\pm 1&&\\&\ddots &\\&&\pm 1\end{matrix}}\\\end{bmatrix}}$ ${\ displaystyle {\ begin {bmatrix} {\ begin {matrix} R_ {1} && \\ & \ ddots & \\ && R_ {k} \ end {matrix}} & 0 \\ 0 & {\ begin {matrix} \ pm 1 && \\ & \ ddots & \\ && \ pm 1 \ end {matrix}} \\\ end {bmatrix}}}$

where R ₁ , ..., R _k - 2x2 matrix of turns; Euler's rotation theorem is a special case of this statement.

Other groups

The orthogonal group is a subgroup of the complete linear group GL ( $n$ ${\ displaystyle n}$ ) Elements of an orthogonal group whose determinant is 1 (this property does not depend on the basis ) form a subgroup - a special orthogonal group $SO(n,Q)$ ${\ displaystyle SO (n, Q)}$ denoted in the same way as the orthogonal group, but with the addition of the letter "S". $SO(n,Q)$ ${\ displaystyle SO (n, Q)}$ , by construction, is also a subgroup of a special linear group $SL(n)$ ${\ displaystyle SL (n)}$ .

Links

Orthogonal group