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The groups of all orthogonal matrices is defined It has a subgroup of matrices with determinant +1, This subgroup is called specail orthogonal group.
$\newcommand{\U}[1]{\underline{#1}}$
Group of Orthogonal Matrices
Let the set of all real $N \times N$, orthogonal, be denoted by $O(N)$. The set \(O(N)\) is a group because
- The product of two orthogonal matrices is again an orthogonal matrix; \item The identity matrix is an orthogonal matrix;
- The inverse of an orthogonal matrix is again an orthogonal matrix.
The group $O(3)$ is a non Abelian group since the order of multiplying two matrices is important.
Special Ortohogonal Group \(SO(N)\)
An orthogonal matrix has determinant \(\pm1\). The set of all \(N\times N\) orthogonal matrices with determinant equal to 1, is a group by itself and will be denoted by \(SO(N)\). This group is called special orthogonal group in \(N\) dimensions, special refers to he restriction on the determinant.
For a vector \(\vec{x}=(x_1, x_2\ldots, x_N)\),
we use the matrix notation
\begin{equation} \U{x}\longrightarrow \U{x}'=\U{R}\U{x}, \qquad\qquad \U{x}\longrightarrow \U{y}'=\U{R}\U{y} \end{equation}
and if \(\U{R}\) is an orthogonal matrix then we have
\begin{equation}\label{EQ24} \U{x}^{T} \U{y}={\U{x}^\prime}^T \U{y}' .\end{equation}
Here \(\U{x}^T\) denotes the transpose of the column vector \(\U{x}\).
Conversely, if \eqref{EQ24} is satisfied for all \(\U{x}\) and \(\U(y)\), then \(U(R)\) must be an orthogonal matrix. The set of all \(N\times N\) matrices \(\U(R)\) obeying \eqref{EQ24} for all \(\U{x},\U{y}\) coincides with the orthogonal group \(O(N).\).
