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All orthogonal all \(N\times N\) orthogonal matrices form a group called \(O(N)\). The set of all orthogonal matrices with unit determinant form a subgroup \(SO(N)\) . The group of all proper rotations coincides with \(SO(3)\).
Let the set of all real $N \times N$, orthogonal, be denoted by $O(N)$. The set \(O(N)\) is a group because \begin{itemize}
- The product of two orthogonal matrices is again an orthogonal matrix;
- 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 Orthogonal Group \(SO(N)\)
An orthogonal matrix has determinant \(\pm1\) , because \( O^TO=I \Rightarrow \det O=\pm 1\).
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 \(O(N).\).
For \(N=3\) the group \(SO(3)\) coincides with the group of all proper rotations in three dimensions.
