The linear operator needs to be an element of the orthogonal group for it to be representable as versor, which means it’s full rank and \det = \pm1 (if it has a unit scaling factor).
In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of n×n orthogonal matrices, where the group operation is given by matrix multiplication; an orthogonal matrix...
In linear algebra, an orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors (orthonormal vectors).
One way to express this is
where
Q
T
{\displaystyle Q^{\mathrm {T} }}
is the transpose of Q and
I
{\displaystyle I}
is the identity matrix.
This leads to the equivalent characterization: a matrix...