Conversely, let " Q " be any orthogonal matrix which does not have " 1 as an eigenvalue; then
32.
Going the other direction, the matrix exponential of any skew-symmetric matrix is an orthogonal matrix ( in fact, special orthogonal ).
33.
The matrix is a member of the three-dimensional special orthogonal group,, that is it is an orthogonal matrix with determinant 1.
34.
In finite-dimensional spaces, the matrix representation ( with respect to an orthonormal basis ) of an orthogonal transformation is an orthogonal matrix.
35.
Their cross product is used as the third column in the linear system of equations obtaining a proper orthogonal matrix for the spacecraft attitude given by
36.
The representation of a rotation as a quaternion ( 4 numbers ) is more compact than the representation as an orthogonal matrix ( 9 numbers ).
37.
A related problem in linear algebra is the "'orthogonal Procrustes problem "'of finding the closest orthogonal matrix to any given matrix.
38.
Generalizing to any finite number of dimensions, a rotation matrix A is an orthogonal matrix that differs from the identity matrix in at most four elements.
39.
However, linear algebra includes orthogonal transformations between spaces which may be neither finite-dimensional nor of the same dimension, and these have no orthogonal matrix equivalent.
40.
A similar problem, with interesting applications in shape analysis, is the orthogonal Procrustes problem, which consists of finding an orthogonal matrix which most closely maps to.
