m×morthogonale
An m-by-m orthogonal matrix is a real square matrix A ∈ R^{m×m} that satisfies A^T A = A A^T = I_m, where A^T is the transpose and I_m is the identity matrix. This condition is equivalent to A^T = A^{-1}, so orthogonal matrices preserve inner products and norms: for any vectors x, y in R^m, (Ax)·(Ay) = x·y and ||Ax|| = ||x||.
Orthogonal matrices form the group O(m) under matrix multiplication; the subgroup of determinant +1, denoted SO(m),
Real eigenvalues of an orthogonal matrix are restricted to ±1, while generally the eigenvalues lie on the
Typical examples include the identity I_m, rotation and reflection matrices, and permutation matrices. Because A is
Applications of orthogonal matrices are widespread: in numerical linear algebra they underlie Gram–Schmidt processes and QR
As a Lie group, O(m) has dimension m(m−1)/2, and its determinant-one subgroup SO(m) inherits the same dimension.