Orthogonal Matrix Definition and Properties
In linear algebra, a square matrix is defined as orthogonal if its columns and rows constitute an orthonormal set with respect to the standard Euclidean inner product. The fundamental theorem establishes that such matrices preserve vector lengths and orthogonality under transformation, effectively representing rigid rotations or reflections in Euclidean space. This concept functions strictly within the theoretical subfield of finite-dimensional vector spaces, serving as a specific case where matrix inversion is equivalent to transposition ($Q^{-1} = Q^T$).
Orthogonal Matrix Definition and Properties
Shows that a matrix with orthonormal columns satisfies Q^T Q = I and that multiplying vectors by an orthogonal matrix preserves their lengths and the angles between them.