= Orthogonality = '''Orthogonality''' is a generalization of perpendicularity. '''Orthonormality''' is a related concept, requiring that the components be [[Calculus/UnitVector|unit vectors]]. See also [[Calculus/Orthogonality|vector orthogonality]]. <> ---- == Orthogonality == The notation for orthogonality is ''⊥'', as in ''x ⊥ y''. For a subspace S to be orthogonal to a subspace T, every vector in S must be orthogonal to every vector in T. [[LinearAlgebra/NullSpace|Null spaces]] are a trivial example. For a given matrix '''''A''''', its null space (i.e., ''N('''A''')'') contains the vectors that are not in the row space (i.e., ''R('''A''')''). Therefore it is orthogonal. Similarly, ''N('''A'''^T^)'' is orthogonal to the column space of '''''A''''' (i.e., ''C('''A''')'') ---- == Orthonormality == If a matrix is composed of [[LinearAlgebra/Orthonormalization|orthonormal columns]], then it can be called a '''matrix with orthonormal columns'''. These are usually denoted as '''''Q'''''. These have several important properties: * '''''Q'''^T^'''Q''' = '''I''''' * The [[LinearAlgebra/Projection|projection matrix]] is given as '''''P''' = '''QQ'''^T^''. The second follows from the first. Recall that, when projecting ''b'' into ''C('''A''')'', the projection matrix is given as '''''P''' = '''A'''('''A'''^T^'''A'''^-1^)'''A'''^T^''. This comes from the linear system '''''A'''^T^'''A'''x̂ = '''A'''^T^b'' and requiring that ''p = '''P'''b''. For a matrix '''''Q''''' with orthonormal columns, the first property simplifies the linear system to ''x̂ = '''Q'''^T^b''. Therefore, '''''P''' = '''QQ'''^T^''. If such a matrix with orthonormal columns is ''also'' square, then it can be called an '''orthogonal matrix'''. These have several important properties: * '''''Q'''^T^'''Q''' = '''QQ'''^T^ = '''I''''' * '''''Q'''^T^ = '''Q'''^-1^'' * The [[LinearAlgebra/Determinant|determinant]] is always 1 or -1 * The projection matrix is given as '''''P''' = '''I''''', indicating that ''b'' must be in ''C('''A''')''. ---- CategoryRicottone