= Orthogonality = '''Orthogonality''' is an important property for relating two vectors, or two subspaces, or a vector and a plane. The math notation is ⊥, as in ''x ⊥ y''. '''Orthonormality''' is an expanded concept, requiring that the components be unit vectors. <> ---- == Vectors == The concept of '''orthogonality''' is a generalization of '''perpendicularity''' in 2-dimensional space. Two vectors are proven to be orthogonal if they obey the [[LinearAlgebra/Distance#n_Dimensions|Pythagorean theorem]]. In the case of two vectors, this test simplifies to ''x^T^y = 0''. ''x^T^x + y^T^y = (x+y)^T^(x+y)'' expands to ''x^T^x + y^T^y = x^T^x + y^T^y + x^T^y + y^T^x'', simplifies to ''0 = x^T^y + y^T^x'', is trivially proven to be equivalent to ''0 = 2x^T^y'', and finally simplifies to ''0 = x^T^y''. ---- == Subspaces == For a subspace S to be orthogonal to a subspace T, every vector in S must be orthogonal to every vector in T. ---- == Vectors and Planes == The [[LinearAlgebra/NullSpaces|null space]] of a matrix '''''A''''' contains the vectors that are not in the row space. These vectors cancel out; they are not a linear combination of the rows; if the row space is a plane, then these vectors are not on that plane. The null space of '''''A''''' (a.k.a. ''N('''A''')'') is '''orthogonal''' to the row space of '''''A''''' (a.k.a. ''R('''A''')''). The null space of '''''A'''^T^'' (a.k.a. ''N('''A'''^T^)'') is orthogonal to the column space of '''''A''''' (a.k.a. ''C('''A''')''). Commonly this means that the row and column spaces are planes while the null spaces of '''''A''''' and '''''A'''^T^'' are vectors, but that isn't always true. ---- == Matrices == If a matrix is composed of orthonormal columns, then it is a '''matrix with orthonormal columns'''. These are usually denoted as '''''Q'''''. This has an important property: '''''Q'''^T^'''Q''' = '''I'''''. The [[LinearAlgebra/Projections#Matrices|projection]] of '''''A''''' if '''''A''''' is a matrix with orthonormal columns simplifies from '''''P''' = '''A'''('''A'''^T^'''A'''^-1^)'''A'''^T^'' into '''''P''' = '''QQ'''^T^''. Correspondingly, the system of normal equations simplifies from '''''A'''^T^'''A'''x̂ = '''A'''^T^b'' into ''x̂ = '''Q'''^T^b''. If a matrix with orthonormal columns is ''also'' square, only then can it be called an '''orthogonal matrix'''. This has an additional important property: '''''Q'''^T^ = '''Q'''^-1^''. For example, if '''Q''' is square, then the projection matrix further simplifies to '''''P''' = '''I'''''. ---- CategoryRicottone