= 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]].

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== Description ==

'''Orthogonality''' is an extension of perpendicularity to higher dimensions of Euclidean space, and also to arbitrary [[LinearAlgebra/InnerProduct|inner product spaces]]. To notate that ''x'' is orthogonal to ''y'', use ''x ⊥ y''.

'''Orthonormality''' is a further constraint: the orthogonal vectors are also [[Calculus/UnitVector|unit vectors]].

For a two vectors (which are ''not'' the zero vector) to be orthogonal, their inner product should be equal to 0. For a set of vectors to be orthogonal, every possible pair from the set should be orthogonal.



=== Matrices ===

A matrix is effectively a set of vectors. If the columns are orthonormal, then it can be called a '''matrix with orthonormal columns'''. These are usually denoted as '''''Q'''''.

Matrices with orthonormal columns 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 and ''only'' if such a matrix is square, it can be called an '''orthogonal matrix'''. These have several further 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''')''.



=== Subspaces ===

For two subspaces to be orthogonal, every vector in the span of one should be orthogonal to every vector in the span of the other.

For example, consider a plane in ''R^3^''. A plane is a subspace spanned by 2 vectors. The subspace that is orthogonal to a plane must be spanned by 1 vector, i.e. it is a line. A plane and a line can be checked for orthogonality by comparing each of the vectors spanning the plane for orthogonality with the single vector spanning the line.

For another example, [[LinearAlgebra/NullSpace|null spaces]] are orthogonal by definiton. For any subspace '''''A''''', the vectors spanning ''N('''A''')'' are precisely those that are not in ''R('''A''')'', therefore they are orthogonal. Similarly, the vectors spanning ''N('''A'''^T^)'' are precisely those that are not in ''C('''A''')'', therefore they are orthogonal.



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