= 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.

<<TableOfContents>>

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== Vectors ==

The concept of '''orthogonality''' is a generalization of '''perpendicularity''' from 2-dimensional space.

Put simply, vectors ''a'' and ''b'' are proven to be orthogonal if their [[LinearAlgebra/VectorMultiplication#Dot_Product|dot product]] is 0.

More precisely: assuming orthogonality, vectors ''a'' and ''b'' will satisfy the Pythagorean theorem. The hypotenuse is [[LinearAlgebra/Distance|Euclidean distance]]: ''(a+b)^T^(a+b)''. Simplifying from there:

''a^T^a + b^T^b = (a+b)^T^(a+b)''

''a^T^a + b^T^b = a^T^a + b^T^b + a^T^b + b^T^a''

''0 = a^T^b + b^T^a''

''0 = 2(a^T^b)''

''0 = a^T^b''

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== Subspaces ==

For a subspace S to be orthogonal to a subspace T, every vector in S must be orthogonal to every vector in T.

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== 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.

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== 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'''''.




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