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'''Orthogonality''' is a generalization of perpendicularity. '''Orthonormality''' is a related concept, requiring that the components be unit vectors. '''Orthogonality''' is a generalization of perpendicularity. '''Orthonormality''' is a related concept, requiring that the components be [[Calculus/UnitVector|unit vectors]].
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For a subspace S to be orthogonal to a subspace T, every vector in S must be orthogonal to every vector in T. [[LinearAlgebra/NullSpaces|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''')'') 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''')'')
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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 an important property: '''''Q'''^T^'''Q''' = '''I'''''. 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^''.
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The [[LinearAlgebra/Projections#Matrices|projection]] of such a matrix '''''A''''' is normally given as '''''P''' = '''A'''('''A'''^T^'''A'''^-1^)'''A'''^T^''. For such a matrix '''''Q''''' that has orthonormal columns, the projection is given as '''''P''' = '''QQ'''^T^''. Correspondingly, the system of normal equations simplifies from '''''A'''^T^'''A'''x̂ = '''A'''^T^b'' into ''x̂ = '''Q'''^T^b''. 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^''.
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If such a matrix with orthonormal columns is ''also'' square, then it can be called an '''orthogonal matrix'''. These have an additional important property: '''''Q'''^T^ = '''Q'''^-1^''. If such a matrix with orthonormal columns is ''also'' square, then it can be called an '''orthogonal matrix'''. These have several important properties:
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For an orthogonal matrix '''''Q''''', the projection matrix is given as '''''P''' = '''I'''''.
  * '''''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''')''.

Orthogonality

Orthogonality is a generalization of perpendicularity. Orthonormality is a related concept, requiring that the components be unit vectors.

See also vector orthogonality.

Contents

  1. Orthogonality
    1. Orthogonality
    2. Orthonormality

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. 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(AT) is orthogonal to the column space of A (i.e., C(A))

Orthonormality

If a matrix is composed of orthonormal columns, then it can be called a matrix with orthonormal columns. These are usually denoted as Q. These have several important properties:

The second follows from the first. Recall that, when projecting b into C(A), the projection matrix is given as P = A(ATA-1)AT. This comes from the linear system ATAx̂ = ATb and requiring that p = Pb. For a matrix Q with orthonormal columns, the first property simplifies the linear system to x̂ = QTb. Therefore, P = QQT.

If such a matrix with orthonormal columns is also square, then it can be called an orthogonal matrix. These have several important properties:

  • QTQ = QQT = I

  • QT = Q-1

  • The determinant is always 1 or -1

  • The projection matrix is given as P = I, indicating that b must be in C(A).

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LinearAlgebra/Orthogonality (last edited 2025-09-24 20:19:45 by DominicRicottone)