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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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| 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 [[LinearAlgebra/Projection|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''. |
Orthogonality
Orthogonality is a generalization of perpendicularity. Orthonormality is a related concept, requiring that the components be unit vectors.
See also vector orthogonality.
Contents
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 an important property: QTQ = I.
The projection of such a matrix A is normally given as P = A(ATA-1)AT. For such a matrix Q that has orthonormal columns, the projection is given as P = QQT. Correspondingly, the system of normal equations simplifies from ATAx̂ = ATb into x̂ = QTb.
If such a matrix with orthonormal columns is also square, then it can be called an orthogonal matrix. These have an additional important property: QT = Q-1.
For an orthogonal matrix Q, the projection matrix is given as P = I.