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A unit vector means that the vector has a length of 1, which is also known as normalized. Orthogonal means that the vectors are all perpendicular to each other. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an orthonormal basis.
A set of mutually orthonormal vectors in a Hilbert space is called an orthonormal system. An orthonormal basis is an orthonormal system with the additional property that the linear span of S {\displaystyle S} is dense in H {\displaystyle H} . [ 6 ]
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q T Q = Q Q T = I , {\displaystyle Q^{\mathrm {T} }Q=QQ^{\mathrm {T} }=I,} where Q T is the transpose of Q and I is the identity matrix .
A set of vectors in an inner product space is called pairwise orthogonal if each pairing of them is orthogonal. Such a set is called an orthogonal set (or orthogonal system). If the vectors are normalized, they form an orthonormal system. An orthogonal matrix is a matrix whose column vectors are orthonormal to each other.
The Gram–Schmidt process takes a finite, linearly independent set of vectors = {, …,} for k ≤ n and generates an orthogonal set ′ = {, …,} that spans the same -dimensional subspace of as . The method is named after Jørgen Pedersen Gram and Erhard Schmidt , but Pierre-Simon Laplace had been familiar with it before Gram and Schmidt. [ 1 ]
The concept of orthogonality may be extended to a vector space over any field of characteristic not 2 equipped with a quadratic form .Starting from the observation that, when the characteristic of the underlying field is not 2, the associated symmetric bilinear form , = ((+) ()) allows vectors and to be defined as being orthogonal with respect to when (+) () = .
In other words, the set of all orthonormal frames is a right ()-torsor. The orthonormal frame bundle of , denoted (), is the set of all orthonormal frames at each point in the base space . It can be constructed by a method entirely analogous to that of the ordinary frame bundle.
Let stand for ,, or . The Stiefel manifold () can be thought of as a set of n × k matrices by writing a k-frame as a matrix of k column vectors in . The orthonormality condition is expressed by A*A = where A* denotes the conjugate transpose of A and denotes the k × k identity matrix.