K-frame
Encyclopedia
In linear algebra
, a branch of mathematics
, a k-frame is an ordered set
of k linearly independent vectors in a space; thus k ≤ n, where n is the dimension of the vector space, and if k = n an n-frame is precisely an ordered basis.
If the vectors are orthogonal, or orthonormal, the frame is called an orthogonal frame, or orthonormal frame
, respectively.
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a k-frame is an ordered set
Ordered set
In order theory in mathematics, a set with a binary relation R on its elements that is reflexive , antisymmetric and transitive is described as a partially ordered set or poset...
of k linearly independent vectors in a space; thus k ≤ n, where n is the dimension of the vector space, and if k = n an n-frame is precisely an ordered basis.
If the vectors are orthogonal, or orthonormal, the frame is called an orthogonal frame, or orthonormal frame
Orthonormal frame
In Riemannian geometry and relativity theory, an orthonormal frame is a tool for studying the structure of a differentiable manifold equipped with a metric...
, respectively.
Properties
- The set of k-frames (particularly the set of orthonormal k-frames) in a given space X is known as the Stiefel manifoldStiefel manifoldIn mathematics, the Stiefel manifold Vk is the set of all orthonormal k-frames in Rn. That is, it is the set of ordered k-tuples of orthonormal vectors in Rn. It is named after Swiss mathematician Eduard Stiefel...
, and denoted Vk(X). - A k-frame defines a parallelotope (a generalized parallelepipedParallelepipedIn geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, its definition encompasses all four concepts...
); the volume can be computed via the Gram determinant.