Search results
Results from the WOW.Com Content Network
A linear isometry also necessarily preserves angles, therefore a linear isometry transformation is a conformal linear transformation. Examples. A linear map from to itself is an isometry (for the dot product) if and only if its matrix is unitary. [10] [11] [12] [13]
Thus a unitary operator is a bounded linear operator that is both an isometry and a coisometry, [1] or, equivalently, a surjective isometry. [2] An equivalent definition is the following: Definition 2. A unitary operator is a bounded linear operator U : H → H on a Hilbert space H for which the following hold: U is surjective, and
A discrete isometry group is an isometry group such that for every point of the space the set of images of the point under the isometries is a discrete set. In pseudo-Euclidean space the metric is replaced with an isotropic quadratic form ; transformations preserving this form are sometimes called "isometries", and the collection of them is ...
In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length. There are four types: translations , rotations , reflections , and glide reflections (see below § Classification ).
In linear algebra, the restricted isometry property (RIP) characterizes matrices which are nearly orthonormal, at least when operating on sparse vectors. The concept was introduced by Emmanuel Candès and Terence Tao [1] and is used to prove many theorems in the field of compressed sensing. [2]
It is a linear isometry at the tangent space of every point on (), that is, it is an isometry on the infinitesimal patches. It preserves the curvature tensor at the tangent space of every point on B r ( x ) {\displaystyle B_{r}(x)} , that is, it preserves how the infinitesimal patches fit together.
In mathematics, a unitary transformation is a linear isomorphism that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation.
Any element of E(n) is a translation followed by an orthogonal transformation (the linear part of the isometry), in a unique way: (+) where A is an orthogonal matrix or the same orthogonal transformation followed by a translation: x ↦ A x + c , {\displaystyle x\mapsto Ax+c,} with c = Ab