partial isometry

Partial isometry is a generalizationPlanetmathPlanetmath of an isometry. Before defining what a partial isometry is, let’s recall two familiar concepts in linear algebra: an isometry and the adjointPlanetmathPlanetmathPlanetmath of a linear map.

  1. 1.

    An isometry T is a linear automorphismPlanetmathPlanetmathPlanetmathPlanetmath over an inner product spaceMathworldPlanetmath V which preserves the inner productMathworldPlanetmath of any two vectors: x,y=Tx,Ty.

  2. 2.

    The adjoint T* of a linear transformation T is linear transformation such that Tx,y=x,T*y, for any pair of vectors x,yV.

If V is non-singularPlanetmathPlanetmath with respect to the inner product , and that the adjoint T* of a linear transformation T exists, it is not hard to show that

T is an isometry if and only if TT*=I=T*T.

In other words, T* is the inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of T.

More generally, in a ring with involution *, an isometry (or an unitary element) is a unit (both a left unit and a right unit) a whose productPlanetmathPlanetmath with its adjoint a* is 1 (i.e. its inverse is its adjoint). Now, if a is not a unit, this product aa* will not be 1. The next best thing to hope for is that the product will be an idempotentPlanetmathPlanetmath. But because aa* is self-adjointPlanetmathPlanetmath, this idempotent is in fact a projectionMathworldPlanetmath. This is how a partial isometry is defined. Formally,

let R be a ring with involution *, an element aR is a partial isometry if aa* and a*a are both projections.

Given a partial isometry a, the projections a*a and aa* are respectively called the initial projection and final projection of a.

Examples. Under this definition, 0 is a partial isometry, and so is any isometry.

This definition can be readily applied to specific (more familiar) situations. For example, if the ring in question is the ring of linear endomorphisms over a Euclidean space (real or complex), then a partial isometry is just a map such that its restrictionPlanetmathPlanetmath to the complementary subspace of its kernel is an isometry. If we look at the case when the space is 3 dimensional over the reals, and taking the standard basis, the matrix


corresponds to a partial isometry whose kernel is a line L. Its restriction to the complement of L corresponds to the matrix


which is an isometry (rotationMathworldPlanetmath).

Remark. If the ring R is a Baer *-ring, an element a is a partial isometry iff aa*a=a (so a*aa*=a*; a and a* are generalized inverses of one another).

Title partial isometry
Canonical name PartialIsometry
Date of creation 2013-03-22 15:50:50
Last modified on 2013-03-22 15:50:50
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Definition
Classification msc 47C10
Defines unitary element
Defines initial projection
Defines final projection