The adjoint of a linear transformation is linear transformation such that , for any pair of vectors .
If is non-singular with respect to the inner product and that the adjoint of a linear transformation exists, it is not hard to show that
is an isometry if and only if .
In other words, is the inverse of .
More generally, in a ring with involution , an isometry (or an unitary element) is a unit (both a left unit and a right unit) whose product with its adjoint is 1 (i.e. its inverse is its adjoint). Now, if is not a unit, this product will not be 1. The next best thing to hope for is that the product will be an idempotent. But because is self-adjoint, this idempotent is in fact a projection. This is how a partial isometry is defined. Formally,
let be a ring with involution , an element is a partial isometry if and are both projections.
Given a partial isometry , the projections and are respectively called the initial projection and final projection of .
Examples. Under this definition, 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 restriction 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 . Its restriction to the complement of corresponds to the matrix
which is an isometry (rotation).
Remark. If the ring is a Baer *-ring, an element is a partial isometry iff (so ; and are generalized inverses of one another).
|Date of creation||2013-03-22 15:50:50|
|Last modified on||2013-03-22 15:50:50|
|Last modified by||CWoo (3771)|