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partial isometry
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(Definition)
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Partial isometry is a generalization of an isometry. Before defining what a partial isometry is, let's recall two familiar concepts in linear algebra: an isometry and the adjoint of a linear map.
- An isometry $T$ is a linear automorphism over an inner product space $V$ which preserves the inner product of any two vectors: $\langle x,y\rangle = \langle Tx, Ty\rangle$
- The adjoint $T^*$ of a linear transformation $T$ is linear transformation such that $\langle Tx,y\rangle = \langle x, T^*y\rangle$ for any pair of vectors $x,y\in V$
If $V$ is non-singular with respect to the inner product $\langle \cdot,\cdot \rangle$ 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 inverse 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 product 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 idempotent. But because
$aa^*$ is self-adjoint, this idempotent is in fact a projection. This is how a partial isometry is defined. Formally,
let $R$ be a ring with involution $*$ an element $a\in R$ 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 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
$A = \begin{pmatrix} 0 & 0 & 0 \\ 0 & \sin\theta & -\cos\theta \\ 0 & \cos\theta & \sin\theta \end{pmatrix}$
corresponds to a partial isometry whose kernel is a line $L$ Its restriction to the complement of $L$ corresponds to the matrix
$B = \begin{pmatrix} \sin\theta & -\cos\theta \\ \cos\theta & \sin\theta \end{pmatrix}$
which is an isometry (rotation).
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).
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"partial isometry" is owned by CWoo.
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| Also defines: |
unitary element, initial projection, final projection |
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Cross-references: generalized inverses, iff, Baer *-ring, rotation, complement, line, matrix, standard basis, kernel, complementary subspace, restriction, map, complex, real, Euclidean space, endomorphisms, ring, self-adjoint, idempotent, product, right, unit, ring with involution, inverse, non-singular, vectors, inner product, preserves, inner product space, automorphism, linear map, adjoint, concepts in linear algebra, isometry
There are 4 references to this entry.
This is version 4 of partial isometry, born on 2006-04-14, modified 2006-04-18.
Object id is 7830, canonical name is PartialIsometry.
Accessed 4417 times total.
Classification:
| AMS MSC: | 47C10 (Operator theory :: Individual linear operators as elements of algebraic systems :: Operators in $^*$-algebras) |
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Pending Errata and Addenda
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