Isometries

We continue with our program of investigating the analogy between numbers and transformations. When does a complex number have absolute value one? Clearly a necessary and sufficient condition is that ; guided by our heuristic principle, we are led to consider linear transformations for which , or, equivalently, for which . (We observe that on a finite-dimensional vector space either of the two conditions and implies the other; see Section: Inverses , Theorems 1 and 2.) Such transformations are called orthogonal or unitary according as the underlying inner product space is real or complex. We proceed to derive a couple of useful alternative characterizations of them.

Theorem 1. The following three conditions on a linear transformation on an inner product space are equivalent to each other.

Proof. If (1) holds, then for all and , and, in particular, for all ; this proves both the implications (1) (2) and (2) (3). The proof can be completed by showing that (3) implies (1). If (3) holds, that is, if for all , then Section: Polarization , Theorem 2 is applicable to the (self-adjoint) transformation ; the conclusion is that (as desired). ◻

Since (3) implies that for all and (the converse implication (4) (3) is also true and trivial), we see that transformations of the type that the theorem deals with are characterized by the fact that they preserve distances. For this reason we shall call such a transformation an isometry . Since, as we have already remarked, an isometry on a finite-dimensional space is necessarily orthogonal or unitary (according as the space is real or complex), use of this terminology will enable us to treat the real and the complex cases simultaneously. We observe that (on a finite-dimensional space) an isometry is always invertible and that ( ) is an isometry along with .

In any algebraic system, and in particular in general vector spaces and inner product spaces, it is of interest to consider the automorphisms of the system, that is, to consider those one-to-one mappings of the system onto itself that preserve all the structural relations among its elements. We have already seen that the automorphisms of a general vector space are the invertible linear transformations. In an inner product space we require more of an automorphism, namely, that it also preserve inner products (and consequently lengths and distances). The preceding theorem shows that this requirement is equivalent to the condition that the transformation be an isometry. (We are assuming finite-dimensionality here; on infinite-dimensional spaces the range of an isometry need not be the entire space. This unimportant sacrifice in generality is for the sake of terminological convenience; for infinite-dimensional spaces there is no commonly used word that describes orthogonal and unitary transformations simultaneously.) Thus the two questions "What linear transformations are the analogues of complex numbers of absolute value one?" and "What are the most general automorphisms of a finite-dimensional inner product space?" have the same answer: isometries. In the next section we shall show that isometries also furnish the answer to a third important question.

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