Editing Partial isometry (section)

== Characterization in finite dimensions ==
In [[dimension (vector space)|finite-dimensional]] [[vector space]]s, a [[matrix (mathematics)|matrix]] <math>A</math> is a partial isometry [[if and only if]] <math> A^* A</math> is the projection onto its support. Contrast this with the more demanding definition of [[isometry]]: a matrix <math>V</math> is an isometry if and only if <math>V^* V=I</math>. In other words, an isometry is an [[injective]] partial isometry.

Any finite-dimensional partial isometry can be represented, in some choice of [[basis (linear algebra)|basis]], as a matrix of the form <math>A=\begin{pmatrix}V & 0\end{pmatrix}</math>, that is, as a matrix whose first <math>\operatorname{rank}(A)</math> columns form an isometry, while all the other columns are identically 0. 

Note that for any isometry <math>V</math>, the Hermitian conjugate <math>V^*</math> is a partial isometry, although not every partial isometry has this form, as shown explicitly in the given examples.