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11.6: Polar decomposition

( \newcommand{\kernel}{\mathrm{null}\,}\)

Continuing the analogy between C and L(V), recall the polar form of a complex number z=|z|eiθ, where |z| is the absolute value or modulus of z and eiθ lies on the unit circle in R2. In terms of an operator TL(V), where V is a complex inner product space, a unitary operator U takes the role of eiθ, and |T| takes the role of the modulus. As in Section11.5, TT0 so that |T|:=TT exists and satisfies |T|0 as well.

Theorem 11.6.1. For each TL(V), there exists a unitary U such that

T=U|T|.

This is called the polar decomposition of T.

Proof. We start by noting that

Tv2=|T|v2,

since Tv,Tv=v,TTv=TTv,TTv. This implies that null(T)=null(|T|). By the Dimension Formula, this also means that dim(range(T))=dim(range(|T|)). Moreover, we can define an isometry S:range(|T|)range(T) by setting

S(|T|v)=Tv.

The trick is now to define a unitary operator U on all of V such that the restriction of U onto the range of |T| is S, i.e.,
U|range(|T|)=S.

Note that null(|T|)range(|T|), i.e., for vnull(|T|) and w=|T|urange(|T|),

w,v=|T|u,v=u,|T|v=u,0=0

since |T| is self-adjoint.

Pick an orthonormal basis e=(e1,,em) of null(|T|) and an orthonormal basis f=(f1,,fm) of (range(T)). Set ˜Sei=fi, and extend ˜S to all of null(|T|) by linearity. Since null(|T|)range(|T|), any vV can be uniquely written as v=v1+v2, where v1null(|T|) and v2range(|T|). Now define U:VV by setting Uv=˜Sv1+Sv2. Then U is an isometry. Moreover, U is also unitary, as shown by the following calculation application of the Pythagorean theorem:

Uv2=˜Sv1+Sv22=˜Sv12+Sv22=v12+v22=v2.

Also, note that U|T|=T by construction since U|null(|T|) is irrelevant.




This page titled 11.6: Polar decomposition is shared under a not declared license and was authored, remixed, and/or curated by Isaiah Lankham, Bruno Nachtergaele, & Anne Schilling.

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