Polar Decomposition

Okay, let’s take the singular value decomposition and do something really neat with it. Specifically, we’ll start with an endomorphism $T:A\rightarrow A$ and we’ll write down its singular value decomposition

$\displaystyle T=V\Sigma W^*$

where $V$ and $W$ are in the unitary (orthogonal) group of $A$ . So, as it turns out, $U=VW^*$ is also unitary. And $P=W\Sigma W^*$ is positive-semidefinite (since $\Sigma$ is). And, of course, since $W^*W=1_A$ , we can write

$\displaystyle T=V\Sigma W^*=VW^*W\Sigma W^*=UP$

That is, any endomorphism can be written as the product of a unitary transformation and a positive-semidefinite one.

Remember that unitary transformations are like unit complex numbers, while positive-semidefinite transformations are like nonnegative real numbers. And so this “polar decomposition” is like the polar form of a complex number, where we write a complex number as the product of a unit complex number and a nonnegative real number.

We can recover the analogy like we did before, by taking determinants. We find

$\displaystyle\det(T)=\det(U)\det(P)=e^{i\theta}r$

since the determinant of a unitary transformation is a unit complex number, and the determinant of a positive-semidefinite transformation is a nonnegative real number. If $T$ is nonsingular, so $P$ is actually positive-definite, then $\det(P)$ will be strictly positive, so the determinant of $T$ will be nonzero.

We could also define $P'=W\Sigma W^*$ , so $T=P'U$ . This is the left polar decomposition (writing the positive-definite part on the left), where the previous form is the right polar decomposition

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August 19, 2009 - Posted by John Armstrong | Algebra, Linear Algebra

7 Comments »

What if I don’t want to ” take the singular value decomposition”? Your entire argument collapses.

NS

Comment by notedscholar | August 19, 2009 | Reply
That’s the nice thing about mathematics. My argument doesn’t collapse just because you hold your breath and stamp your feet.

Comment by John Armstrong | August 19, 2009 | Reply
Is the decomposition T=UP unique in any sense?

Comment by Å | August 20, 2009 | Reply
The positive-definite part $P$ is unique (I’ll be able to say more about that after today’s post), and from that point $U$ is also unique exactly when $T$ (and thus $P$ ) is nonsingular.

Comment by John Armstrong | August 20, 2009 | Reply
[...] Uniqueness of Polar Decomposition I asserted in a comment that the polar decomposition of a transformation has certain uniqueness properties. First, we can always uniquely recover the [...]

Pingback by The Uniqueness of Polar Decomposition « The Unapologetic Mathematician | August 21, 2009 | Reply
Right. But it does collapse if we don’t grant your premises.

NS

Comment by notedscholar | August 21, 2009 | Reply
[...] used this to show in particular if is an endomorphism on an inner product space we can write where is unitary and [...]

Pingback by Decompositions Past and Future « The Unapologetic Mathematician | August 24, 2009 | Reply

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