# Polar decomposition

This post proves the existence of polar decomposition.

The polar decomposition is a generalization of the polar form of complex numbers .

Theorem 1 (Existence of polar decomposition)Let be an complex matrix. Then , where is a positive semi-definite Hermitian matrix, and is unitary. Furthermore, is uniquely determined by .

Lemma 1There exists a unique positive semi-definite square root for a positive semi-definite Hermitian matrix .

**Proof**

Existence: By spectral theorem, there exists unitary such that is diagonal. Take the non-negative square root of each diagonal entry (remember that a Hermitian matrix admits real eigenvalues only), we get a diagonal matrix , such that . Then .

Uniqueness: Suppose that . BY spectral theorem, there exists an orthonormal eigenbasis for the left multiplication map by , i.e. for . Squaring, we get for all Since is uniquely determined by (its eigenvalues), are also uniquelydetermined by , asserting the uniqueness of positive semi-definite square root.

**Proof of Theorem 1
**

Notice that is a Hermitian matrix. By spectral theorem, there exists an orthonomal eigenbasis for the left multiplication map by . Let . If is the complex inner product, we have

This means that is an orthogonal set. Extend this to an orthogonal basis, and make it orthonomal: , i.e. if , then .

Then it is trivial to see that the map can be decomposed to . The first operation is a unitary one because it is a change of orthonomal bases. The second operation is a mere scaling up, so it is positive semi-definite.

**Remark**

If we suppose that such a decomposition exists, then , i.e. is the positive semi-definite Hermitian square root of , thus is unique.