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 1 There 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.
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