A polar decomposition of a bounded operator 
on a Hilbert
space 
is a factorization 
,
where 
is the unique positive operator square root
of 
and 
is a partial isometry. Here 
is the adjoint operator.
There is a unique such 
satisfying 
. Its initial subspace
is the orthogonal complement of 
, and its final subspace is
the closure of the range of
.
If 
is invertible, 
is unitary (Courtney
et al. 2024, pp. 20-21).
For a square matrix, the factor 
can instead be chosen to be a unitary
matrix even when 
is singular. In that case
it is not unique and does not satisfy the canonical kernel
condition above (Higham 2020).
-Algebras. GOALS lecture notes,
pp. 4 and 20-21, 2024.