Let
A
=
U
P
A=UP
be a polar decomposition of an
n
×
n
n\times n
complex matrix
A
A
. Then for every unitarily invariant norm
|
|
|
⋅
|
|
|
|||\cdot |||
, it is shown that
\[
|
|
|
|
U
P
−
P
U
|
2
|
|
|
≤
|
|
|
A
∗
A
−
A
A
∗
|
|
|
≤
‖
U
P
+
P
U
‖
|
|
|
U
P
−
P
U
|
|
|
,
|||\, |UP-PU|^2||| \le |||A^*A-AA^*|||\le \|UP+PU\|\,|||UP-PU|||,
\]
where
‖
⋅
‖
\|\cdot \|
denotes the operator norm. This is a quantitative version of the well-known result that
A
A
is normal if and only if
U
P
=
P
U
UP=PU
. Related inequalities involving self-commutators are also obtained.