Digital Signal Processing Reference
In-Depth Information
A
−
1
B
1
/
2
A
(
)
(2.13)
turns out to be equal to the one in (
2.9
). This matrix was introduced in [
15
]asthe
geometric mean of
A
and
B
.
In 1979 T. Ando published a very important paper [
1
] that brought the geometric
mean to the attention of a large community. Among other things, Ando showed that
is positive there is
a maximum, and this maximum is equal to the geometric mean. In other words,
2 block matrix
AX
XB
among all Hermitian
X
for which the 2
×
max
X
AX
XB
0
A
#
B
=
:
≥
.
(2.14)
Ando used this characterisation to prove several striking results about convexity
of some matrix functions that are important in matrix analysis and quantum theory.
He highlighted the inequality between the harmonic, geometric and arithmetic means:
A
−
1
−
1
B
−
1
+
A
+
B
≤
A
#
B
≤
,
(2.15)
2
2
and the fact that
A
#
B
is a jointly concave function of
A
and
B
.
(2.16)
We remark here that the matrix
AX
XB
is positive if and only if there exists a
A
1
/
2
KB
1
/
2
contraction
K
such that
X
The maximal
X
is characterised by the
fact that the
K
occurring here is unitary. In other words
=
.
A
1
/
2
UB
1
/
2
A
#
B
=
,
(2.17)
where
U
is unitary, and this condition determines
A
#
B
.
The paper of Ando was followed by the foundational work of Kubo and Ando [
19
]
where an axiomatic framework is laid down for a general theory of binary matrix
means.
With the success of this work it was natural to look for a good definition of a
geometric mean of more than two positive matrices. This turned out to be a tricky
problem resisting solution for nearly 25 years. Once again the arithmetic and the
harmonic means of
m
positive matrices can be defined in the obvious way as
⎛
⎞
−
1
m
m
1
m
1
m
⎝
A
−
1
j
⎠
A
j
and
,
j
=
1
j
=
1
