A DIFFERENTIAL GEOMETRIC APPROACH TO THE GEOMETRIC MEAN
OF SYMMETRIC POSITIVE-DEFINITE MATRICES
∗
MAHER MOAKHER
†
Submitted to: SIAM J. MATRIX ANAL. APPL.
Abstract. In this paper we introduce metric-based means for the space of positive-definite matrices. The mean associated with the
Euclidean metric of the ambient space is the usual arithmetic mean. The mean associated with the Riemannian metric corresponds to
the geometric mean. We discuss some invariance properties of the Riemannian mean and we use differential geometric tools to give a
characterization of this mean.
Key words. Geometric mean , Positive-definite symmetric matrices, Riemannian distance, Geodesics.
AMS subject classifications. 47A64, 26E60, 15A48, 15A57
1. Introduction. Almost 2500 years ago, the ancient Gree ks defined a list of ten (actually eleven) distinct
“means” [14, 21]. All these means are constructed using geometric proportions. Among these, are the well known
arithmetic, geometric, and harmonic (originally called “subcontrary”) means. These three principal means, which
are used particularly in the works of Nicomachus of Gerasa and Pappus, are the only ones that survived in common
usage.
The arithmetic, geometric and harmonic means, originally defined for two positive numbe rs, generalize natu-
rally to a finite set of positive numbers. In fact, for a set of m positive numbers, {x
k
}
1≤k≤m
, the arithmetic mean is
the positive number ¯x =
1
m
P
m
k=1
x
k
. The arithmetic mean has a variational property; it minimizes the sum of the
squared distances to the given points x
k
(1.1) ¯x = arg min
x>0
m
X
k=1
d
e
(x, x
k
)
2
,
where d
e
(x, y) = |x − y| represents the usual Euclidean distance in IR. Their geometric mean which is given by
˜x =
m
√
x
1
x
2
···x
k
also has a variational property; it minimizes the sum of the squared hyperbolic distances to the
given points x
k
(1.2) ˜x = arg min
x>0
m
X
k=1
d
h
(x
k
, x)
2
,
∗
This work was partially supported by the Swiss National Science Foundation.
†
Laboratory for Mathematical and Numerical Modeling in Engineering Science, National Engineering School at Tunis, Tunis El-Manar
University, ENIT-LAMSIN, B.P. 37, 1002 Tunis-Belv´ed`ere, Tunisia, (Maher.Moakher@enit.rnu.tn).
1
2 M. MOAKHER
where d
h
(x, y) = |log x − log y| is the hyperbolic distance
1
between x and y. Their harmonic mean is simply given
by the inverse of the arithmetic mean of their inverses, i.e., ˆx = [
1
m
P
m
k=1
(x
k
)
−1
]
−1
, and thus it has a variational
characterization as well.
The arithmetic mean has been widely used to average elements of linear Euclidean spaces. Depending on the
application, it is usually referred to as the average, the barycenter or the center of mass. The use of the geometric
mean on the other hand has been limited to positive numbers and positive integrable functions [13, 6]. In 1975,
Anderson and Trapp [2], and Pusz and Woronowicz [20] introduced the harmonic and geometric means for a pair of
positive operators on a Hilbert space. Thereafter, an extensive theory on operator means originated. It has been
shown that the geometric mean of two positive-definite operators shares many of the properties of the geometric mean
of two positive numbers. A recent paper by Lawson and Lim [16] surveys e ight shared properties. The geometric
mean of positive operators has been mainly used as a binary operation.
In [24], there was a discussion about how to define the geometric mean of more than two Hermitian semi-
definite matrices. There have been attempts to use iterative proce dures but none seemed to work when the matrices
do not commute. In [1] there is a definition for the geometric mean of a finite set of operators, however, the given
definition is not invariant under reordering of the matrices. The present author, while working with means of a finite
number of 3-dimensional rotation matrices [18] discovered that there is a close connection between the Riemannian
mean of two rotations and the geometric mean of two Hermitian definite matrices. This observation motivated the
present work on the generalization of the geometric mean for more than two matrices using metric-based means.
In an abstract setting, if M is a Riem annian manifold with metric d(·, ·). Then by analogy to (1.1) and (1.2), a
plausible definition of a mean associated with d(·, ·) of m points in M is given by
(1.3) M(x
1
, . . . , x
k
) := arg min
x∈M
m
X
k=1
d(x
k
, x)
2
.
Note that this definition do es not guarantee that the mean is unique.
As we have seen, for the set of positive real numbers, which is at the same time a Lie group and an open
convex cone
2
, different notions of mean can be associated with different metrics. In what follows, we will extend
these metric-based means to the cone of positive-definite transformations. The methods and ideas used in this
paper carry over to the complex counterpart of the space considered here, i.e., the convex cone of Hermitian definite
transformations. We here concentrate on the real space just for simplicity of exposition but not for any fundamental
reason.
The remainder of this paper is organized as follows. In § 2 we gather all the necessary background from
1
We borrow this terminology from the hyperbolic geometry of the Poincar´e upper half-plane. In fact, the hyperbolic length of the
geodesic segm ent joining the points P (a, y
1
) and Q(a, y
2
), y
1
, y
2
> 0 is | log
y
1
y
2
|, (see [22, 25]).
2
Here and throughout we use the term ope n convex cone, or simply cone, when we really mean the interior of a convex cone.
GEOMETRIC MEAN 3
differential geometry and optimization on manifolds that will be used in the sequel. Further information on this
condensed material can be found in [8, 4, 10, 23, 25]. In § 3 we give a Riemannian-metric based notion of mean for
positive-definite matrices. We discuss some invariance properties of this mean and show that in the case where two
matrices are to be averaged, this mean coincides with the geometric mean.
2. Preliminaries. Let M(n) be the set of n-by-n real matrices and GL(n) be its subset containing only
non-singular matrices. GL(n) is a Lie group, i.e., a group which is also a differentiable manifold and for which
the operations of group multiplication and inverse are smooth. The tangent space at the identity is called the
corresponding Lie algebra and denoted by gl(n). It is the space of all linear transformations in IR
n
, i.e., M(n).
In M(n) we shall use the Euclidean inner product, known as the Frobenius inner product and defined by
hA, Bi
F
= tr(A
T
B), where tr(·) stands for the trace and the superscript
T
denotes the transpose. The associated
norm kAk
F
= hA, Ai
1/2
F
, is used to define the Euclidean distance on M(n)
(2.1) d
F
(A, B) = kA − Bk
F
.
2.1. Exponential and logarithms. The exponential of a matrix in gl(n) is given as usual by the conve rgent
series
(2.2) exp A =
∞
X
k=0
1
k!
A
k
.
We remark that the product of the exponentials of two matrices A and B is equal to exp(A + B) only when A and
B commute.
Logarithms of A in GL(n) are solutions of the matrix equation exp X = A. When A does not have eigenvalues
in the (closed) negative real line, there exists a unique real logarithm, called the principal logarithm and denoted by
Log A, whose spectrum lies in the infinite strip {z ∈ lC : −π < Im(z) < π} of the complex plane [8]. Furthermore,
if, for any given matrix norm k·k, kI − Ak < 1, where I denotes the identity transformation in IR
n
, then the series
−
∞
X
k=1
(I − A)
k
k
converges to Log A and therefore one can write
(2.3) Log A = −
∞
X
k=1
(I − A)
k
k
.
We note that in general Log(AB) 6= Log A + Log B. We here recall the important fact [8]
(2.4) Log(A
−1
BA) = A
−1
(Log B)A.
This fact is also true when Log in the above is replaced by an analytic matrix function.
In what follows, we give the following result which is essential in the development of our analysis.
4 M. MOAKHER
Proposition 2.1. Let X(t) be a real matrix-valued function of the real variable t. We assume that, for all t
in its domain, X(t) is an invertible matrix which does not have eigenvalues on the closed negative real line. Then
d
dt
tr
Log
2
X(t)
= 2 tr
Log X(t)X
−1
(t)
d
dt
X(t)
.
Proof. We recall the following facts:
(i) tr(AB) = tr(BA).
(ii) tr
R
b
a
M (s)ds
=
R
b
a
tr(M (s))ds.
(iii) Log A commutes with [(A − I)s + I]
−1
.
(iv)
R
1
0
[(A − I)s + I]
−2
ds = (I − A)
−1
[(A − I)s + I]
−1
1
0
= A
−1
.
(v)
d
dt
Log X(t) =
R
1
0
[(X(t) − I)s + I]
−1
d
dt
X(t) [(X(t) − I)s + I]
−1
ds.
The facts (i), (ii), (iii) and (iv) are easily checked. See [9] for a proof of (v).
Using the above we have
d
dt
tr
[Log X(t)]
2
(i)
= 2 tr(Log X(t)
d
dt
Log X(t))
(v)
= 2 tr
Log X(t)
Z
1
0
[(X(t) − I)s + I]
−1
d
dt
X(t) [(X(t) − I)s + I]
−1
ds
= 2 tr
Z
1
0
Log X(t) [(X(t) − I)s + I]
−1
d
dt
X(t) [(X(t) − I)s + I]
−1
ds
(ii)
= 2
Z
1
0
tr
Log X(t) [(X(t) − I)s + I]
−1
d
dt
X(t) [(X(t) − I)s + I]
−1
ds
(i)
= 2
Z
1
0
tr
[(X(t) − I)s + I]
−1
Log X(t) [(X(t) − I)s + I]
−1
d
dt
X(t)
ds
(iii)
= 2
Z
1
0
tr
Log X(t) [(X(t) − I)s + I]
−2
d
dt
X(t)
ds
= 2 tr
Log X(t)
Z
1
0
[(X(t) − I)s + I]
−2
ds
d
dt
X(t)
(iv)
= 2 tr
Log X(t)X
−1
(t)
d
dt
X(t)
.
2.2. Gradient and geodesic convexity. For a real-valued function f (x) defined on a Riemannian manifold
M , the gradient ∇f is the unique tangent vector u at x such that
(2.5) hu, ∇fi =
d
dt
f(γ(t))
t=0
,
where γ(t) is a geodesic emanating from x in the direction of u, and h·, ·i denotes the Riemannian inner product on
the tangent space.
A subset A of a Riemannian manifold M is said to be convex if the shortest geodesic curve between any two
points x and y in A is unique in M and lies in A . A real-valued function defined on a convex subset A of M is
GEOMETRIC MEAN 5
said to b e convex if its restriction to any geodesic path is convex, i.e., if t 7→
ˆ
f(t) ≡ f(exp
x
(tu)) is convex over its
domain for all x ∈ M and u ∈ T
x
(M ), where exp
x
is the exponential map at x.
2.3. The cone of the positive-definite symmetric matrices. We denote by
S(n) = {A ∈ M(n), A
T
= A}
the space of all n × n symmetric matrices, and by
P(n) = {A ∈ S(n), A > 0}
the s et of all n × n positive-definite symmetric matrices. Here A > 0 means that the quadratic form x
T
Ax > 0 for
all x ∈ IR
n
\{0}. It is well known that P(n) is an open convex cone, i.e., if P and Q are in P(n), so is P + tQ for
any t > 0.
We recall that the exponential map from S(n) to P(n) is one-to-one and onto. In other words, the exponential
of any symmetric matrix is a positive-definite symmetric matrix and the inverse of the exponential (i.e., the principal
logarithm) of any positive-definite symmetric matrix is a symmetric matrix.
As P(n) is an open subset of S(n), for each P ∈ P(n) we identify the set T
P
of tangent vectors to P(n) at P
with S(n). On the tangent space at P we define the positive-definite inner product and corresponding norm
(2.6) hA, Bi
P
= tr(P
−1
AP
−1
B), kAk
P
= hA, Ai
1/2
P
,
that depend on the point P . The positive definiteness is a consequence of the positive definiteness of the Frobenius
inner product, for hA, Ai
P
= tr(P
−1/2
AP
−1/2
P
−1/2
AP
−1/2
) =
P
−1/2
AP
−1/2
, P
−1/2
AP
−1/2
.
Let [a, b] be a closed interval in IR, and Γ : [a, b] → P(n) be a sufficiently smooth curve in P(n). We define
the length of Γ by
(2.7) L(Γ) :=
Z
b
a
r
D
˙
Γ(t),
˙
Γ(t)
E
Γ(t)
dt =
Z
b
a
q
tr(Γ(t)
−1
˙
Γ(t))
2
dt.
We note that the length L(Γ) is invariant under congruent transformations, i.e., Γ 7→ CΓC
T
, where C is any fixed
element of ∈ GL(n). As
d
dt
Γ
−1
= −Γ
−1
˙
ΓΓ
−1
, one can readily see that this length is also invariant under inversion.
The distance between two matrices A and B in P(n) considered as a differentiable manifold is the infimum
of lengths of curves connecting them
(2.8) d
P(n)
(A, B) := inf {L(Γ) | Γ : [a, b] → P(n) with Γ(a) = A, Γ(b) = B}.
This metric makes P(n) a Riemannian manifold which is of dimension
1
2
n(n + 1). The geodesic emanating from I
in the direction of S, a (symmetric) matrix in the tangent space, is given explicitly by e
tS
[17]. Using unvariance
under congruent transformations, the geodesic P (t) such that P (0) = P and
˙
P (0) = S is therefore given by
P (t) = P
1/2
e
tP
−1/2
SP
−1/2
P
1/2
.
