数理解析研究所講究録
第 1839 巻 2013 年 31-39

31A brief introduction of the Karcher mean
東洋大学

山崎

丈明 (Takeaki Yamazaki)
Toyo University

Abstract. The purpose of this paper is to introduce

some recent topics about Karcher mean. The Karcher
mean is a kind of geometric mean of several matrices,
and an extension of the well-known geometric mean of
two-matrices. In this paper, we shall introduce definition, a way of computation, some inequalities of

Karcher mean, and then we shall introduce how toextend the Karcher mean from matrices to operators.

1. INTRODUCTION
In 1975, W. Pusz and S.L. Woronowicz [21] have defined geometricmean of two-operators. It looks little bit complicated, but it has many
good properties. Then, it has been extended to the theory of operator means by Kubo-Ando [15]. It is known that operator means have
one-to-one correspondence to operator monotone functions. To extend
the theory of operator means to several variable is one of the natural
question. Some extensions have been obtained, however, there were notknown any go $od$ extension of geometric mean of two-operators (we will
explain about “good extension”, below). In 2004, Ando-Li-Mathias [3]have obtained a new definition of geometric mean of several matrices.
It has at least ten good properties, for example, monotonicity, joint
concavity, permutation invariance and arithmetic-geometric means inequality. Then many authors studied about geometric mean of several
variables. Until now we obtain three kinds of the definitions of thegeometric mean of several variables. The one is defined by Ando-LiMathias (we call ALM mean for short). It has been defined by theiteration method. The second one is defined by Bini-Meini-Poloni [5]and Izumino-Nakamura [13] (we call BMP mean), independently, it
is a refinement of ALM mean. It has at least ten same properties of
ALM mean. But it is better to computation than ALM mean [5, 13].
The third one is called Karcher mean or Riemannian geometric mean
[4, 20, 16]. It is defined by the geometrical way, however the sameten properties of ALM or BMP means are satisfied. Moreover, someinequalities of Karcher mean are obtained [12, 23]. In this paper, weshall introduce recent topics about Karcher mean.
This paper is organized as follows: In section 2, we shall introduce
some notations which are used later. In Section 3, we shall introduce
some topics of Karcher mean, definition, basic properties, and somerecent results. The Karcher mean is only defined for matrices, firstly.
To extend the Karcher mean of several matrices to operators, we have32
to consider the power mean. In Section 4, we will introduce the definition and some properties of the power mean. And the last section,
we will introduce how to extend the Karcher mean of several matrices to operators. This paper is a survey paper for introducing recent
topics about matrix means and matrix inequalities, so we will omit
introducing proofs.
2. PRELIMINARY

Let
be the set of all $m-by-m$ matrices, and let
be the set of
all positive invertible matrices in. The famous geometric mean of
two-matrices is defined as follows [21]: Let $A,$
. Then geometric
$A\#
B$
mean
is defined by$\mathbb{M}_{m}$$\mathbb{P}_{m}$$\mathbb{M}_{m}$$B\in \mathbb{P}_{m}$$A\# B=A^{\frac{1}{2}}(A^{\frac{-1}{2}BA^{\frac{-1}{2})^{\frac{1}{2}}A^{\frac{1}{2}}}}.$

Also, weighted geometric mean is known as

for $t\in[O, 1].$

$A\#_{t}B=A^{\frac{1}{2}}(A^{\frac{-1}{2}BA^{\frac{-1}{2})^{t}A^{\frac{1}{2}}}}$

Geometric mean can be defined for positive invertible operators. Let
be a complex Hilbert space, and let
be the algebra of all
bounded linear operators on . We write as a set of all positive.
invertible operators in$B(\mathcal{H})$$\mathcal{H}$$\mathcal{H}$$\mathbb{P}$$B(\mathcal{H})$A vector $\omega=(w_{1}, \cdots, w_{n})\in[0,1]^{n}$ is called a probability vector if
and only if
and $w_{i}>0$ . Let
be the set of all probability
vector.
For $A,$, the Riemannian metric is defined by $\delta_{2}(A, B)=$, wheremeans the trace norm, i.e.,$\sqrt{traceX^{*}X}$
for
Let
and
be the spectral of $X$ which are$A\prec(\log)B$ is definedarranged in the decreasing order. For $A,$
as follows:$\sum_{i=1}^{n}w_{i}=1$$\triangle_{n}$$B\in \mathbb{P}_{m}$$\Vert\log A^{\frac{-1}{2}BA^{\frac{-1}{2}}}\Vert_{2}$$\Vert X\Vert_{2}=$$\Vert_{2}$$\Vert$$X\in \mathbb{M}_{m}.$$X\in \mathbb{P}_{m}$$\lambda_{m}(X)$$\lambda_{1}(X),$$\ldots,$$B\in \mathbb{P}_{m},$for $k=1,2,$$\prod_{i=1}^{k}\lambda_{i}(A)\leq\prod_{i=1}^{k}\lambda_{i}(B)$$\prod_{i=1}^{n}\lambda_{i}(A)=\prod_{i=1}^{n}\lambda_{i}(B)$We call the above relation$\log$$\ldots,$$m-1,$.

-majorisation.

3. KARCHER MEAN
In this section, we shall introduce the definition and basic propertiesof Karcher mean.

33
Definition 1 (Karcher mean [4, 20, 16]). Let
and. Then the Karcher mean (Riemannian geometric mean)$\mathbb{A}=(A_{1}, \cdots, A_{n})\in \mathbb{P}_{m)}^{n}$$\omega\in\triangle_{n}$$\Lambda(\omega, \mathbb{A})$is defined by$\Lambda(\omega, \mathbb{A})fi=ar\min_{\in \mathbb{P}_{m}}\sum_{i=1}^{n}w_{i}\delta_{2}^{2}(A_{i}, X)$.

It is easy to check that for $A,$$\Lambda((1-t, t);A, B)=A\#_{t}B$ . If
, then we write it
for short. As in Definition
1, Karcher mean is defined by geometrical way, the idea is originatedfrom differential geometry [6]. Because the set of all positive invertible$Y\rangle=traceYX^{*}$ for $X,$ $Y\in$matrices equipped with inner productis the Riemannian manifold, and the Riemannian metric is thegeodesic for the Riemannian manifold.
$B\in \mathbb{P}_{m},$$\omega=(\frac{1}{n}, \cdots, \frac{1}{n})\in\triangle_{n}$$\Lambda(\mathbb{A})$$\langle X,$$\mathbb{M}_{m}$The Karcher mean has at least ten good properties. The following properties $(P1)-(P10)$ are satisfied for Karcher mean [4, 20, 16].
Moreover ALM [3] and BMP [5, 13] means also satisfy them. Letand(Pl) Commutative case. If
commute with each other,then$\mathbb{A}=(A_{1}, \ldots, A_{n}),$ $\mathbb{B}=(B_{1}, \ldots, B_{n})\in \mathbb{P}_{m}^{n}$$\omega=(w_{1}, \ldots, w_{n})\in\triangle_{n}.$$A_{n}$$A_{1},$$\ldots,$$\Lambda(\omega;\mathbb{A})=\prod_{i=1}^{n}A_{i}^{w_{i}}.$(P2) Joint homogeneity. Let$a_{1},$$\ldots,$

be positive numbers. Then$a_{n}$.

$\Lambda(\omega;a_{1}A_{1}, \ldots, a_{n}A_{n})=(\prod_{i=1}^{n}a_{i}^{w_{i}})\Lambda(\omega;\mathbb{A})$(P3) Permutation invariance. Let
Then$\sigma$be a permutation on $(1, 2, \cdots, n)$ .

$\Lambda(\omega;\mathbb{A})=\Lambda(w_{\sigma(1)}, \ldots, w_{\sigma(n)};A_{\sigma(1)}, \ldots, A_{\sigma(n)})$(P4) Monotonicity. For each $i=1,2,$$\ldots,$$\Lambda(\omega;\mathbb{A})\leq\Lambda(\omega;\mathbb{B})$(P5) Continuity. For each $i=1,2,$
as $karrow+\infty$ , then$\ldots,$$\Lambda(\omega;A_{1}^{(k)}, \ldots, A_{n}^{(k)})arrow\Lambda(\omega;\mathbb{A})$

$n$

$n$

, assume.

$A_{i}\leq B_{i}$. Then.

, if the sequences$A_{i}^{(k)}arrow A_{i}$as $karrow+\infty$(P6) Joint concavity. For $0\leq\lambda\leq 1,$$\lambda\Lambda(\omega;\mathbb{A})+(1-\lambda)\Lambda(\omega;\mathbb{B})\leq\Lambda(\omega;\lambda \mathbb{A}+(1-\lambda)\mathbb{B})$.

(P7) Congruence invariance. For any invertible matrix $S,$$\Lambda(\omega;S^{*}A_{1}S, \ldots, S^{*}A_{n}S)=S^{*}\Lambda(\omega;\mathbb{A})S.$(P8) Self-duality.
$\Lambda(\omega;A_{1}^{-1}, \ldots, A_{n}^{-1})^{-1}=\Lambda(\omega;\mathbb{A})$.

34(P9) Determinantial identity.
$\det\Lambda(\omega;\mathbb{A})=\prod_{i=1}^{n}(\det A_{i})^{w_{i}}.$(P10) Arithmetic-geometric-harmonic means inequality.
$( \sum_{i=1}^{n}w_{i}A_{i}^{-1})^{-1}\leq\Lambda(\omega;\mathbb{A})\leq\sum_{i=1}^{n}w_{i}A_{i}.$The Karcher mean has been defined by geometrical way, however, it
is known two types of characterizations of the Karcher mean. The firstone is as follows:

Theorem 1 ([20, 16]). Let. Then the Karcher meanandis the unique positive solution of$\mathbb{A}=(A_{1}, \ldots, A_{n})\in \mathbb{P}_{m}^{n}$$\triangle_{n}$(3.1)$\Lambda(\omega;\mathbb{A})$$\omega=(w_{1}, \ldots, w_{n})\in$$\sum_{i=1}^{n}w_{i}\log(x\frac{-1}{2}A_{i}x\frac{-1}{2})=0.$

holds. We
Especially, $\sum_{i=1}^{n}w_{i}\log A_{i}=0$ if and only ifcall (3.1) the Karcher equation [14].
The second characterization is follows: Let. Suppose that we choose a natural number
and$i_{k}\in\{1, \ldots, n\}$ in the k-th independent trial with the probability
as follows:
and let $X_{k}=A_{i_{k}}$ . Define$\Lambda(\omega;\mathbb{A})=I$$\mathbb{A}=(A_{1}, \ldots, A_{n})\in \mathbb{P}_{m}^{n},$$\omega=(w_{1}, \ldots, w_{n})\in\triangle_{n}$$w_{i_{k}},$$\{S_{k}\}$$S_{1}=X_{1}, S_{2}=S_{1}\# X_{2}=X_{1}\# X_{2},$$S_{3}=S_{2}\#_{\frac{1}{3}}X_{3}=(X_{1}\# X_{2})\#_{\frac{1}{3}}X_{3}, \ldots,$$S_{k}=S_{k}\#_{\frac{1}{k+1}}X_{k+1}.$

Theorem 2 ([16, 22]). Almost always$\lim_{karrow\infty}S_{k}=\Lambda(\omega;\mathbb{A})$

Theorem 3 (No dice approach [11]). Let
byDefine the matrix sequenceholds.

$\mathbb{A}=(A_{1}, \ldots, A_{n})\in \mathbb{P}_{m}^{n}.$$\{X_{k}\}$$\{X_{k}\}=\{A_{1}, A_{2}, \ldots, A_{n}, A_{1}, A_{2}, \ldots, A_{n}, A_{1}, A_{2}, \ldots\}.$Then$\lim_{karrow\infty}S_{k}=\Lambda(\mathbb{A})$always holds.
There are a lot of matrix inequalities related to geometric mean of
two matrices. However, ALM and BMP means can not extend them.
Until now, Karcher mean can extend them to inequalities for several
variables.

35
Theorem 4 ([23]). Let. Then$\mathbb{A}=(A_{1}, \cdots, A_{n})\in \mathbb{P}_{m}^{n}$, and $\omega=(w_{1}, \cdots, w_{n})\in$$\triangle_{n}$$\sum_{i=1}^{n}w_{i}\log A_{i}\leq 0\Rightarrow\Lambda(\omega;\mathbb{A})\leq I.$

Remark. Theorem 4 is an extension of the well-known result shownin [1, 9, 8] as a two-matrices case: Let $A,$
. Then$B\in \mathbb{P}_{m}$(3.2)$\log B\leq\log A\Rightarrow(A^{\frac{1}{2}}BA^{\frac{1}{2}})^{\frac{1}{2}}\leq A.$

Moreover the above (3.2) is well known result as the most essential
inequality of the famous Furuta inequality [7]. Hence, we obtain an

extension of FUruta inequality for several variables by using Theorem
4.
Theorem 5 ([12]). Let
and $q>0$ . Then$\mathbb{A}=(A_{1}, \ldots, A_{n})\in \mathbb{P}_{m}^{n}$$A_{i}^{q}\geq A_{n}^{q}>0(i=1, \ldots, n-1)$

implies$\Lambda(\omega;A_{1}^{-p_{1}}, \ldots, A_{n-1}^{-p_{n-1}}, A_{n}^{p_{n}})\leq A_{n}^{q}\leq A_{i}^{q}$

for allp,
and$i=1,$$\geq 0,$$\ldots,$$n-1$ and $p_{n}>q$ , where$\hat{\omega}=(\frac{1}{p_{1}+q}, \ldots,\frac{1}{p_{n-1}+q},\frac{n-1}{p_{n}-q})$$\omega=\frac{\hat{\omega}}{\Vert\hat{\omega}\Vert_{1}}.$

In fact, Furuta inequality can be obtained as a two-matrices case of
Theorem 5, that is, if $B\leq A$ , then$A^{\frac{-r}{2}(A^{\frac{r}{2}}B^{p}A^{\frac{r}{2}})^{\frac{1+r}{p+r}A^{\frac{-r}{2}}}}= \Lambda((\frac{p-1}{p+r},\frac{1+r}{p+r});A^{-r}, B^{p})\leq B\leq A$

holds for $p\geq 1,$ $r\geq 0.$
The Karcher mean extends so-called Ando-Hiai inequality. Let. For.
Theorem 6 ([23]). Let, and $\omega=(w_{1}, \cdots, w_{n})\in$. Then$\mathbb{A}=$$(A_{1}, \ldots, A_{n})\in \mathbb{P}_{m}^{n}$$p\in \mathbb{R},$$\mathbb{A}^{p}$$:=(A_{1}^{p}, \ldots, A_{n}^{p})$$\mathbb{A}=(A_{1}, \cdots, A_{n})\in \mathbb{P}_{m}^{n}$$\triangle_{n}$$\Lambda(\omega;\mathbb{A})\leq I\Rightarrow\Lambda(\omega;\mathbb{A}^{p})\leq Iforp\geq 1.$

In fact, we can obtain Ando-Hiai inequality in the two-matrices case[2],
for $p\geq 1$
Using the anti-symmetric tensor technique to Theorem 6, the following extension of the main result in [2] holds:
Theorem 7 ([10]). Let, and $\omega=(w_{1}, \cdots, w_{n})\in$. Then$A\#_{\alpha}B\leq I\Rightarrow A^{p}\#_{\alpha}B^{p}\leq I$$\mathbb{A}=(A_{1}, \cdots, A_{n})\in \mathbb{P}_{m}^{n}$$\triangle_{n}$$\Lambda(\omega;\mathbb{A})\prec(log)\Lambda(\omega;\mathbb{A}^{p})^{\frac{1}{p}}$holds for all $p\in(0,1)$ .
Moreover the following $log$ -majorisation result holds.
$\Lambda(\omega;\mathbb{A})\prec(\log)\exp(\sum_{i=1}^{n}w_{i}\log A_{i})$36
4. POWER MEANS
As in the definition of Karcher mean, the Karcher mean is definedfor only positive definite matrices. To define the Karcher mean forbounded linear operators, we need to consider the following power meanof several operators in$\mathbb{P}.$Definition 2 (Power mean, [18, 17]). Let$t\in(0,1], the \omega-$ weighted powerby the unique positive solution of

and
order
of
of

$\mathbb{A}\in \mathbb{P}^{n}$$mean P_{t}(\omega;\mathbb{A})$$\omega\in\triangle_{n}$$t$$\mathbb{A}$. For

is defined$X= \sum_{i=1}^{n}w_{i}(X\#_{t}A_{i}) (orI=\sum_{i=1}^{n}w_{i}(x\frac{-1}{2}A_{i}X^{\frac{-1}{2}})^{t})$For $t\in[-1,0)$ , define$P_{t}(\omega;\mathbb{A})=P_{-t}(\omega;\mathbb{A}^{-1})^{-1}.$

are shown in [18]. For a provExistence and uniqueness of, we write it
for short. By
ability vector
if and only if
the definition of power mean, we obtain$P_{t}(\omega;\mathbb{A})$$\omega=(\frac{1}{n}, \ldots, \frac{1}{n})\in\triangle_{n}$$P_{t}(\mathbb{A})$$\sum_{i=1}^{n}w_{i}A^{t}=I$holds, easily.
Power mean satisfies the following properties [18]: Let $a=(a_{1}, \ldots, a_{n})\in$
and $M$ be an invertible operator oncommute with each other,(Pl) Commutative case. Ifthen$P_{t}(\omega;\mathbb{A})=I$$(0, \infty)^{n},$$\mathbb{A},$$\mathbb{B}\in \mathbb{P}^{n},$$\omega\in\Delta_{n}$$\mathcal{H}.$$A_{n}$$A_{1},$$\ldots,$$P_{t}( \omega;\mathbb{A})=(\sum_{i=1}^{n}w_{i}A_{i}^{t})^{\frac{1}{t}}$(P2) Scalar multiple. Let$a_{1},$$\ldots,$$a_{n}$

be positive numbers. Then

$P_{t}( \omega;a\mathbb{A})=(\sum_{i=1}^{n}w_{i}a_{i}^{t})^{\frac{1}{t}}P_{t}(\omega a^{t};\mathbb{A})$.
where(P3) Permutation invariance. Let,$\omega a^{t}=(a_{1}^{t}w_{1}, \ldots, a_{n}^{t}w_{n})$$\sigma$be a permutation on $(1, 2, \cdots, n)$ .

Then$P_{t}(\omega;\mathbb{A})=P_{t}(w_{\sigma(1)}, \ldots, w_{\sigma(n)};A_{\sigma(1)}, \ldots, A_{\sigma(n)})$(P4) Monotonicity. For each $i=1,2,$$P_{t}(\omega;\mathbb{A})\leq P_{t}(\omega;\mathbb{B})$(P5) Continuity. For each $i=1,2,$
as $karrow+\infty$ , then$\ldots,$$n$$P_{t}(\omega;A_{1}^{(k)}, \ldots, A_{n}^{(k)})arrow P_{t}(\omega;\mathbb{A})$(P6) Joint concavity. For$n$$\ldots,$, assume.

$A_{i}\leq B_{i}$. Then.

, if the sequences$A_{i}^{(k)}arrow A_{i}$as $karrow+\infty$$0\leq\lambda\leq 1,$$(1-\lambda)P_{|t|}(\omega;\mathbb{A})+\lambda P_{|t|}(\omega;\mathbb{B})\leq P_{|t|}(\omega;(1-\lambda)\mathbb{A}+\lambda \mathbb{B})$.

37(P7) Congruence invariance.
$P_{t}(\omega;M\mathbb{A}M^{*})=MP_{t}(\omega;\mathbb{A})M^{*}$(P8) Self-duality.
$P_{t}(\omega;\mathbb{A}^{-1})^{-1}=P_{-t}(\omega;\mathbb{A})$(P9) Determinantial inequalities. If.

$A_{1},$$A_{n}\in \mathbb{P}_{m}$$\ldots,$, then.

$\det P_{-|t|}(\omega;\mathbb{A})\leq\prod_{i=1}^{n}\det A_{i}^{w_{i}}\leq\det P_{|t|}(\omega;\mathbb{A})$(P10) Arithmetic-power-harmonic means inequality.
$( \sum_{i=1}^{n}w_{i}A_{i}^{-1})^{-1}\leq P_{t}(\omega;\mathbb{A})\leq\sum_{i=1}^{n}w_{i}A_{i}.$It is obvious thatandare arithmetic and harmonic means, respectively. Moreover power mean interpolates thesemeans. For $G,$ $H$ :, we define$P_{1}(\omega;\mathbb{A})$$P_{-1}(\omega;\mathbb{A})$$\triangle_{n}\cross \mathbb{P}^{n}arrow \mathbb{P}$$G\leq H$

for all$\omega\in\triangle_{n}$

and

if

$G(\omega;\mathbb{A})\leq H(\omega;\mathbb{A})$$\mathbb{A}\in \mathbb{P}^{n}.$Theorem 8 ([18, 17, 19]). For $0<s\leq t\leq 1,$$\mathcal{H}=P_{-1}\leq P_{-t}\leq P_{-s}\leq\cdots\leq P_{s}\leq P_{t}\leq P_{1}=\mathcal{A}.$Furthermore, the limit of power means, then itwith the Karcher mean, i. e.,$\lim_{tarrow 0}P_{t}(\omega;\mathbb{A})$exists.

$coincide\mathcal{S}$$\mathbb{P}_{m}^{n}$If$\mathbb{A}\in$$\lim_{tarrow 0}P_{t}(\omega;\mathbb{A})=\Lambda(\omega;\mathbb{A})$.

Here we shall consider matrix inequalities similar to Theorem 4. Define$Q_{t}( \omega;\mathbb{A})=(\sum_{i=1}^{n}w_{i}A_{i}^{t})^{\frac{1}{t}}$Theorem 9 ([19]). Let. Then(1) forWe call$Q_{t}(\omega;\mathbb{A})$$\mathbb{A}=(A_{1}, \ldots, A_{n})\in \mathbb{P}_{m}^{n}$quasi-arithmetic mean.

and let $\omega=(w_{1}, \ldots, w_{n})\in$$\triangle_{n}$;$t\in(O, 1], Q_{t}(\omega;\mathbb{A})\leq I\Rightarrow P_{t}(\omega;\mathbb{A})\leq I$(2) for $t\in[-1,0),$

It is well known that$Q_{t}(\omega;\mathbb{A})\geq I\Rightarrow P_{t}(\omega;\mathbb{A})\geq I.$$\lim_{t\searrow 0}Q_{t}(\omega;\mathbb{A})=\exp(\sum_{i=1}^{n}w_{i}\log A_{i})$. Hence by

Theorem 8, we can consider Theorem 9 as a limit of Theorem 4 asfollows:$\lim_{t\searrow 0}Q_{t}(\omega;\mathbb{A})=\exp(\sum_{i=1}^{n}w_{i}\log A_{i})\leq I\Rightarrow\lim_{t\searrow 0}P_{t}(\omega;\mathbb{A})=\Lambda(\omega;\mathbb{A})\leq I.$

Moreover we obtain that for. However it is not known whether this norm inequality holds for all unitarily
invariant norm or not.
$t\in(0,1], 1P_{t}(\omega;\mathbb{A})\Vert\leq\Vert Q_{t}(\omega;\mathbb{A})\Vert$

38
Theorem 10 (Ando-Hiai property, [19]). Let$\omega=(w_{1}, \ldots, w_{n})\in\triangle_{n}andt\in(O, 1]$ . Then
implies(1)
for all $p\geq 1,$
implies(2)
for all $p\geq 1.$$\mathbb{A}=(A_{1}, \ldots, A_{n})\in \mathbb{P}_{m}^{n}$$P_{t}(\omega;\mathbb{A})\leq I$,$P_{\frac{t}{p}}(\omega;\mathbb{A}^{p})\leq I$$P_{-t}(\omega;\mathbb{A})\geq I$$P_{-\frac{t}{p}}(\omega;\mathbb{A}^{p})\geq I$

Theorem 10 can be also considered as a limit of Theorem 6.

5. KARCHER MEAN FOR INFINITE DIMENSIONAL OPERATORS
As in the definition of Karcher mean, it is defined in only matrices.
In this section, we shall introduce a way to extend the Karcher meanof -matrices to -operators. The main idea is to define Karcher mean
as a unique positive solution of Karcher equation.
$n$

$n$

Theorem 11 ([17]). For any, the Karcher equation$\mathbb{A}=(A_{1}, \ldots, A_{n})\in \mathbb{P}^{n}$

and $\omega=(w_{1}, \ldots, w_{n})\in$$\triangle_{n}$$\sum_{i=1}^{n}w_{i}\log(x\frac{-1}{2}A_{i}x\frac{-1}{2})=0$

has positive definite solution.
Especially,tion.

$\Lambda(\omega;\mathbb{A})=\lim_{tarrow 0}P_{t}(\omega;\mathbb{A})$

Theorem 12 ([17]). For any, the Karcher equation

is a solution of the Karcher equa-$\mathbb{A}=(A_{1}, \ldots, A_{n})\in \mathbb{P}^{n}$and$\omega=(w_{1}, \ldots, w_{n})\in$$\triangle_{n}$$\sum_{i=1}^{n}w_{i}\log(x\frac{-1}{2}A_{i}x\frac{-1}{2})=0$

has unique positive definite solution.

Definition 3 ([17]). Let. Then the Karcher mean
solution of the Karcher equation

and $\omega=(w_{1}, \ldots, w_{n})\in$
is defined by the unique positive$\mathbb{A}=(A_{1}, \ldots, A_{n})\in \mathbb{P}^{n}$$\triangle_{n}$$\Lambda(\omega;\mathbb{A})$$\sum_{i=1}^{n}w_{i}\log(X^{i^{1}}A_{i}X^{\frac{-1}{2})=0}.$In this case, the Karcher mean also satisfies all ten properties (Pl)(P10) mentioned above [17].

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DEPARTMENT OF ELECTRICAL, ELECTRONIC AND COMPUTER ENGINEERING,TOYO UNIVERSITY, KAWAGOE 350-8585, JAPAN-mail address: t-yamazaki@toyo.jp$E$