数理解析研究所講究録
第 1893 巻 2014 年 122-125122正定値行列の幾何構造について
On geometric structure of positive definite matrices
大阪教育大学教養学科情報科学

藤井淳一

Jun Ichi Fujii
Departments of Arts and Sciences (Information Science)
Osaka Kyoiku University

In this note, from the viewpoint of Corach-Porta-Recht [3, 4], we discuss a Riepositive definite matrices $C(n)$ by Bhatia-Holbrookmannian geometry for the[2], say the CPRBH geometry: The principal fiber bundle is the regular matrices
with the unitary groupas the structure one and the projection$\pi(X)=XX^{*}$ . The fiber at $A\in C(n)$ is $\pi^{-1}(A)=\sqrt{A}\mathcal{U}(n)$ and the Rimannian. It was shown in [4] that the path of
metric $g_{A}(X, Y)=$ tr $(A^{-1}XA^{-1}Y)$ at
the geometric means$n\cross n$$\mathcal{G}=\mathcal{G}(n)$$\mathcal{U}(n)$$A\in \mathcal{C}(n)$$A\#_{t}B=A^{\frac{1}{2}}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^{t}A^{\frac{1}{2}}$is the geodesic from at $t=0$ to at $t=1.$
The manifold $C(n)$ is a homogeneous spacewith the involution $\sigma(T)=$. The differential $d\sigma(Z)=-Z^{*}$ forforis theCartan involution with the Cartan decomposition as a Lie group and a Lie algebra;
$B$

$A$

$\mathcal{G}(n)/\mathcal{U}(n)$$(T^{*})^{-1}$$Z\in \mathcal{T}(\mathcal{G}(n))=\mathcal{M}_{n}$$T\in \mathcal{G}(n)$$\mathcal{G}(n)=\mathcal{U}(n)C(n)$,【 $(n)=u(n)\oplus \mathcal{T}C(n)=u(n)\oplus iu(n)$$\mathfrak{g}$where the Lie algebra $u(n)$ is the skew-hermitian matrices and the tangent bundle
is the hermitian ones. In fact,
is the Cartan involution since$d\sigma$$\mathcal{T}C(n)$$-B(X, d\sigma(X))=tradX$ ad $X^{*}=2ntrXX^{*}-2trXtrX^{*}\geqq 0$

where is the Killing form.
It is related to the connection in : The vertical space in the tangent space
is $\sqrt{A}Uu(n)$ and the horizontal one isfor some unitary . In fact, for an
invertible matrix , the orthogonal decomposition at $T=\sqrt{A}U$ is$B$$\mathcal{T}\pi^{-1}(A)$$\mathcal{G}$$U$$\sqrt{A}U\mathcal{T}C(n)$$G$$G= \frac{T(T^{-1}G-G^{*}(T^{*})^{-1})}{2}+\frac{T(T^{-1}G+G^{*}(T^{*})^{-1})}{2}$Thereby the horizontal lifthorizontal condition is$\Gamma$of$\gamma$should satisfy that$\dot{\Gamma}\Gamma^{*}=\Gamma\dot{\Gamma}^{*}$$\Gamma^{-1}\dot{\Gamma}$

is hermitian, i.e., the123
Moreover, as P\’alfia [12] pointed, $C(n)$ is a symmetric space with the symmetry
at $A\in C(n)$ satysfying $s_{A}(B)=AB^{-1}A$ . The Cartan decomposition shows that asymmetric spaceis the real form and its dual symmetric$s_{A}$$\mathcal{U}(n)=\mathcal{U}(n)\cross \mathcal{U}(n)/\triangle \mathcal{U}(n)$space

is $C(n)$ itself whereis the diagonal subspace andis thecomplexification of. This shows that it is not compact and the sectional curvature
is non-positive, that is $C(n)$ is a $CAT$ $(O)$ -space. Let and be geodesics. If$\mathcal{U}(n)_{\mathbb{C}}/\mathcal{U}(n)$$\triangle \mathcal{U}(n)$$\mathcal{U}(n)_{\mathbb{C}}$$\mathcal{U}(n)$$\delta$$\gamma$$d( \gamma(1/2), \delta(1/2))\leq\frac{d(\gamma(1),\delta(1))}{2}$always holds, then it is said that Busemann curvatures are non-positive. If$d^{2}(Z, \gamma(t))\leqq(1-t)(d^{2}(Z, \gamma(0))+td^{2}(Z, \gamma(1)))-t(1-t)d^{2}(\gamma(0), \gamma(1))$

always holds, it is said that Alexandrov curvatures are non-positive. This inequality iscalled Courbure n\’egative one or semi-parallelogram law for the case $t=1/2$ ([1]). In
the Riemannian case, they are equivalent to nonpositivity of sectional curvature [10].
Moreover, $C(n)$ is (simply connected) complete space, it is called Hadamard manifold.
Then it is known that $F(t)=d(\gamma(t), \delta(t))$ is convex.
Since every symmetric space is geodesically complete (hence we also have that it iscomplete as a metric space), the extended curve$a$$\gamma(t)=A\natural_{t}B=A^{\frac{1}{2}}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^{t}A^{\frac{1}{2}}$for $t\in(-\infty, \infty)$ is the geodesic including $A\#_{t}B$ . Then we have the parallel translatealong the geodesic is given by

Theorem. One of the horizontal lift of the geodesic$\gamma(t)=A\natural_{t}B$is$\Gamma(t)=A^{\frac{1}{2}}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^{\frac{t}{2}}$and the parallel translate
given by

$P_{t}^{s}$from$\gamma(s)$to$\gamma(t)$along$\gamma$in the tangent bundle$P_{t}^{S}X=A^{\frac{1}{2}}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^{\frac{t-\epsilon}{2}}A^{-\frac{1}{2}}XA^{-\frac{1}{2}}(A^{-\frac{1}{2}}BA- \frac{1}{2})^{\frac{t-s}{2}}A^{\frac{1}{2}}.$

Proof. By$\pi(\Gamma(t))=\Gamma(t)\Gamma(t)^{*}=A^{\frac{1}{2}}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^{t}A^{\frac{1}{2}}=\gamma(t)$$\Gamma$

is a lift of . The horizontality follows from the fact that$\gamma$$2\Gamma(t)^{-1}\dot{\Gamma}(t)=2\dot{\Gamma}(t)^{*}\Gamma^{*}(t)^{-1}=\log(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})$,$\mathcal{T}C(n)$is124
is hermitian. The parallel translate of $X$ from$s$

to is$t$$P_{t}^{s}X=\Gamma(t)\Gamma(s)^{-1}X(\Gamma(s)^{-1})^{*}\Gamma(t)^{*}$$=A^{\frac{1}{2}}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^{\frac{t-s}{2}}A^{-\frac{1}{2}}XA^{-\frac{1}{2}}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^{\frac{t-\epsilon}{2}}A^{\frac{1}{2}}.

Condider the triangle closed pathof $XisV^{*}XV$ for$Iarrow^{A^{t}}$$A^{A\# t^{BB^{1-t}}}arrow Barrow I$\square$. Then the parallel translate$V=A^{\frac{1}{2}}A^{-\frac{1}{2}}C^{\frac{1}{2}}A^{\frac{1}{2}}B^{-\frac{1}{2}}=A^{\frac{1}{2}}A^{-\frac{1}{2}}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^{\frac{1}{2}}A^{\frac{1}{2}}B^{-\frac{1}{2}}=(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^{\frac{1}{2}}A^{\frac{1}{2}}B^{-\frac{1}{2}}.$Thus, $V^{*}V=I$ and $\det V=(\det A)^{0}(\det B)^{0}=1$ , so thatany loop by a polygon of geodesics, we have:
Corollary. The holonomy group of $C(n)$ is included by$V\in S\mathcal{U}(n)$$S\mathcal{U}(n)$. Approximating.

Remark. In virtue of the Ambrose-Singer theorem, P\’alfia [12] showed that they coinside, which might be already known.
via the Lie algebra$\epsilon u(n)$In this geometry, the tangent vector at$\gamma(t)$is given by (cf. [9])$S_{t}(A|B)=A^{\frac{1}{2}}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^{t}\log(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})A^{\frac{1}{2}},$in particular, the tangent one at $t=0$ is the relative operator entropy [5, 6]:$S(A|B)=S_{0}(A|B)=A^{\frac{1}{2}}\log(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})A^{\frac{1}{2}}.$For the above lift , the horizontal condition is now$\Gamma$$2\Gamma(t)^{-1}\dot{\Gamma}(t)=2\dot{\Gamma}(t)^{*}\Gamma^{*}(t)^{-1}=A^{-\frac{1}{2}}S(A|B)A^{-\frac{1}{2}}.$

Recently E.Kamei pointed in a seminar talk that the tangent vector at$S_{r}(A|B)=(A\natural_{r}B)(A\natural_{t}B)^{-1}S_{t}(A|B)$$r$.

shows the parallel translate of the tangent vector $S_{t}(A|B)$ to $S_{r}(A|B)$ . In fact, for, we have
$C=A^{-\frac{1}{2}}BA^{-\frac{1}{2}}$$\Gamma(r)\Gamma(t)^{-1}S_{t}(A|B)\Gamma(t)^{-1}\Gamma(r)=A^{\frac{1}{2}}c\frac{r-t}{2}A^{-\frac{1}{2}}S_{t}(A|B)A^{-\frac{1}{2}}c\frac{r-t}{2}A^{\frac{1}{2}}$$=A^{\frac{1}{2}}c \frac{r-t}{2}C^{t}(\log C)c\frac{r-\ell}{2}A^{\frac{1}{2}}=A^{\frac{1}{2}}C^{r}\log CA^{\frac{1}{2}}=S_{r}(A|B)$.

125
In Hadamard manifolds, the parallel geodesics are defined by the boundedness;$d(\gamma(t), \delta(t))<\exists M$for all(it is also called asymptotic). But the parallel translates for the parallel
vectors along parallel geodesics are not always parallel. So, considering flat geometry in $C(n)$ , we need -commutativity ([2]): $A,$andare -commute if matrices
commute. It is equivalent to the commutativity of matrices$t\in \mathbb{R}$$\Gamma$$C^{-\frac{1}{2}}AC^{-\frac{1}{2}},$$B$$C$$\Gamma$$C^{-\frac{1}{2}}BC^{-\frac{1}{2}}$$X^{-\frac{1}{2}}AX^{-\frac{1}{2}}, X^{-\frac{1}{2}}BX^{-\frac{1}{2}}, X^{-\frac{1}{2}}CX^{-\frac{1}{2}}$for some $X.$

参考文献
[1] R.Bhatia, “Positive Definite Matrices”, Princeton Univ. Press, 2007.
[2] R.Bhatia and J.A.R.Holbrook, Riemannian geometry and matrix geometric means, Linear Algebra. Appl. 423 (2006), 594-618.
[3] G.Corach, H.Porta and L.Recht, Geodesics and operator means in the space of positiveoperators. Internat. J. Math. 4 (1993), 193-202.

[4] G. Corach and A.L.Maestripieri, Differential and metrical structure of positive operators,Positivity 3 (1999), 297-315.

[5] J.I.Fujii and E.Kamei, Relative operator entropy in noncommutative information theory,
Math. Japon. 34 (1989), 341-348.
[6] J.I.Fujii and E.Kamei, Uhlmann’s interpolational method for operator means, Math.
Japon. 34 (1989), 541-547.
[7] J.I.Fujii and E.Kamei, Interpolational paths and their derivatives, Math. Japon. 39(1993), 557-560.
[8] J.I.Fujii, The Hiai-Petz geodesic for strongly convex norm is the unique shortest path,
Sci. Math. Japon., 71(2010), 19-26.

[9] T.Furuta, Parametric extensions of Shannon inequality and its reverse one in Hilbert
space operators, Linear Alg. Appl., 381(2004), 219-235.
[10] J.Jost, “Nonpositive Curvature: Geometric and Analytic Aspects”, Springer, 1997.

[11] F.Kubo and T.Ando, Means of positive linear operators, Math. Ann. 246 (1980), 205-224.

[12] M.P\’alfia, Semigroups of operator means and generalized Karcher equations, Preprint.