Banach Journal of Mathematical Analysis ISSN: (electronic)

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1 Banach J Math Anal (009), no, Banach Journal of Mathematical Analysis ISSN: (electronic) ON A GEOMETRIC PROPERTY OF POSITIVE DEFINITE MATRICES CONE MASATOSHI ITO, YUKI SEO, TAKEAKI YAMAZAKI AND MASAHIRO YANAGIDA 4 Communicated by M S Moslehian Abstract We shall discuss the matrix geometric mean for the positive definite matrices The set of all n n matrices with a suitable inner product will be a Hilbert space, and the matrix geometric mean can be considered as a path between two positive matrices In this paper, we shall obtain a matrix geometric mean inequality, and as an application of it, a property of Riemannian metric space is given We also obtain some examples related to our result Introduction Let M n be the set of all n n matrices on C The set M n will be a Hilbert space with the inner product A, B trb A for A, B M n, and the associated norm A (tra A) The set of all Hermitian matrices H n constitutes a real vector space on M n The subset P n of M n consisting of positive definite matrices is an open subset in H n Hence it is a differentiable manifold The tangent space to P n at any of its points A is the space T A P n A H n, identified for simplicity, with H n The inner product on H n leads to a Riemannian metric on the manifold P n At the point A this metric is given by the differential ds A daa [ tr(a da) ] Date: Received: April 009; Accepted: 8 June 009 Corresponding author 000 Mathematics Subject Classification Primary 47A64 Secondary 47A6, 47L5 Key words and phrases Positive matrix, Riemannian metric, geometric mean 64

2 ON A GEOMETRIC PROPERTY OF POSITIVE DEFINITE MATRICES CONE 65 This is a mnemonic for computing the length of a (piecewise) differentiable path in P n If γ : [a, b] P n is such a path, we define its length as L(γ) b For any two points A and B in P n, let a γ (t)γ (t)γ (t) dt δ (A, B) infl(γ) : γ is a path from A to B Bhatia and Holbrook [] show that the infimum is attained at a unique path joining A and B This path is called geodesic from A to B, and it is denoted by [A, B] This gives a metric on P n, and concrete form of δ (A, B) is given as follows: Theorem ([]) Let A and B be any two elements of P n Then there exists a unique geodesic [A, B] joining A and B This geodesic has a parametrization which is natural in the sense that γ(t) A (A BA ) t A, 0 t, δ (A, γ(t)) tδ (A, B) for each t [0, ] Furthermore, the metric is precisely estimated by δ (A, B) log A BA We remark that the geodesic from A to B is known as the generalized geometric mean A t B of A and B, that is, γ(t) A (A BA ) t A A t B for t [0, ] In particular, A B (denoted by A B, simply) is called the matrix geometric mean, simply, and it has many good properties We shall introduce some properties of the generalized geometric mean in the next section The metric δ has an important property which is called exponential metric increasing property as follows: Theorem ([, ]) For each pair of points A, B in P n, we have δ (A, B) log A log B In other words, for any two matrices H and K in H n, δ (e H, e K ) H K So the map exp : (H n, ) (P n, δ ) () increases distances, or is metric increasing It is known that P n is a Riemannian manifold of nonpositive curvature [4] Another essential feature of this geometry is the semiparallelogram law for the metric δ To understand this, recall the parallelogram law in a Hilbert space H Let a and b be any two points in H and let m a+b be their midpoint Given any other points c and d consider the parallelogram, one of whose diagonals is

3 66 M ITO, Y SEO, T YAMAZAKI, M YANAGIDA [a, b] and the other [c, d] The two diagonals intersect at m and the parallelogram law is the equality a b + c d ( a c + b c ) Upon rearrangement this can be written as c m a c + b c a b 4 In the semiparallelogram law this last equality is replaced by an inequality Theorem (The Semiparallelogram Law [, ]) Let A and B be any two points of P n and let M A B be the midpoint of the geodesic [A, B] Then for any C in P n we have δ (C, M) δ (A, C) + δ (B, C) δ (A, B) 4 In the Euclidean space, the distance between the midpoints of two sides of a triangle is equal to half the length of the third side In a space whose metric satisfies the semiparallelogram law this is replaced by an inequality as follows: Theorem 4 ([]) Let A, B and C be any three points in P n Then δ (A B, A C) δ (B, C) By Theorem, the matrix geometric mean is closely related to Riemannian metric on the manifold P n So we can expect that many results of the matrix geometric mean can be applied for the study of Riemannian metric in the manifold P n As a property of the matrix geometric mean, the following result is obtained by Ando Li Mathias in []: Theorem 5 ([]) For positive definite matrices A, B, C, D, A B C D implies (A C) (B D) A B Roughly speaking, Theorem 5 says that for four points A, B, C, D P n, if the diagonals [A, B] and [C, D] have the same midpoint G A B C D in a quadrilateral ADBC, then the line joining their midpoints in opposite sides [A, C] and [D, B] pass through the point G In this paper, we shall discuss an extension of Theorem 5 In section, we shall introduce some properties of matrix means In section, we will give an extension of Theorem 5 In section 4, we give examples related to the result in section Matrix mean In this section, we shall introduce some properties of matrix means Let M be a binary operation M : P n P n P n Then M is called a matrix mean if and only if the following conditions hold: () M(A, B) M(C, D) if 0 A C and 0 B D (monotonicity),

4 ON A GEOMETRIC PROPERTY OF POSITIVE DEFINITE MATRICES CONE 67 () XM(A, B)X M(XAX, XBX) holds for X 0 (transformer inequality), () M is upper semicontinuous, and (4) M(I, I) I We remark that if X is invertible, then the transformer inequality () can be replaced into the transformer equality It is well known that the matrix arithmetic and geometric means are typical examples of matrix mean In this paper, we write the matrix arithmetic and the geometric means by A B A + B and A B A (A BA ) A, respectively More generally, the generalized matrix arithmetic mean A t B and the geometric mean A t B are defined as follows: For t [0, ], A t B ( t)a + tb and A t B A (A BA ) t A We remark that if t [0, ], then A (A BA ) t A is not a matrix mean, but we write it A t B for the sake of convenience Here we introduce some properties of the generalized matrix arithmetic and geometric means Firstly, the famous arithmetic geometric means inequality holds, ie, A t B A t B for t [0, ] () Next property is shown by operator concavity of x r for r [0, ] A r t B r (A t B) r for r [0, ] and t [0, ] () We remark that in the case r, the sign of inequality in () is reversed, ie, A t B (A t B) for t [0, ] () In general, a matrix mean does not satisfy the commutative law, but the matrix arithmetic and the geometric means satisfy A t B B t A and A t B B t A (4) for t [0, ], (A t B B t A for t [0, ] also holds) The former equality is obvious, and the latter one is given by (AB A) AB(BA B) BA for A, B 0 and real number in [7, Lemma ] We remark that an argument over general matrix means is in [5, 8] Results In this section, we shall give an extension of Theorem 5 as follows: Theorem Let A, B, C, D be positive invertible matrices with 0 < mi A, B, C, D MI for some positive numbers m and M If A B C D G holds for a positive number (0, ), then for each β [0, ], (h 0 + ) 0 (h 0 + ) G (A β C) (B β D) 0 G () 4h 0 4h 0

5 68 M ITO, Y SEO, T YAMAZAKI, M YANAGIDA holds, where h M m and 0 min, By putting in Theorem, we obtain a slight extension of Theorem 5 as follows: Corollary Let A, B, C, D be positive invertible matrices If A B C D G holds, then for each β [0, ], holds (A β C) (B β D) G () Roughly speaking, Corollary says that when we consider a quadrilateral ADBC with the same midpoints G in each diagonals, for each internal ratio of β to β for β [0, ], the line joining their internally dividing point in opposite sides [A, C] and [D, B] pass through the midpoint G We would like to remark that for any (0, ), in the center of the left hand side in () can not be replaced into Concrete example and discussion will appear in the next section From the viewpoint of Riemannian metric, Theorem can be written by the following form: Theorem Let A, B, C, D be positive invertible matrices with 0 < mi A, B, C, D MI for some positive numbers m and M If A B C D G holds for a positive number (0, ), then for each β [0, ], (h 0 + ) 0 δ ((A β C) (B β D), G) n log 4h 0 holds, where h M and m 0 min, To prove Theorem, we need the following lemma which is a kind of reverse inequality of () Lemma ([6, 9]) Let A and B be positive invertible matrices satisfying 0 < m I A M I and 0 < m I B M I for some positive numbers 0 < m < M and 0 < m < M Then (h + ) A t B (A t B ) () 4h holds for t [0, ], where h max M m, M m It is a known result, but for the reader s convenient, we give a proof Proof Let X A BA, and X λde λ be the spectral decomposition of X Since h max M m, M m, then we have λ h We remark that for t (0, ) and h > 0, since h +, h h ( t) +t( t)(h+ h )+t ( h ( h ) t ) (h + ) (h + ) + (4) 4h 4h

6 ON A GEOMETRIC PROPERTY OF POSITIVE DEFINITE MATRICES CONE 69 Moreover λ + λ h + h for h λ h Then for t [0, ], ( t) + tx ( t) + tλde λ ( t) + tλ ( t) + tλ ( t) + tλ de λ ( t) + t( t)(λ + λ ) + t ( t) + tλ de λ ( t) + t( t)(h + h ) + t ( t) + tλ de λ (h + ) ( t) + tλ de λ by (4) 4h ( + h) ( t) + tx 4h Hence we have ( t) + tx ( + h) 4h ( t) + tx Multiplying both sides of this inequality by A, we have ( t)a + tb ( + h) ( t)a + tb, 4h that is, () Proof of Theorem Firstly, we prove the case (0, ] (ie, 0 ) A B G A (A BA ) A G (A BA ) A GA Hence B A (A GA ) A A G Similarly, by (4), we have C D G D C G C D G

7 70 M ITO, Y SEO, T YAMAZAKI, M YANAGIDA Then G (A β C) (B β D)G G A β (D G) (A (G AG ) β (G DG (G AG ) β (G DG G) β D I) ) G (G AG I) β G DG (G AG ) β G DG (X β Y ) (X β Y ) by putting X G AG, Y G DG (X β Y ) (X β Y ) by () (X β Y ) (X β Y ) by () and (X β Y ) 4h (h + ) 4h (h + ) (X β Y ) ()+( ) (h + ) I, 4h (X β Y ) where the last inequality holds since I X, Y hi ensures h where h 0 max h h X β Y (h 0 + ), h h 4h 0 (X β Y ), [0, ] h by Lemma Hence we have (h + ) (A β C) (B β D) G (5) 4h Now, we shall prove the lower bound of () By 0 < mi A, B, C, D MI, we have 0 < M I A, B, C, D m m I, M h Hence by M m (5), we obtain (A β C ) (B β D (h + ) ) G 4h Taking the inverse of both sides, (h + ) (A β C) (B β D) G, that is, we get the desired inequality 4h

8 ON A GEOMETRIC PROPERTY OF POSITIVE DEFINITE MATRICES CONE 7 Next, we prove the case [, ) (ie, 0 ) By A B C D G, we have B A D C G Since (0, ], (h + + ) (h + ) G (B β D) (A β C) G 4h 4h So that by (4), (h 0 + ) 0 (h 0 + ) 0 G (A β C) (B β D) G 4h 0 4h 0 4 Counterexamples In the previous section, we prove Corollary which is an extension of a result by Ando Li Mathias But we do not show that whether in the center of the left hand side in () can be replaced into for or not One might think that Corollary can be extended to more general form The aim of this section is to give a counterexample of the problem as follows: Proposition 4 There exist positive invertible matrices A, B, C and D satisfying A B C D and (A C) (B D) A B for any (0, ) To prove the above proposition is a little bit complicated, so we shall give a concrete counterexample in the case, firstly Example 4 There exist positive invertible matrices A, B, C and D satisfying Proof Let A Then But A ( B C ), B A (A C) (B D) D and (A C) ) ( B C D, C (B D) A G 4 4 B and D G The above example is not a counterexample for the general case (0, ) Next, we shall prove Proposition 4, ie, general case (0, ) To give a proof of it, we prepare the following discussions: Proposition 4 For (0, ), assume that positive invertible matrices A, B, C, D with A B C D G satisfy (A C) (B D) G Then there exist positive invertible matrices X and Y such that X Y (X Y )

9 7 M ITO, Y SEO, T YAMAZAKI, M YANAGIDA Proof Let A B C D G Then B A G and C D G Then we have G (A C) (B D) G G [ A (D G) (A (G AG ) (G DG (G AG ) (G DG ) (G AG ) (G DG ) ] G) D I) G (G AG I) (G DG ) (G AG ) (G DG (G AG so that (A C) (B D) G is equivalent to (G AG ) (G DG ) (G AG Let X G A G ) ) ) (G DG ), (G DG ) I and Y (G DG ) Then it is equivalent to (X Y ) (X Y ) I (4) By the way, for positive invertible matrices S and T, Hence (4) is equivalent to It completes the proof X Y S T I T S (X Y ) (X Y ) Hence by Proposition 4, we have only to consider a counterexample of for a positive number r with r X r Y r (X Y ) r (4) Proposition 44 For positive matrices X and Y with det(x) det(y ) Then X + Y X Y det(x + Y ) It is a well-known formula (see []), but we shall give a proof for the reader s convenient Proof Let D X Y X Then by the Cayley Hamilton theorem, we have Then by we obtain D trace(d)d + I 0 D (D + I) trace(d) + 0, D D + I trace(d) +

10 ON A GEOMETRIC PROPERTY OF POSITIVE DEFINITE MATRICES CONE 7 Hence X Y X D X X + Y trace(d) + Here by det(x Y ), we can get X Y X + Y det(x + Y ) Then we shall give a counterexample to prove Proposition 4 based on the above discussions Example 45 Let X 0 0 and Y Then (4) does not hold for all positive number r Proof It is easy to see det(x) det(y ) Then there exist real numbers and β r such that X Y (X + Y ), hence (X Y ) r r (X + Y ) r, X r Y r β r (X r + Y r ) by Proposition 44 Put (X Y ) r r (X + Y ) r X (r) X (r), X (r) X (r) X r Y r β r (X r + Y r Y (r) Y ) (r) Y (r) Y (r) If (4) holds for some r, then we have X (r) Y (r) and X (r) Y (r), so that, X (r) X (r) Y (r) Y (r) (4) Hence we have only to check that (4) does not hold for all positive number r with r Put a unitary matrix U as U Then we have 0 Y U U 0 Hence we obtain X r + Y r ( ) r By putting r in (44), we get X + Y r + r r r (44) + 7 5

11 74 M ITO, Y SEO, T YAMAZAKI, M YANAGIDA Let V be a unitary matrix as follows: ( ) V 4 Then we have and then (X + Y ) r k(r) V (X + Y )V ( ) , ( (6 + ) r + ( )(6 ) r ( )(6 + ) r (6 ) r ( )(6 + ) r (6 ) r ( )(6 + ) r + (6 ) r where k(r) ( ) r (4 ) (45) If X r Y r (X Y ) r for some r > 0 with r, then by (4), (44) and (45) r + r (6 + ) r + ( )(6 ) r ( )(6 + ) r (6 ) r, (46) if and only if ( r + )( )(λ r ) ( r )λ r + ( ) 0, where λ 6 + 6, if and only if ( ) (λ) r r + λ r ( ) 0 Let f(r) ( ) (λ) r r + λ r ( ) Then we have only to show f(r) 0 for any r > 0 such that r by the above argument By calculation, f (r) ( ) (λ) r log(λ) r log + λ r log λ (λ) r ( ) log(λ) λ r log + r log λ Put g(r) ( ) log(λ) λ r log + r log λ Then g (r) λ r log λ log + r log log λ log log λ (λ r r ) > 0 for r > 0 since < λ < Therefore f (r) (λ) r g(r) is strictly increasing for r 0, so that f(r) is a convex function for r 0 Hence f(r) 0 for any r > 0 such that r since f(0) f() 0 Remark 46 The above calculation is a bit complicated, but only to prove the existence of a counterexample, there is an easier calculation as follows: For some r > 0, we assume that X r Y r (X Y ) r (4) holds for any positive invertible matrices X and Y Then by putting X X r and Y Y r, we have (X Y ) r X r Y r ),

12 ON A GEOMETRIC PROPERTY OF POSITIVE DEFINITE MATRICES CONE 75 Hence (4) also holds for So we have only to consider the case r > If (4) r holds for all X, Y > 0 and some r >, then we obtain X r Y r (X r Y r ) r (X Y ) r, that is, (4) holds for r By the same way, (4) holds for r n Hence we have to check it for all sufficiently large r Here, from the left and right hand sides of (46), we have r + lim r r lim r lim r 4 r +, (6 + ) r + ( )(6 ) r ( )(6 + ) r (6 ) r lim r ( 6+ 6 )r Hence we obtain that (46) does not hold for all sufficiently large r since the above limit points are different It completes the proof Consequently we can prove Proposition 4 as follows: Proof of Proposition 4 By Example 45, there exist positive invertible matrices X and Y satisfying X Y (X Y ) for all (0, ) Therefore, by scrutinizing the proof of Proposition 4, we can get desired matrices A, B, C and D for a given positive invertible matrix G as follows: A G X G, D G Y G, B A G and C D G References T Ando, CK Li and R Mathias, Geometric means, Linear Algebra Appl, 85 (004), 05 4 R Bhatia, Positive definite matrices, Princeton University Press, New Jersey, (007) R Bhatia and J Holbrook, Riemannian geometry and matrix geometric means, Linear Algebra Appl, 4 (006), G Corach, H Porta and L Recht, Geodesics and operator means in the space of positive operators, Int J Math, 4 (99), JI Fujii and E Kamei, Uhlmann s interpolational method for operator means, Math Japon, 4 (989), JI Fujii, M Nakamura, J Pečarić and Y Seo, Bounds for the ratio and difference between parallel sum and series via Mond-Pečarić method, Math Inequal Appl, 9 (006), T Furuta, Extension of the Furuta inequality and Ando Hiai log majorization, Linear Algebra Appl, 9 (995), F Kubo and T Ando, Means of positive linear operators, Math Ann, 46 (980), J Pečarić and J Mićić, Chaotic order among means of positive operators, Sci Math Jpn, 7 (00), Department of Integrated Design Engineering, Maebashi Institute of Technology, Maebashi 7-086, Japan address: m-ito@maebashi-itacjp

13 76 M ITO, Y SEO, T YAMAZAKI, M YANAGIDA Faculty of Engineering, Shibaura Institute of Technology, Saitama , Japan address: yukis@sicshibaura-itacjp Department of Mathematics, Kanagawa University, Yokohama -8686, Japan address: yamazt6@kanagawa-uacjp 4 Department of Mathematical Information Science, Faculty of Science, Tokyo University of Science, Tokyo 6-860, Japan address: yanagida@rskagutusacjp