Matrix theory fundamentals
Let N be the set of all natural numbers For each n * N, we denote by M n the set of all n×n complex matrices,H n is the set of alln×n Hermitian matrices,H + n is the set ofn×n positive semi-deịnite matrices,P n is the cone of positive deịnite matrices inM n , andDnis the set of density matrices which are the positive deịnite matrices with trace equal to one Denote byIandO the identity and zero elements ofM n , respectively This thesis deals with problems for matrices, which are operators in ịnite-dimensional Hilbert spacesH We will indicate if the case is inịnite-dimensional.
Recall that for two vectors x = (xj), y = (yj) * C n the inner product 8x, y9 of x and y is deịned as8x, y9 c �胅 j xjyứj Now let A be a matrix in M n , the conjugate transpose or the adjointA 7 ofAis the complex conjugate of the transposeA T We have,8Ax, y9=8x, A 7 y9. Deịnition 1.1.1 A matrixA= (aij) n i,j=1 *M n is said to be:
(ii) invertible if there exists an matrixB of ordern×nsuch thatAB =In In this situation
Ahas a unique inverse matrixA 21 *M n such thatA 21 A 21 =I n
(vi) positive semi-deịnite if8Ax, x9 g0for allx*C n
(vii) positive deịnite if8Ax, x9>0for allx*C n \{0}.
Deịnition 1.1.2(Lơownerếs Order, [86]) LetAandB be two Hermitian matrices of same order n We say thatAgB if and only ifA2Bis a positive semi-deịnite matrix.
Deịnition 1.1.3 A complex numberằis said to be an eigenvalue of a matrix Acorresponding to its non-zero eigenvectorxif
The multiset of the eigenvalues ofAis denoted bySp(A)and called the spectrum ofA.
There are several conditions that characterize positive matrices Some of them are listed in theorem below [10].
(i) Ais positive semi-deịnite if and only if it is Hermitian and all its eigenvalues are nonneg- ative Moreover,Ais positive deịnite if and only if it is Hermitian and all its eigenvalues are positive.
(ii) A is positive semi-deịnite if and only if it is Hermitian and all its principal minors are nonnegative Moreover, A is positive deịnite if and only if it is Hermitian and all its principal minors are positive.
(iii) A is positive semi-deịnite if and only if A = B 7 B for some matrix B Moreover, A is positive deịnite if and only ifB is nonsingular.
(iv) A is positive semi-deịnite if and only ifA = T 7 T for some upper triangular matrix T. Further,T can be chosen to have nonnegative diagonal entries IfA is positive deịnite, thenT is unique This is called the Cholesky decomposition ofA Moreover,Ais positive deịnite if and only ifT is nonsingular.
(v) Ais positive semi-deịnite if and only if A = B 2 for some positive matrix B Such a B is unique We writeB = A 1/2 and call it the (positive) square root ofA Moreover,Ais positive deịnite if and only ifB is positive deịnite.
(vi) Ais positive semi-deịnite if and only if there existx1, , xn inHsuch that aij =8xi, xj9.
Ais positive deịnite if and only if the vectorsx j ,1fj fn, are linearly independent.
Let A * M n , we denote the eigenvalues of A by ằ j (A), for j = 1,2, , n For a matrix
A *M n , the notationằ(A) c(ằ1(A),ằ 2 (A), ,ằ n (A))means thatằ 1 (A)g ằ 2 (A)g g ằn(A) Theabsolute value of matrixA*M n is the square root of matrixA 7 Aand denoted by
We call the eigenvalues of|A|by thesingular valueofAand denote assj(A), forj = 1,2, , n. For a matrix A * M n , the notation s(A) c (s 1 (A), s 2 (A), , s n (A)) means that s 1 (A) g s2(A)g .gsn(A).
There are some basic properties of the spectrum of a matrix.
(ii) IfAis a Hermitian matrix thenSp(A)¢R.
(iii) A is a positive semi-deịnite (respectively positive deịnite) if and only ifAis a Hermitian matrix andSp(A)¢R g0 (respectively Sp(A)¢R + ).
Thetraceof a matrixA= (aij)*M n , denoted byTr(A), is the sum of all diagonal entries, or, we often use the sum of all eigenvaluesằi(A)ofA, i.e.,
Related to the trace of the matrix, we recall the Araki-Lieb-Thirring trace inequality [18] used consistently throughout the thesis.
Theorem 1.1.1 LetAandB be two positive semi-deịnite matrices, and letq >0, we have
B r 2 A r B r 2 �胀 q r �胆 fTr�胆�胁
B r 2 A r B r 2 �胀 q r �胆 gTr�胆�胁
ThedeterminantofAis denoted and deịned by det(A) = �胅 Ã* S n
�胅n j=1 ằ j whereS n is the set of all permutationsÃof the setS={1,2, , n}.
Proposition 1.1.3 LetA, B * H n withằ(A) = (ằ1,ằ 2 , ,ằ n )andằ(B) = (à1, à2, , àn).
(i) IfA >0andB >0, thenAgBif and only ifB 21 gA 21
(ii) IfAgB, thenX 7 AX gX 7 BX for everyX *M n
(iii) IfAgB, thenằ j gà j for eachj = 1,2, , n.
A function�胀á�胀:M n ³Ris said to be a matrix norm if for allA, B *M n and"³ *Cwe have:
(ii) �胀A�胀= 0if and only ifA= 0.
(iv) �胀A+B�胀 f �胀A�胀+�胀B�胀.
In addition, a matrix norm is said to be sub-multiplicative matrix norm if
�胀AB�胀 f �胀A�胀á�胀B�胀.
A matrix norm is said to be a unitarily invariant norm if for every A * M n , we have
�胀U AV�胀=�胀A�胀for allU, V *U n unitary matrices It is denoted as�胀| á |�胀.
These are some important norms overM n
The operator norm ofA, deịned by
The Ky Fank-norm is the sum of all singular values, i.e.,
The Schattenp-norm is deịned as
Whenp= 2, we have the Frobenius norm or sometimes called the Hilbert-Schmidt norm :
Letx = (x1, x2, , xn)andy = (y1, y2, , yn)be in R n Let x ³ = �胀 x [1] , x [2] , , x [n] �胀 denote a rearrangement of the components ofxsuch thatx[1] �背x[2] �背 .�背x[n] We say thatx is majorized byy, denoted byxzy, if
We say thatxis weakly majorized byyif
Ifx >0(i.e.,xi > 0fori= 1, , n) andy > 0, we say thatxislog-majorized byy, denoted byxz log y, if
In other words,xz log yif and only iflogxzlogy.
MatrixP * M n is called aprojectionifP 2 =P One says thatP is aHermitian projection if it is both Hermitian and a projection; P is an orthogonal projection if the range of P is orthogonal to its null space The partial ordering is very simple for projections IfP andQare projections, then the relationP f Q means that the range ofP is included in the range of Q.
An equivalent algebraic formulation isP Q =P The largest projection inM n is the identityI and the smallest one is 0 Therefore0 f P f I for any projectionP * M n Assume thatP andQare projections on the same Hilbert space Among the projections which are smaller than
P andQthere is a maximal projection, denoted byP 'Q, which is the orthogonal projection onto the intersection of the ranges ofP andQ.
Theorem 1.1.2 [45] Assume thatP andQare orthogonal projections Then
Matrix function and matrix mean
Now let us recall the spectral theorem which is one of the most important tools in functional analysis and matrix theory.
Theorem 1.2.1(Spectral decomposition, [9]) Letằ 1 >ằ 2 >ằ k be eigenvalues of a Hermi- tian matrix A Then
A�胅k j=1 ằ j Pj, wherePj is the orthogonal projection onto the subspace spanned by the eigenvectors associated to the eigenvalueằ j
For a real-valued functionf deịned on some intervalK ÂR, and for a self-adjoint matrix
A * M n with spectrum in K, the matrixf(A)is deịned by means of the functional calculus, i.e.,
A�胅k j=1 ằ j Pj =ú f(A) :�胅k j=1 f(ằj)Pj.
Or, ifA =Udiag (ằ 1 , ,ằ n )U 7 is a spectral decomposition ofA(whereU is some unitary), then f(A) :=Udiag (f(ằ 1 ),ỏ ỏ ỏ , f(ằ n ))U 7
We are now at the stage where we will discuss matrix/operator functions Lơowner was the ịrst to study operator monotone functions in his seminal papers [63] in 1930 In the same time, Kraus investigated the notion operator convex function [55].
Deịnition 1.2.1([63]) A continuous functionf deịned on an intervalK(K ÂR)is said to be operator monotone of ordernonKif for two Hermitian matricesAandB inM n with spectras inK, one has
Iff is operator monotone of any orders thenf is calledoperator monotone.
Theorem 1.2.2(Lơowner-Heinzếs Inequality, [86]) The functionf(t) = t r is operator monotone on[0,>)for 0 f r f 1 More speciịcally, for two positive semi-deịnite matrices such that
Deịnition 1.2.2([55]) A continuous functionf deịned on an intervalK(K ÂR)is said to be operator convex of ordern onK if for any Hermitian matricesAandB inM n with spectra in
K, and for all real numbers0fằ f1, f(ằA+ (12ằ)B)fằf(A) + (12ằ)f(B).
Iff is operator convex of any ordernthenfis calledoperator convex If2fis operator convex then we callf is operator concave.
Theorem 1.2.3([10]) Functionf(t) =t r in[0,>)is operator convex whenr *[21,0]*[1,2].
More speciịcally, for any positive semi-deịnite matricesA, Band for anyằ *[0,1],
Another important example is the function f(t) = logt, which is operator monotone on (0,>)and the functiong(t) =tlogtis operator convex The relations between operator mono- tone and operator convex via the theorem below.
Theorem 1.2.4 ([9]) Let f be a (continuous) real function on the interval [0,³) Then the following two conditions are equivalent:
(ii) The functiong(t) = f(t) t is operator monotone on(0,³).
Deịnition 1.2.3([10]) Letf(A, B)be a real valued function of two matrix variables Then,f is calledjointly concave, if for all0f³f1, f(³A1 + (12³)A2,³B1+ (12³)B2)g³f(A1, B1) + (12³)f(A2, B2) for allA1, A2, B1, B2.If2f is jointly concave, we sayf isjointly convex.
We will review very quickly some basic concepts of the Frôechet differential calculus, with special emphasis on matrix analysis LetX, Y be real Banach spaces, and let L(X, Y) be the space of bounded linear operators fromX toY Let U be an open subset ofX A continuous mapf fromU to Y is said to be differentiable at a pointu ofU if there existsT * L(X, Y) such that limv³0
It is clear that if such aT exists, it is unique If f is differentiable atu, the operator T above is called the derivative of f at u We will use for it the notation Df(u), of "f(u) This is sometimes called the Frôechet derivative Iff is differentiable at every point ofU, we say that it is differentiable onU One can see that, iff is differentiable atu, then for everyv *X,
This is also called the directional derivative off atuin the directionv.
Iff1, f2are two differentiable maps, thenf1 +f2is differentiable and
The composite of two differentiable mapsf andg is differentiable and we have the chain rule
One important rule of differentiation for real functions is the product rule: (f g) 2 = f 2 g+gf 2
If f and g are two maps with values in a Banach space, their product is not deịned - unless the range is an algebra as well Still, a general product rule can be established Letf, gbe two differentiable maps fromX intoY1, Y2, respectively LetB be a continuous bilinear map from
Y1ìY2 intoZ Letì be the map fromX toZ deịned asì(x) =B(f(x), g(x)) Then for all u, vinX
The product rule for differentiation states that the derivative of a product of two functions f(x) and g(x) is given by f'(x)g(x) + f(x)g'(x) A special case of this rule arises when f(x) = g(x) = L(Y), the algebra of bounded operators in a Banach space Y In this case, the product of two operators is defined as the usual product of their matrix representations, and the product rule simplifies to the form f'(x)g(x) + f(x)g'(x).
Higher-order Fréchet derivatives can be identified with multilinear maps For a differentiable map f from X to Y, its first-order derivative Df(u) at each point u is an element of the Banach space L(X, Y) This defines a map Df from X to L(X, Y) that assigns to each u the linear map Df(u).
If this map is differentiable at a pointu, we say thatfis twice differentiable atu The derivative of the mapDf at the pointuis called the second derivative off atu It is denoted asD 2 f(u). This is an element of the spaceL(X,L(X, Y)) LetL2(X, Y)be the space of bounded bilinear maps fromX×X intoY The elements of this space are mapsf fromX×X intoY that are linear in both variables, and for whom there exists a constantcsuch that
�胀f(x1, x2)�胀 fc�胀x1�胀 �胀x2�胀 for all x1, x2 * X The inịmum of all such c is called �胀f�胀 This is a norm on the space
L2(X, Y), and the space is a Banach space with this norm If×is an element ofL(X,L(X, Y)), let ÷ ×(x1, x2) = [×(x1)] (x2) forx1, x2 *X.
Then×÷ * L2(X, Y) It is easy to see that the map× ³ ×÷is an isometric isomorphism Thus the second derivative of a twice differentiable mapffromXtoY can be thought of as a bilinear map fromX×X toY It is easy to see that this map is symmetric in the two variables; i.e.,
D 2 f(u) (v1, v2) = D 2 f(u) (v2, v1) for allu, v1, v2 Derivatives of higher order can be deịned by repeating the above procedure. Thepth derivative of a mapffromXtoY can be identiịed with ap-linear map from the space
X×X×á á á×X(pcopies) intoY A convenient method of calculating thepth derivative of f is provided by the formula
For the convenience of readers, let us provide some examples for the derivatives of matrices. Example 1.2.1 In these examplesX =Y =L(H).
(B1, B2) =B1B2+B2B1. (ii) Letf(A) =A 21 for each invertibleA Then
(iii) Letf(A) =A 22 for each invertibleA Then
In connections with electrical engineering, Anderson and Dufịn [3] deịned theparallel sum of two positive deịnite matricesAandBby
The harmonic meanis2(A : B)which is the dual of thearithmetic meanA'B = A+B
2 In this period time, Pusz and Woronowicz [69] introduced thegeometric meanas
They also proved that the geometric mean is the unique positive solution of the Riccati equation
In 2005, Moakher [65] conducted a study, and then in 2006, Bhatia and Holbrook [14] investi- gated the structure of the Riemannian manifoldH + n They showed that the curve ³(t) = A�胉tB =A 1/2 �胀
A 1/2 (t *[0,1]) is the unique geodesic joining A and B, and called t-geometric mean or weighted geometric mean The weighted harmonic and the weighted arithmetic means are deịned by
The well-known inequality related to these quantities is the harmonic, geometric, and arithmetic means inequality [47, 60] , that is,
A!tB fA�胉tB fA' t B.
These three means are Kubo-Ando means LetÕs collect the main content of the Kubo-Ando means theory in the general case [54] For x > 0andt g 0, the function Ç(x, t) = x(1 +t) x+t is bounded and continuous on the extended half-line [0,>] The Lơowner theory ([9, 45]) on operator-monotone functions states that the mapm�胀³f, deịned by f(x) �胅
[0,>] Ç(x, t)dm(t)forx >0, establishes an afịne isomorphism from the class of positive Radon measures on[0,>]onto the class of operator-monotone functions In the representation abvove,f(0) = inf x f(x) =m({0}) andinf x f(x)/x=m({>}).
Theorem 1.2.5 [Kubo-Ando] For each operator connection Ã, there exists a unique operator monotone functionf :R + ³R + , satisfying f(t)In=InÃ(tIn), t >0, and forA, B >0the formula
AÃB =A 1 2 f(A 2 1 2 BA 2 1 2 )A 1 2 holds, with the right hand side deịned via functional calculus, and extended to A, B g 0 as follows
�胈³0(A+�胈In)Ã(B+�胈In).
We callf the representing function ofÃ.
The next theorem follows from the integral representation of matrix monotone functions and from the previous theorem.
Theorem 1.2.6 The map,m�胀³Ã, deịned by
1 +t t {(tA) :B}dm(t) where a=m({0})andb=m({>}), establishes an afịne isomorphism from the class of positive Radon measures on[0,>]onto the class of connections.
IfP andQare two projections, then the explicit formulation forPÃQis simpler.
Theorem 1.2.7 IfÃis a mean, then for every pair of projectionsP andQ
PÃQ=a(P 2P 'Q) +b(Q2P 'Q) +P 'Q, where a = 1Ã0 and b = lim x³>(1Ãx)/x.
An immediate consequence of the above theorem is the following relation for projectionsP andQ
Let f be the representing function ofà Since xf(x 21 )is the representing function of the transposeà 2 , thenà is symmetric if and only iff(x) = xf(x 21 ) The next theorem gives the representation for a symmetric connection.
Theorem 1.2.8 The map,n�胀³Ã, deịned by
1 +t 2t {(tA) :B+A : (tB)}dn(t) wherec=n({0}), establishes an afịne isomorphism from the class of positive Radon measures on the unit interval[0,1]onto the class of symmetric connections.
In recent years, many researchers have paid attention to different distance functions on the setP n of positive deịnite matrices Along with the traditional Riemannian metricdR(A, B) �胂 n
(whereằ i (A 21 B)are eigenvalues of the matrixA 21/2 BA 21/2 ), there are other important functions Two of them are the Bures-Wasserstein distance [13], which are adapted from the theory of optimal transport : db(A, B) = �胀
, and the Hellinger metric or Bhattacharya metric [11] in quantum information : dh(A, B) =�胀
Notice that the metric dh is the same as the Euclidean distance between A 1/2 and B 1/2 , i.e.,
Recently, Minh [43] introduced the Alpha Procrustes distance as follows: For³>0and for two positive semi-deịnite matricesAandB, db,³ = 1 ³db(A 2³ , B 2³ ).
He showed that the Alpha Procrustes distances are the Riemannian distances corresponding to a family of Riemannian metrics on the manifold of positive deịnite matrices, which encom- pass both the Log-Euclidean and Wasserstein Riemannian metrics Since the Alpha Procrustes distances are deịned based on the Bures-Wasserstein distance, we also call them theweighted Bures-Wasserstein distances In that òow, in this chapter we can deịne theweighted Hellinger metricfor two positive semi-deịnite matrices as follows: dh,³(A, B) = 1 ³dh(A 2³ , B 2³ ), then investigate its properties within this framework.
The results of this chapter are taken from [32].
Weighted Hellinger distance
Deịnition 2.1.1 For two positive semi-deịnite matricesAandB and for³>0, the weighted Hellinger distance betweenAandBis deịned as dh,³(A, B) = 1 ³dh(A 2³ , B 2³ ) = 1 ³(Tr(A 2³ +B 2³ )22 Tr(A ³ B ³ )) 1 2 (2.1.1)
It turns out that dh,³(A, B) is an interpolating metric between the Log-Euclidean and the Hellinger metrics We start by showing that the limit of the weighted Hellinger distance as ³ tends to 0 is the Log-Euclidean distance We also show that the weighted Bures-Wasserstein and weighted Hellinger distances are equivalent (Proposition 2.1.2).
Proposition 2.1.1 For two positive semi-deịnite matricesAandB, ³³0limd 2 h,³ (A, B) = ||log(A)2log(B)|| 2 F
Proof We rewrite the expression ofd h,³ (A, B)as d 2 h,³ (A, B) = 1 ³ 2 d 2 h (A 2³ B 2³ )
�胁 (logA) 2 + (logB) 2 + 2 logA.logB�胁
A ³ B ³ 2A ³ 2B ³ +I =³ 2 logAálogB+á á á Consequently, d 2 h,³ (A, B) = ||A ³ 2I|| 2 F ³ 2 + ||B ³ 2I|| 2 F ³ 2 22 Tr(logA.logB)
F. Tending³to zero, we obtain d 2 h,³ (A, B) =||logA|| 2 F +||logB|| 2 B 22�胄 logA,logB�胄
It is interesting to note that the weighted Bures-Wasserstein and weighted Hellinger distances are equivalent.
Proposition 2.1.2 For two positive semi-deịnite matricesAandB, db,³(A, B)fdh,³(A, B)f:
Proof According the Araki-Lieb-Thirring inequality [43] , we have
2 we obtain the following Tr(A ³ B 2³ A ³ ) 1/2 gTr(A ³ B ³ ).
In other words, db,³(A, B)fdh,³(A, B).
WithÃ,Ã *Dn, we have d 2 h (Ã,Ã) = 222 Tr(Ã 1/2 Ã 1/2 )f424 Tr((Ã 1/2 ÃÃ 1/2 ) 1/2 ) = 2d 2 b (Ã,Ã), or,
In the above inequality replaceÃwith A 2³
2 Tr�胆 (A ³ B 2³ A ³ ) 1/2 �胆 f Tr(A 2³ ) 1/2 Tr(B 2³ ) 1/2 + Tr(A ³ B ³ ) f 1
The above inequality is equivalent to
In-betweenness property
In 2016, Audenaert [5] introduced the in-betweenness property of matrix means We say that a matrix meanÃsatisịes thein-betweenness propertywith respect to the metricdif for any pair of positive deịnite operatorsAandB, d(A, AÃB)fd(A, B).
In [34], the authors introduced and studied the in-sphere property of matrix means Dinh, Franco and Dumitru also published several papers [26, 28] on geometric properties of the matrix power meanàp(t;A, B) := (tA p + (12t)B p ) 1/p with respect to different distance functions They also considered the case of the matrix power mean in the sense of Kubo-Ando [54] which is deịned as
Pp(t, A, B) = A 1/2 �胀 tI+ (12t)(A 21/2 BA 21/2 ) p �胀1/p
In this section, we focus our study on the in-betweenness properties of the matrix power means with respect to the weighted Bures-Wasserstein and weighted Hellinger distances As a consequence of the equivalence, using the operator convexity and concavity of the power func- tions, we show that the matrix power mean satisịes the in-betweenness property with respect todh,³ (Theorem 2.2.1) anddb,³(Theorem 2.2.2) We also show that among symmetric means, the arithmetic mean is the only one that satisịes the in-betweenness property in the weighted Bures-Wasserstein and weighted Hellinger distances.
Now we are ready to show that the matrix power meansà p (t;A, B)satisfy the in-betweenness property indh,³ anddb,³.
Theorem 2.2.1 Let0< p/2f³fpand0ftf1 Then dh,³(A, àp(t;A, B))fdh,³(A, B), for allA, B *H + n.
Proof We have d 2 h,³ (A, àp(t;A, B)) = 1 ³ 2 Tr�胁
. Therefore, the above result follows if
Tr�胁 à 2³ p (t;A, B)22A ³ à ³ p (t;A, B)�胁 fTr�胁
By the operator convexity of the mapx�胀³x 2³/p , when p
2 f³ fp, à 2³ p (t;A, B) = �胁 tA p + (12t)B p �胁2³/p ftA 2³ + (12t)B 2³
Thus, the desired result follows if
By the operator concavity of the mapx�胀³x ³/p , when p
2 f³fp, à ³ p (t;A, B) =�胁 tA p + (12t)B p �胁³/p gtA ³ + (12t)B ³
Therefore, the distance monotonicity follows if
Tr�胆 t(A 2³ 2B 2³ )22A ³ �胁 tA ³ + (12t)B ³ �胁�胆 f 22 Tr(A ³ B ³ ), or tTr�胁
A 2³ +B 2³ 22A ³ B ³ �胁 g0, which is from AM-GM inequality.
Theorem 2.2.2 Let0< p/2f³fpand1/2ftf1 Then, db,³(A, àp(t;A, B))fdb,³(A, B), for allA, B *H + n.
Proof Firstly, we show that for any positive semi-deịnite matricesAandB, forp/2f ³ fp and1/2ft f1, db,³(A, àp(t;A, B))fdh,³(A, àp(t;A, B))f:
By the Araki-Lieb-Thirring inequality, we have
By the operator convexity of the functionx�胀³ x 2³/p and the operator concavity of the function x�胀³x ³/p , we obtain d 2 b,³ (A, àp(t;A, B)) f 1 ³ 2 Tr�胆
A 2³ +tA 2³ + (12t)B 2³ 22A ³ �胁 tA ³ + (12t)B ³ �胁�胆
From here, applying the square root function to both sides witht*[ 1 2 ,1], we have db,³(A, àp(t;A, B))f:
In [28, Theorem 2] the authors proved that the matrix Kubo-Ando power meanPp(t, A, B) satisịes the in-betweenness property which follows from the fact that the function g(t) Tr(A 1/2 Pp(t;A, B) 1/2 ) is concave Note thatPt(A, B) 7= Pt(B, A), i.e.,Pt is not symmetric. However, for the symmetric means we may have the following result whose proof is adapted from [22].
Theorem 2.2.3 Letà be a symmetric mean and assume that one of the following inequalities holds for any pair of positive deịnite matricesAandB: dh,³(A, AÃB)fdh,³(A, B) (2.2.2) or db,³(A, AÃB)fdb,³(A, B) (2.2.3)
Proof By Theorem 1.2.6 and 1.2.8, the symmetric operator meanà is represented as follows:
(0,>) ằ+ 1 ằ {(ằA) :B+A: (ằB)}dà(ằ), (2.2.4) whereA, B g0,ằ g0andàis a positive measure on(0,>)withã+à((0,>)) = 1, and the parallel sumA :B is given byA:B = (A 21 +B 21 ) 21 , whereAandB are invertible.
For two orthogonal projections P, Qacting on a Hilbert space H, let us denote by P 'Q their inịmum which is the orthogonal projection on the subspaceP(H)+Q(H) IfP 'Q= 0, then by Theorem 1.2.7,
Let us consider the following orthogonal projections
�胄 cos 2 ằ cosằsinằ cosằsinằ sin 2 ằ
Notice thatQằ ³P asằ³0andQằ'P = 0.From the projections above, it is easy to see that the inequality (2.2.2) becomes dh,³(P,ã(P +Qằ)/2)fdh,³(P, Qằ).
Since this is true for allằ >0, we can take a limit asằ ³0 + to obtain dh,³(P,ãP)fdh,³(P, P) whose equality occurs if and only ifã= 1 This shows thatà= 0andÃis the arithmetic mean. The statement ford h,³ can be proved similarly.
In this chapter, we introduce a new distance called the weighted Hellinger distance and investigate its properties This distance is constructed based on MinhÕs approach when he con- structed the weighted Bures distance The weighted Bures distance is an extended version with one parameter of the Bures distance In the next chapter, we introduce a new quantum diver- gence called the ³-z-Bures Wasserstein divergence, which is considered as an extension with two parameters of the Bures distance.
It is well-known that in the Riemannian manifold of positive deịnite matrices, the weighted geometric mean A�胉tB = A 1/2 (A 21/2 BA 21/2 ) t A 1/2 is the unique geodesic joining A and B, whereA, B *P n Fort = 1/2,A�胉1/2B is called the geometric mean ofAandB It is obvious thatA�胉1/2B is a matrix generalization of the geometric mean: abof positive numbersaandb. LetA1, A2,ỏ ỏ ỏ , Am be positive deịnite matrices In 2004, Moakher [65] and then Bhatia and Holbrook [14] studied the following least squares problem minX>0
�胅m i=1 ã 2 2 (X, Ai), (3.0.1) whereã 2 (A, B) = ||log(A 21 B)||2 is the Riemannian distance betweenAandB They showed that (3.0.1) has a unique solution which is called the Karcher mean of A 1 , A 2 ,á á á , A m In literature, this mean has different names such as: Frôechet mean, Cartan mean, Riemannian center of mass It turns out that the solution of (3.0.1) is the unique positive deịnite solution of the Karcher equation
In [60], Lim and Palịa showed that the solution of (3.0.2) is nothing but the limit of the solution of the following matrix equation ast ³0,
Recently, Franco and Dumitru [38] introduced the so-called Rôenyi power means of matrices. More precisely, for0