We survey some results concerning the problem of finding the complex hermitian matrix or matrices of least supremum norm with variable diagonal. Some qualitative general results are given and more specific descriptions are shown for the 3 x 3 case. We also comment some results and examples concerning this approximation problem.Fil: Andruchow, Esteban. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Mata Lorenzo, Luis E.. Universidad Simón Bolívar; VenezuelaFil: Mendoza, Alberto.. Universidad Simón Bolívar; VenezuelaFil: Recht, Lázaro. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Simón Bolívar; VenezuelaFil: Varela, Alejandro. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentin

Andruchow, Esteban

Mata Lorenzo, Luis E.

Mendoza, Alberto.

Recht, Lázaro

Varela, Alejandro

Revista de la Unión Matemática Argentina

English

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

Minimal hermitian matrices with fixed entries outside the diagonal

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REVISTA DE LAUNIÓN MATEMÁTICA ARGENTINAVolumen 49, Número 2, 2008, Páginas 17–28MINIMAL HERMITIAN MATRICES WITH FIXED ENTRIESOUTSIDE THE DIAGONALE. ANDRUCHOW, L. E. MATA-LORENZO, A. MENDOZA, L. RECHT, AND A. VARELADedicated to the memory of Mischa Cotlar. Teacher and friend.Abstract. We survey some results concerning the problem of finding thecomplex hermitian matrix or matrices of least supremum norm with variablediagonal. Some qualitative general results are given and more specific de-scriptions are shown for the 3 × 3 case. We also comment some results andexamples concerning this approximation problem.Contents1. Notation and Preliminaries 172. Previous results 203. Description of minimal hermitian matrices of 3× 3 204. Minimal matrices in a class 225. Different cases 236. The topology of the set of minimal matrices of 3× 3 257. An example in 4× 4 using minimal matrices 26References 271. Notation and PreliminariesThe problem of finding a complex hermitian matrix of least supremum normwith variable diagonal originally aroused when describing short curves in certainflag manifolds. A finite dimensional flag manifold is one such that its elements arechains of vector spaces included strictly. That is{0}  Vr  Vs...  Vk ⊂ Cnwhere dimVi = i. The metric considered here is the invariant Finsler metric thatwill be described further on.We will be interested in the special case of complete flags, that is when k = n.2000 Mathematics Subject Classification. 15A57, 47A58, 47C15.Key words and phrases. short curves, flag manifolds, matrix nearness, nearest diagonal.1718 ANDRUCHOW, MATA-LORENZO, MENDOZA, RECHT AND VARELAThe flags could be seen from different points of view. For example if we choosebases for the respective subspaces (that can be taken orthonormal), that is ifVi = [w1, · · · , wi]we can describe the previous chain listing the elements of the base in order (loosingthe unicity of the representation).They can also be described choosing sets of orthonormal projections of rank onewhich sum gives the identity, that is:{P1, P2, · · · , Pn} such thatn∑i=1Pi = ICnYet another way of representing flag manifolds is by the quotient between thegroup of n× n unitary complex matrices U over the subgroup of the unitary diag-onals Udiag:U /Udiag ={[v] : x ∈ [v]⇔x = u.v, with u ∈ Udiag}.Therefore the elements of U /Udiag are the classes [v] with elements of the formd.v =0BBB@λ1 0 · · · 00 λ2 · · · 0....... . ....0 0 · · · λn1CCCA .0BBB@v1,1 v1,2 · · · v1,nv2,1 v2,2 · · · v2,n....... . ....vn,1 vn,2 · · · vn,n1CCCA =0BBB@λ1 v1,1 λ1 v1,2 · · · λ1 v1,nλ2 v2,1 λ2 v2,2 · · · λ2 v2,n....... . ....λn vn,1 λn vn,2 · · · λn vn,n1CCCAWe will use the following notationMn×n(C), Mhn×n(C), Mahn×n(C)for the n × n algebra of matrices of complex coefficients, the hermitian and theanti-hermitian respectively.We will also denote withDn×n, Dhn×n, Dahn×n,the subagebra of the diagonal matrices of Mn×n(C), the hermitian (that is real)and anti-hermitian respectively. Observe that Dahn×n = iDhn×n.We will consider the quotient spacesMhn×n(C)/Dhn×n , Mahn×n(C)/Dahn×nwith the usual quotient norm for M ∈Mhn×n(C):| [M ] |q = infD∈Dhn×n||M +D ||, (1)where || · || is the usual operator (supremum) norm (respectively M ∈ Mahn×n(C)and D ∈ Dahn×n for the anti-hermitian case).The quotient space Mahn×n(C)/Dahn×n can be identified with the tangent of thespace U /Udiag in the base point 1, since U /Udiag is an homogeneous space underthe natural action of left multiplication of elements of U :T (U /Udiag )[1] = T (U)[1] /T (Udiag )[1]∼= Mahn×n(C)/Dahn×n.Rev. Un. Mat. Argentina, Vol 49-2MINIMAL HERMITIAN MATRICES 19Moreover, the Finsler metric considered in T (U /Udiag )[1] coincides with thequotient norm (1) of Mahn×n(C)/Dahn×n.This will let us link geometric results to the problem we are interested to describein this article. For instance, in theorem 1 of the next section, the existence ofmatrices D reaching the minimum of the norm quotient (1) is related with theexistence of curves in U /Udiag of the shortest posible length. In section 7, anexample in 4×4 of infinite different curves of the shortest length joining arbitrarilyclose points is shown.Operator approximation problems consist of finding, for a given operator, theelement in some special class nearest to it, when distance is measured with a norm.These problems have been treated in the case of hermitian, positive and unitaryapproximants using different norms in [5], [6], [7], and others. The survey article[8] is related to matrix nearness. There, explicit formulas of operator approxima-tion solutions are presented. Uniqueness results and algorithms for computing orestimating the minimal norm attained are also described, as well as the matrix ormatrices sought in different contexts. Nevertheless, in that paper, the operator orsupremum norm is not considered.The problem of finding the minimum of ‖M + D‖ for a given matrix M ∈Mhn×n(C) among all the diagonal matrices D ∈ Dhn×n, and finding the matrix ormatricesD that realize the minimum, is indeed an operator approximation problem.It has a trivial translation to the problem of finding a real diagonal matrix D′ thatsatisfies that M +D′ ≥ 0 and that ‖M +D′‖ is minimum.In the n × n case, some bounds of this minimum were obtained in [3]. In thatwork the calculation of this minimum is related to the estimation of bounds of thenorm of the operator O : Mn×n(C)→Mn×n(C) that for any n× n matrix replacesall its diagonal entries by zeroes.Definition 1. We will call a matrix Z ∈Mhn×n(C) minimal hermitian if‖Z‖ ≤ ‖Z +D‖, for all D ∈ Dhn×n.or equivalently if‖Z‖ = |[Z]|q.for [Z] ∈Mhn×n(C)/Dhn×n. In a similar way it can be defined a minimal antiher-mitian matrix.Remark 1. If Z ∈Mhn×n(C) is a minimal hermitian matrix and||Z|| = λ (> 0)then(1) i.Z is a minimal antihermitian matrix.(2) ±λ are eigenvalues of Z.(3) The diameter of the spectrum of Z is 2λ.(4) If Z is of 3× 3 then σ(Z) = {λ,−λ, µ} with µ = Tr(Z).Rev. Un. Mat. Argentina, Vol 49-220 ANDRUCHOW, MATA-LORENZO, MENDOZA, RECHT AND VARELA2. Previous resultsIn Theorem I of [4], Durán, Mata-Lorenzo and Recht proved a result that, inthe context of matrices, can be stated as follows:Theorem 1. Let [M ] ∈ U /Udiag and [X ] ∈ Mahn×n(C)/Dahn×n∼= T (U /Udiag )[M ].If there exists a antihermitian minimal matrix Z ∈Mahn×n(C) that projects in [X ],that is, [Z] = [X ] and ||Z|| = |[X ]|q, then the curveγ(t) = LetZ · [M ] = [etZM ]has minimum length in the class of all the curves in U /Udiag that join γ(0) withγ(t) for each t with |t| ≤ pi2||Z|| .Remark 2. Note that this theorem implies that, in order to find the curve ofminimum length such that γ˙(0) = [X ], we have to find the matrix Z with the sameoff-diagonal entries as X and with a diagonal that makes Z a minimal hermitianmatrix.This result shows the importance of the set of minimal matrices in the study ofthe shortest curves (in the geometric sense) in these homogeneus spaces.With the ideas of Section 5 of [4], resctricted to the context of matrices, thefollowing characterization of minimal hermitian matrices can be obtained:Theorem 2. A hermitian matrix Z ∈Mhn×n is minimal, if and only if, there existsa positive matrix P ∈Mn×n(C) such that,• P.Z2 = λ2 P , where ||Z|| = λ.• The diagonal elements of the product P.Z are all zero.3. Description of minimal hermitian matrices of 3× 3The previous theorem about minimal matrices together with auxiliary results inC3 (see [1]), allow the following characterization:Theorem 3. Let Z ∈ Mh3×3(C) with ||Z|| = λ > 0. Then Z is minimal, if andonly if, there exist two unitary eigenvectors,(1) v+ for the eigenvalue λ, and(2) v− for the eigenvalue −λ,such that their respective coordinates have the same absolute value|(v+)i| = |(v−)i|, with 1 ≤ i ≤ 3.Remark 3. Under these hypotheses, it can be proved that the eigenvectors v+ andv− are triangular, that is, |v1|2, |v2|2and |v3|2can represent the sides of a triangle(the sum of any two of them is always greater than the other one, see figure 1).Rev. Un. Mat. Argentina, Vol 49-2MINIMAL HERMITIAN MATRICES 21Parametrization of the set of minimalmatrices of 3× 3Definition 2. Let M ∈Mh3×3(C). We will say that M is of extremal type if thereexist(1) η ∈ [0, 2pi),(2) λ > 0,(3) µ ∈ R with |µ| ≤ λ,such that M is some of the following three matrices:µ 0 00 0 λ ei η0 λ e−i η 0 ; 0 0 λ e−i η0 µ 0λ ei η 0 0 ; 0 λ e−i η 0λ ei η 0 00 0 µDefinition 3. Let M ∈ Mh3×3(C). We will say that M is of non extremal typeif there exist:(1) η, ξ ∈ [0, 2pi),(2) λ, µ ∈ R with λ > 0, |µ| ≤ λ,(3) α, β, χ ∈ R≥0, with:α+ β + χ = 12 ,α+ β > 0, β + χ > 0,α+ χ > 0.such thatM = µ 2α n12 n31n12 2β n23n31 n23 2χ+ λ 0 m12 m31m12 0 m23m31 m23 0where:8>>>>>><>>>>>>:n12 =−2αβ ± i√2αβ χp(α + χ)(β + χ)e−iηn31 =−2αχ± i√2αβ χp(α+ β)(β + χ)e−iξn23 =−2β χ± i√2αβ χp(α+ β)(α+ χ)e−i(ξ−η)8>>>>>><>>>>>>:m12 =χ± i√2αβ χp(α + χ)(β + χ)e−iηm31 =β ± i√2αβ χp(α + β)(β + χ)e−iξm23 =α± i√2αβ χp(α + β)(α + χ)e−i(ξ−η)for one of the two corresponding choices of the signs.Remark 4. For matrices of both types (extremals and non extremals), the param-eters λ > 0 and µ give the norm of M , ||M || = λ, and the trace of M , Tr(M) = µ.In the previous Theorem 3 we have seen that M ∈ Mh3×3(C), with ‖M‖ = λ isminimal, if and only if, there exist unitary eigenvalues v+ and v− of λ and −λ suchthat their coordinates have equal absolute value.The following theorem gives a description of all the minimal hermitian matricesof 3× 3 in terms of the parameters α, β, χ, η, ξ, λ and µ.Rev. Un. Mat. Argentina, Vol 49-222 ANDRUCHOW, MATA-LORENZO, MENDOZA, RECHT AND VARELAFigure 1. Construction of α, β and χ appearing in theparametrization.Theorem 4. [Parametrization theorem] Let Z ∈ Mh3×3(C), then, Z is mini-mal, if and only if, verifies any of the following mutually exclusive cases:(1) The eigenvector v+ of λ has azero coordinate and Z is of ex-tremal type.(2) The eigenvector v+ of λ has nozero coordinates and Z is of nonextremal type.4. Minimal matrices in a classEvery matrix M ∈Mh3×3(C) can be writen in the form,M =a x y¯x¯ b zy z¯ c ,where a , b , c ∈ R and x , y , z ∈ C.We already observed that, if M is minimal hermitian (not null), the eigenvaluesof M are λ = ‖M‖, −λ and µ = Tr(M) (with |µ| ≤ λ).If we consider the characteristic polynomial of M :det(M − ΛI) = −Λ3 + uΛ2 + vΛ + wthese properties impose necessary conditions on the coefficients u, v and w:u v + w = 0v = ‖M‖2u = Tr(M)u2 ≤ vThen for specific class of a matrix [M0] (that is for x, y and z fixed)M0 =0 x y¯x¯ 0 zy z¯ 0 ,we consider the real manifold of R3 given by the equation:∆ := u v + w = 0Rev. Un. Mat. Argentina, Vol 49-2MINIMAL HERMITIAN MATRICES 23or, in terms of the matrix coefficients a, b and c:∆ := (a+ b)(a+ c)(b+ c)− (a+ b)|x|2 − (a+ c)|y|2 − (b+ c)|z|2 − 2Re(x y z) = 0.Let us call with ∆ this manifold.Every minimal matrix M of [M0] must belong to this manifold and must mini-mizeλ2 := ‖M‖2 = |x|2 + |y|2 + |z|2 − ab− ac− bcwith the restriction u2 ≤ v.To simplify the expression of the map ∆, the following linear change of variablescan be introduced,a = (r + s− t)/2 , b = (t+ r − s)/2 , c = (s+ t− r)/2.The equations above change to give a new description of ∆∆ := r s t− r |x|2 − s |y|2 − t |z|2 + 2Re(x y z) = 0,and a new expression for the function to minimize λ2,λ2(r, s, t) =14(r2 + s2 + t2)−12(r s+ r t+ s t) + |x|2 + |y|2 + |z|2.5. Different casesTo find the minimal matrix (or matrices) in the class [M0] we have to minimizeλ2 on ∆. We shall consider four cases depending on the triple (x, y, z). Figuresrepresenting ∆ using the r, s and t variables are shown in each case. It can beproved that only in the fourth case there might be multiple minima in the givenclass [M0]. Two rounded surfaces, shown in the first three figures, do not belongto ∆, they represent the bounding surfaces µ = ±λ in between which the (unique)minimum is located.(1) When Im(x y z) 6= 0.In this case the surface∆ is regu-lar (a smooth manifold) and themethod of Lagrange multiplierscan be used to find the uniqueminimum in the class. In the fig-ure to the right, the middle por-tion represents the componentsatisfying u2 ≤ v, and the darkpoint indicates the minimum.(2) When Im(x y z) = 0 and Re(x y z) 6= 0.Rev. Un. Mat. Argentina, Vol 49-224 ANDRUCHOW, MATA-LORENZO, MENDOZA, RECHT AND VARELAIn this case the surface ∆ is notregular, has one singular pointwhich is the unique minimumin the class. In the figure tothe right, two components of ∆touch at the singular point whichis the minimum.(3) When exactly one coordinate of (x, y, z) is null.In this case the surface ∆ is reg-ular; the class has a unique min-imum at the origin, (r, s, t) =(0, 0, 0) = (a, b, c). Observe thatin the figure the vertical t-axislies in ∆.(4) When exactly two coordinates of (x, y, z) are null.In this case the surface ∆ is notregular along two curves, the twobranches of the hyperbola shownin the figure, and the class hasmultiple minima, represented bythe segment shown in the figurejoining the two branches of thehyperbola.Remark 1. Despite this last case, there is no multiplicity of minimal curves ofmatrices in 3 × 3 and all multiple minimal matrices Z produce the same curveγ(t) = [etZM ] for M ∈ U /Udiag .Rev. Un. Mat. Argentina, Vol 49-2MINIMAL HERMITIAN MATRICES 25If the matrixM is real with zero diagonal and we suppose that there is a diagonalD that reaches the minimum of ||M + D|| and such that M + D has all of itseigenvalues of equal absolute value (that is, the spectrum of M +D is {±λ} and λor −λ has double multiplicity), then a precise formula of D can be found in termsof the entries of M (this proposition is proved in [10] motivated by results of [9]).We state it here as the following remark:Remark 2. Let M =0 c ac 0 ba b 0, with a, b, c 6= 0, be a matrix in R3×3 andD ∈ R3×3 be a diagonal such that ||M + D|| is minimum and the eigenvalues ofM +D have equal absolute value. Then D must be of the formD =−12 (bca+ abc− acb) 0 00 − 12 (−bca+ abc+ acb) 00 0 − 12 (bca− abc+ acb) .6. The topology of the set of minimal matrices of 3× 3Using the theorem of parametrization and considering the setsΣ = { (α, β, χ) ∈ R3 : α+ β + χ =12, α ≥ 0, β ≥ 0, χ ≥ 0 } ,andC ={(µ , λ) ∈ R2 : λ > 0, |µ| ≤ λ},we defineW = Σ× S1 × S1 × C,where S1 is the unitary circle in the complex plane and∇ = W+ ⊔W−,is the disjoint union of two copies of W .Let us consider in ∇ the smallest equivalent relation ‘∼’ that identifies theelements of ∇ that give the same matrix in the theorem of parametrization. Then,it can be proved that the set ∇ /∼ is homemophic to the set of minimal hermitianmatrices.The following is a representation of the set of minimal hermitian matrices through∇ /∼ .Rev. Un. Mat. Argentina, Vol 49-226 ANDRUCHOW, MATA-LORENZO, MENDOZA, RECHT AND VARELA7. An example in 4× 4 using minimal matricesWe have remarked that in 3× 3, despite of the existence of matrices that allowinfinite minimizing diagonals, there are no multiplicity of short curves. This is notthe case in 4× 4.Let us consider the manifoldS ⊂ P(4) = U4×4/(Udiag)4×4defined byS = (SU(2)× SU(2))/(SU(2)× SU(2))diag∼=(SU(2)/SU(2)diag)×(SU(2)/SU(2)diag)where SU(2) is the special unitary group (unitary complex matrices with determi-nant equal to 1) and with the Finsler metric in P(4) as in the 3× 3 case.Using a suitable description (see [2] and [11] for details) we can consider S ∼=S2 × S2, where S2 is the unit sphere in R3. This gives a better geometrical viewof the example. Let us outline which are the properties of the curves mentioned atthe beginning of the section.Let N = (N,N) ∈ S2 × S2 be the point whose coordinates are both the NorthPole, N ∈ S2. Let Q = (Q1, Q2) ∈ S2 × S2 be any point such that Q1 has higherlatitude than Q2 in S2 (Q1 is closer to N than Q2).Q is going to be fixed so that Q2 is above the equator line (andQ1 is even higher). In [2], using the characterizations of mini-mal matrices seen previously, a family of minimal curves Γβ(t) =(γ1,β(t), γ2(t)), for t ∈ [0, 1], was constructed, all joining N to Q,with the following properties.• The curve γ2(t) in S2 will trace the smaller arc of the greatcircle that contains N and Q2.• The family of curves γ1,β(t) will vary continuously with theparameter β.• Each of the curves of the family γ1,β(t) will parameterize thesmaller arc of some circle in S2 that joins N to Q1; the arcswill not be great circles but for β = 0.• The curve γ2(t) runs over a great circle in S2.• The curve γ1,β(t) varies continuously with the parameter β.• The curve γ1,β(t) has constant speed in S2.• The curve γ2(t) has constant speed in S2.The following is a representation of these curves in S2 × S2:Rev. Un. Mat. Argentina, Vol 49-2MINIMAL HERMITIAN MATRICES 27References[1] Andruchow, Esteban; Mata-Lorenzo, Luis E.; Recht, Lázaro; Mendoza, Alberto;Varela, Alejandro. Minimal matrices and the corresponding minimal curves on flagmanifolds in low dimension, preprint.[2] Andruchow, Esteban; Mata-Lorenzo, Luis E.; Recht, Lázaro; Mendoza, Alberto;Varela, Alejandro. Infinitely many minimal curves joining arbitrarily close points ina homogeneous space of the unitary group of a C∗-algebra. Rev. Un. Mat. Argentina46 (2005), no. 2, 113–120 (2006).[3] Bhatia, R., Choi, M.D., Davis, C. Comparing a matrix to its off-diagonal part. TheGohberg anniversary collection, Vol. I (Calgary, AB, 1988), 151–164, Oper. TheoryAdv. Appl., 40, Birkhauser, Basel, 1989.[4] Durán, C.E., Mata-Lorenzo, L.E., Recht, L., Metric geometry in homogeneous spacesof the unitary group of a C∗-algebra: Part I–minimal curves, Adv. Math. 184 No. 2(2004), 342-366.[5] Fan, K., Hoffman, A.J., Some metric inequalities in the space of matrices, Proc.Amer. Math. Soc., 6 (1955) 11-116.[6] Halmos, P.R., Positive approximants of operators, Indiana Univ. Math. J., 21 (1972),951–960.[7] Halmos, P.R., Spectral approximants of normal operators, Proc. Edinburgh Math.Soc., 19 (1974), 51–58.[8] Higham, N.J., Matrix nearness problems and applications, Applications of MatrixTheory, Oxford University Press, 1989, 1–27.[9] Mata-Lorenzo, L.E., Recht, L. Convexity properties of Tr[(a ∗ a)n], Linear Algebraand its Applications 315 (2000), 25-38.Rev. Un. Mat. Argentina, Vol 49-228 ANDRUCHOW, MATA-LORENZO, MENDOZA, RECHT AND VARELA[10] Varela, A., Villagra, M.P., Algunos resultados de minimización de matrices de dia-gonal variable, preprint.[11] Whittaker, E. T. A Treatise on the Analytical Dynamics of Particles and RigidBodies, Cambridge University Press, London, 1988.Esteban AndruchowInstituto de Ciencias,Universidad Nacional de Gral. Sarmiento,J. M. Gutierrez 1150, (1613) Los Polvorines,Argentinaeandruch@ungs.edu.arLuis E. Mata-LorenzoUniversidad Simón Bolívar,Apartado 89000, Caracas 1080A, Venezuelalmata@usb.veAlberto MendozaUniversidad Simón Bolívar,Apartado 89000, Caracas 1080A, Venezuelajacob@usb.veLázaro RechtUniversidad Simón Bolívar,Apartado 89000, Caracas 1080A, Venezuelarecht@usb.veAlejandro VarelaInstituto de Ciencias,Universidad Nacional de Gral. Sarmiento,J. M. Gutierrez 1150, (1613) Los Polvorines,Argentinaavarela@ungs.edu.arRecibido: 10 de abril de 2008Aceptado: 15 de mayo de 2008Rev. Un. Mat. Argentina, Vol 49-2

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Minimal hermitian matrices with fixed entries outside the diagonal

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