Lidskii aditivo y multiplicativo con igualdades

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1 Lidsii aditivo y multiplicativo con igualdades Seminario IAM 26 / 10 / Submajorization and log-majorization Next we briefly describe majorization and log-majorization, two notions from matrix analysis theory that will be used throughout the paper. For a detailed exposition on these relations see [1]. Given x, y R d we say that x is submajorized by y, and write x w y, if x i y i for every I d. If x w y and tr x = d x i = d y i = tr y, then we say that x is majorized by y, and write x y. If the two vectors x and y have different size, we write x y if the extended vectors completing with zeros to have the same size) satisfy the previous relationship. On the other hand we write x y if x i y i for every i I d. It is a standard exercise to show that x y = x y = x w y. Log-majorization between vectors in R d 0 is a multiplicative analogue of majorization in R d. Indeed, given x, y R d 0 we say that x is log-majorized by y, denoted x log y, if x i y i for every I d 1 and d d x i = y i. Our interest in log-majorization is also motivated by the relation of this notion with tracial inequalities for convex functions. It is wnown see [1]) that if x, y R d 0, x log y x w y. Hence, if x, y R d 0 are such that x log y then for every convex and increasing function f : 0, ) R we get that trfx)) trfy)). 2 Aditivo 2.1 El teorema tradicional Theorem 2.1 Teorema de Weyl). Sean A, B Hd). Entonces: λ j A) + λ d B) λ j A + B) λ j A) + λ 1 B) para todo j I d. 1) 1

2 Si se da alguna de las igualdades, existe un x H unitario tal que A + B) x = λ j A + B) x, A x = λ j A) x and B x = λ 1 B) x o bien λ d B) x, Proof. Let u j and v j denote the eigenvectors of A and A + B respectively, corresponding to their eigenvalues arranged in decreasing order. Consider the subspaces S = span{v 1,..., v j } and T = span{u j,..., u d }. Let x S T an unit vector. Then λ j A + B) A + B) x, x = A x, x + B x, x λ j A) + λ 1 B). If we further assume that an equality holds in 1), then we deduce that A + B)x, x = λ j A + B), A x, x = λ j A) and B x, x = λ 1 B). Como x S T, these last facts imply that A + B) x = λ j A + B) x and A x = λ i A) x. The fact that B x, x = λ 1 B) x = B x = λ 1 B) x is nown. La otra sale parecido. Corollary 2.2 Weyl s monotonicity principle). Let A Hd) and B M d C) +. Then If there exists J I d such that λ j A + B) λ j A) for every j I d. 2) λ j A + B) = λ j A) for every j J, then there exists an orthonormal system {x j } j J such that A x j = λ j A) x j and B x j = 0 for every j J. Proof. Inequality 2) follows easily from Thm. 2.1 λ d B) 0). The second part follows by induction on the set J : Fix j 0 J. By Thm. 2.1 again, there exists a unit vector x j0 such that A x j0 = λ j0 A) x j0 and B x j0 = λ d B) x j0 = 0. This proves the case J = 1. If J > 1, consider the space W = {x j0 } C d which reduces A, B and A + B. Let I = {j : j J, j < j 0 } {j 1 : j J, j > j 0 }. The operators A W LW ) sa and B W LW ) + satisfy that λ j A W +B W ) = λ j A W ) for every j I, with I = J 1. By the inductive hypothesis we can find an orthonormal system {x j } j I W which satisfy the desidered properties. Theorem 2.3 Otro de Weyl). Sean A y B Hn). Entonces λa + B) λa) + λb). 3) Es importante el hecho de que, en la suma µa) + µb), se asume que ambos vectores están ordenados de la misma forma. Proof. Sale por el Principio del Máximo de Ky Fan, que decia que si A Hn) entonces λ j A) = max Ax j, x j, para todo I n, donde el máximo se toma sobre todas las -uplas ortonormales {x 1,..., x } en C n. 2

3 Observar que 3) dice que, si A, B Hn) y I n, λ j A + B) λ j A) λ j B), pero eso implica mal que λa + B) λa) λb), porque el orden de la izquierda no es el correcto. La buena prueba sale con el otro Weyl: Theorem 2.4 Lidsii con igualdad). Let A, B Hd). Then λa + B) λa) λb) Por otro lado, una eventual igualdad ) λa + B) λa) = λb) = A and B commute. 4) Proof. We can assume that B is not a multiple of the identity. Let I d be such that λ 1 B) > λ B). Let us denote by B = B λ B) I. By construction λ B ) = 0. Let B + be the positive part of B. Then λb + ) = ) λ 1 B ),..., λ 1 B ), 0 1 d +1 Since B + M dc) + and B B + then Weyl s monotonicity principle implies that λ j A + B ) λ j A + B + ), j I d = λ j A + B ) λ j A + B + ), j J 1 j J 1 para cualquier J 1 I n tal que J 1 = 1. Luego λ j A + B ) λ j A) λ j A + B + ) λ ja) j J 1 j J 1 5) j I d λ j A + B + ) λ ja) 6) = tr A + B + ) tr A 5) = 1 λ j B ), since λ j A + B + ) λ ja) for j I d again by Weyl s monotonicity principle. Sumandoles ahora 1 veces λ B) de cada lado, sale que 1 λ j A + B) λ j A) λ j B) j J 1 para todo J 1 I n tal que J 1 = 1. Eso muestra que λa + B) λa) λb). Suppose that there exists a permutation σ S d such that λ j B) = λ σj) A + B) λ σj) A) for every j I d. Therefore, there exists an increasing sequence {J } d =1 of subsets of I d such that J = and j J λ j A + B) λ j A) = λ j B) for every I d, 7) 3

4 and notice Eq. 7) also holds if we replace B by B. Tambien las desigualdades 6) de arriba siguen valiendo, pero ahora son igualdades con los J adecuados), lo que ensangucha lo del medio. Luego si J c 1 = I d \ J 1, se tiene que j J c 1 λ j A + B + ) λ ja) = 0 Weyl s = λ j A + B + ) = λ ja) for every j J c 1. By Corollary 2.2 there exists an ONS {x j } j J c 1 such that A x j = λ j A) x j and B + x j = 0 for every j J 1 c. All these facts together imply that P def = j J c 1 x j x j = P er B + since λ B ) = 0 and dim er B + = d + 1 = J c 1. and P A = A P, Recall that P is also the spectral projection of B associated to the interval, λ B)], for any I d such that λ 1 B) > λ B). Since the spectral projection of B associated with, λ 1 B)] equals the identity operator, and B is a linear combination of the projections P and I, we conclude that A and B commute. 2.2 Characterization of optimal matching matrices Fix S 0 Hn). Es facil ver que Lidsii implica que si S 1 Hn) entonces λ S 0 ) + λ S 1 ) λs 0 + S 1 ), porque λ S 1 ) = λ S 1 ). In this section we characterize those matrices S 1 M d C) + such that λs 0 + S 1 ) = λ S 0 ) + λ S 1 ) ). 8) If S 1 Hn) satisfies Eq. 8) then S 1 is an optimal matching matrix OMM) for S 0. Theorem 2.5. Let S 0, S 1 Hd) be such that λs 0 + S 1 ) = λs 0 ) + λ S 1 ) ). Then S0 and S 1 commute. Proof. Tae B = S 0 + S 1 and A = S 1. Therefore λa) = λ A) = λ S 1 ), so that λa + B) λa) = λs 0 ) + λ S 1 ). Hence A and B satisfy the assumptions in 4) and they must commute. In this case S 0 and S 1 also commute. Let S 0 M d C) + and let S 1 M d C) + be an OMM for S 0. By Theorem 2.5 there exists a common ONB of eigenvectors for S 0 and S 1. Pero la cosa es mejor: Theorem 2.6 Equality in Lidsii s inequality). Let S 0 M d C) + and let S 1 M d C) + be an OMM for S 0. Let λ = λs 0 ) and µ = λ S 1 ). Then there exists {v i : i I d } a ONB for S 0 and λ such that S 1 = i I d µ i v i v i and S 0 + S 1 = i I d λ i + µ i ) v i v i. 4

5 In order to give a proof we first consider some technical results. We begin by fixing some notations. Let λ R d >0. For every j I d we define the set Lλ, j) = {i I d : λ i = λ j }. If we assume that λ = λ or λ = λ then the sets Lj) are formed by consecutive integers. In the firs case we have that λ i < λ j = > l for every Lλ, i) and l Lλ, j). Given a permutation σ S d and λ R d >0 we denote by λ σ = λ σ1),..., λ σd) ). Observe that λ = λ σ λ = λ σ 1 σ Lλ, j) ) = Lλ, j) for every j I d. 9) The following inequality is well nown see for example [1, II.5.15]): Proposition 2.7 Rearrangement inequality for products of sums). Let λ, µ R d >0 be such that λ = λ and µ = µ. Then d λ i + µ i ) d λ i + µ σi) ) for every permutation σ S d. The following result deals with the case of equality in the last inequality. Proposition 2.8. Let λ, µ R d >0 be such that λ = λ and µ = µ. Let σ S d be such that Moreover, assume that σ also satisfies that: λ + µ) = λ + µ σ ). if r, s I d are such that µ σr) = µ σs) with σr) < σs) then r < s. 10) Then the permutation σ satisfies that λ = λ σ. Proof. For every τ S d let F τ) = d λ i + µ τi) ). By the hypothesis and Proposition 2.7, F σ) = F id) = max τ S d F τ). Assume that λ λ σ 1. In this case there exists j, I d such that µ j < µ and λ σ 1 j) < λ σ 1 ). 11) Indeed, let j 0 be the smallest index such that σ 1 does not restrict to a permutation on Lλ, j 0 ). Then, there exists j Lλ, j 0 ) such that σ 1 j) / Lλ, j 0 ). As σ 1 Lλ, j 0 ) \ {j}) Lλ, j 0 ) there also exists / Lλ, j 0 ) such that σ 1 ) Lλ, j 0 ). They have the required properties: First note that λ σ 1 j) < λ j0 = λ σ 1 ) and then also σ 1 j) > σ 1 ) ) because σ 1 j) can not be in Lλ, j 0 ) nor in Lλ, r) for any r < j 0 where σ 1 acts as a permutation). A similar argument shows that j <. We have used in both cases that the sets Lλ, j) are formed by consecutive integers, since the vector λ is decreasingly ordered. Observe that j < = µ j µ. So it suffices to show that µ j µ. Let us denote by r = σ 1 j) and s = σ 1 ). The previous items show that r > s and σr) < σs). Hence the equality µ j = µ σr) = µ σs) = µ is forbidden by our hypothesis 10). 5

6 So Eq. 11) is proved. Consider now the permutation τ = σ 1 j, ), where j, ) stands for the transposition of the indexes j and. Straightforward computations show that λ σ 1 j) + µ j ) λ σ 1 ) + µ ) λ σ 1 j) + µ ) λ σ 1 ) + µ j ) = λ σ 1 j) λ σ 1 )) µ µ j ) 11) < 0. From the previous inequality we conclude that F id) = F σ) < F τ) F id). This contradiction arises from the assumption λ λ σ 1. Therefore λ = λ σ 1 = λ σ as 9) desired. Remar 2.9. Let λ, µ R d >0 be such that λ = λ and µ = µ. Let τ S d be such that λ + µ) = λ + µ τ ). Then, by considering convenient permutations of the sets Lµ, j) we can always replace τ by σ in such a way that µ σ = µ τ and such that this σ satisfies the condition 10) of Proposition 2.8. Hence, in this case λ + µ) = λ + µ σ ) and the previous result applies. Ahora si va la prueba. Repetimos el enunciado de onda: Theorem 2.6. Let S 0 M d C) + and let S 1 M d C) + be an optimal matching matrix for S 0. Let λ = λs 0 ) and µ = λ S 1 ). There exists {v i : i I d } a ONB for S 0 and λ such that S 1 = i I d µ i v i v i and S 0 + S 1 = i I d λ i + µ i ) v i v i. 12) Proof. Let us assume further that S 0, S 1 are invertible matrices so that λ, µ R d >0. By Theorem 2.5 we see that S 0 and S 1 commute. Then, there exists B = {w i : i I d } an ONB for S 0 and λ such that S 1 w i = µ τi) w i for every i I d, and for some permutation τ S d. Therefore ) 8) λ + µ = λs 0 + S 1 ) = ) λ + µ τ. By Remar 2.9 we can replace τ by σ S d in such a way that µ τ = µ σ, λ + µ) = λ + µ σ ) and σ satisfies the hypothesis 10). Hence, by Proposition 2.8, we deduce that λ σ 1 = λ. Therefore one easily checs that the ONB formed by the vectors v i = w σ 1 i) for i I d i.e. the rearrangement B σ 1 of B) is still a ONB for S 0 and λ, but it now satisfies Eq. 12). In case S 0 or S 1 are not invertible, we can argue as above with the matrices S 0 = S 0 + I and S 1 = S 1 + I. These matrices are invertible and such that S 1 is an optimal matching for S 0. Further, λ S 0 ) = λs 0 ) + 1 and λ S 1 ) = λs 1 ) + 1. Hence, if {v i : i I d } has the desired properties for S 0 and S 1 then this ONB also has the desired properties for S 0 and S 1. 3 Multiplicativo 3.1 El teorema de Lee - Mathias We begin by revisiting the following well nown inequality from matrix theory. Our interest relies in the case of equality. Proposition 3.1 Ostrowsi s inequality). Let S Hd) and let V M d C). Then V V I = λ i V SV ) for every i I d. 13) Moreover, if there exists J I d such that = λ i V SV ) for i J, then there exists an o.n.s. {v i } i I C d such that V v i = v i and S v i = v i for i I. 6

7 Proof. Fix i I d and notice that, by Sylvester s law of inertia, λ i V S I) V ) = 0, since λ i S I) = 0. By Weyl s inequalities we have that 0 = λ i V S I) V ) λ i V SV ) + λ 1 V V ) = λ i V SV ) λ d V V ), 14) which clearly shows Eq. 13), since λ d V V ) 1. The following inequalities are the multiplicative version of Lidsii s inequality: Theorem 3.2 [4]). Let S Hd) and V Gl d). Let J I d, J =, be such that > 0 for i J. Then we have that λ i V V ) i J λ i V SV ) λ i V V ). 15) Proof. We can assume that V c I and that V Gl d) + sino se labura con V y anda). Let V = U D λ U where U Ud) and λ = λ V ). Let 2 d and denote by V = λ 1 V. In this case λ i V ) = λ iv ) λ V ) for every i I d. In particular, λ V ) = 1. We now consider B = U D µ U for µ = λ 1 V ),..., λ 1 V ), 1,..., 1) R d >0). Note that B I and V 1 B = B V 1 I. Let J 1 I d be such that J 1 = 1. Then, by Ostrowsi s inequality we get that λ i V S V ) λ i B V i J 1 i J 1 1 )V S V )V 1 B )) = λ i B S B ). i J 1 Using Ostrowsi s inequality again, we see that λ ib S B ) 1 for every i I d and therefore λ i B S B ) i J 1 i I d λ i B S B ) = detb S B ) dets) 1 = detb) 2 = λ i V 2 ). Asi sale la desigualdad de la derecha en 15) falta arreglar con λ V ) 2 2 de ambos lados). La otra sale tomando inversas adecuadas. 4 Igualdades Los teos anteriores con casos de igualdad. Ostrossy: Let S Hd) and let V M d C) be such that V V I. If there exists J I d such that = λ i V SV ) for i J, then there exists an o.n.s. {v i } i I C d such that V v i = v i and S v i = v i for i I. 7

8 Proof. The first part of the statement is well nown see for example [2, Thm ]). Hence, we prove the second part of the statement by induction on J, the number of elements of J. Assume first that V M d C) + is such that V 2 I. Fix i J and notice that, by Sylvester s law of inertia, λ i V S I) V ) = 0, since λ i S I) = 0. By Weyl s inequalities we have that λ i V SV ) λ d V 2 ) = λ i V SV ) + λ 1 V 2 ) λ i V S I) V ) = 0. 16) Since λ d V 2 ) 1 and λ i V SV ) = i J ) we conclude that λ d V 2 ) = 1. Moreover, by the equality in Eq. 16) and [6, Lemma 6.1.], there exists x C d, x = 1 such that V SV x = λ i V SV ) x and V 2 x = λ 1 V 2 ) x = λ d V 2 ) x = x. Hence V 2 x = x and then V x = x. Thus, V SV x = λ i V SV ) x = Sx = λ i V SV ) x = x. This proves the statement for J = 1. If we assume that J > 1 then we fix v i = x and consider W = {v i }, which reduces both A and V. Therefore an easy inductive argument involving the restrictiond S W and V W shows the general case. If we now consider an arbitrary V M d C) such that V V I then let V = U V be the polar decomposition of V. In this case V SV = U S S V U so that λ i V SV ) = λ i V S V ) for i I d, where V 2 = V V = U V V )U I. These last facts together with the case of equality for positive expansions prove the statement. In order to state our results we introduce the following notion. Definition 4.1. Let S Gl d) + and let V Gl d). We say that V is an upper multiplicative matching UMM) of S resp. lower MM or LMM of S) if there exists a family {J } Id such that J J +1 I d for 1 d 1, J = for I d and such that λ i V SV ) = λ i V V ), I d i J resp. λ i V V ) = λ d+1 iv V ) = λ i V SV ) i J λ i, I S) d ). Theorem 4.2. Let S Gl d) + and let V Gl d) be a UMM or a LMM of S. Then A and V commute. Proof. We can assume that V is not a multiple of the identity. We use the splitting technique considered in [4]. Let V Gl d) be an UMM of S. Assume further that V Gl d) + and let V = U D λ U where U Ud) and D λ M d C) + denotes the diagonal matrix with main diagonal λ = λ i ) i Id with λ i λ i+1 for 1 i d 1. Let 2 d be such that λ 1 > λ. Let V = λ 1 V, which is also an UMM for S. In this case λ i V ) = λ iv ) for λ V ) every i I d. In particular, λ V ) = 1. We now consider B = U D µ U, where µ = λ 1 V ),..., λ 1 V ), 1,..., 1) R d >0). 8

9 Notice that W = erb I) coincides with the span of the o.n.s. {u i } d i=, where each u i denotes the i-th column of the matrix U. In particular dimw = d + 1. Also notice that the orthogonal projection onto W, denoted by P, coincides with the spectral projection of V corresponding to the interval 0, λ V )]. On the other hand, by construction of B, we see that B I and V 1 B = B V 1 I. Let J 1 I d be such that J 1 = 1 and λ i V SV ) i J 1 1 = λ i V 2 ). 17) Then, by Ostrowsi s inequality we get that λ i V S V ) λ i B V i J 1 i J 1 1 )V S V )V 1 B )) = λ i B S B ) i J 1. Using Ostrowsi s inequality again, we see that λ ib S B ) λ i V S V ) i J 1 λ i B S B ) i J 1 = detb S B ) dets) 1 for every i I d and therefore i I d λ i B S B ) 1 = detb) 2 = λ i V 2 ). By Eq. 17) we see that the previous inequalities are actually equalities. Hence, if we let J c 1 = I d \ J 1 then J c 1 = d + 1 and i J c 1 λ i B S B ) = 1 = λ i B S B ) = for i J c 1. By the case of equality in Ostrowsi s inequality in Proposition 3.1 we conclude that there exists an o.n.s. {v i } i J c 1 C d such that B v i = v i and Sv i = for i J c 1. 18) Then we conclude that {v i } i J c 1 is an o.n.b. of W. Hence P = i J v 1 c i v i and, by Eq. 18), we conclude that P and A commute. Finally, since V can be written as a linear combination of the spectral projections P and the identity I, we see that V and A commute in this case. The general case for arbitrary V Gl d) follows from the positive case with the reduction described at the end of the proof of Proposition 3.1. Assume now that V M d C) + is a LMM of S. Then V 1 is an UMM for V SV. Indeed, if J I d is such that λ i V 2 ) = λ i V SV ), i J then we have that 1 ) 1 λ i V 2 ) = λ i )) V 2 λ i V SV ) = = λ i V 1 V SV ) V 1 ). λ i S) λ i V SV ) i J i J 9

10 By the first part of this proof, we conclude that V 1 and V SV commute, which implies that S and V commute. If V Gl d) is arbitrary we conclude that S and V commute with the reduction described at the end of the proof of Proposition 3.1. Theorem 4.3. Let S Gl d) + and let λ R d >0). Then, for every V M d C) such that λv V ) = λ we have that λs) λ log λv SV ) log λs) λ R d >0). 19) Moreover, if λv SV ) = λs) λ ) resp. λv SV ) = λs) λ) then there exists an o.n.b. {v i } i Id of C d such that S = i I d v i v i and V = i I d λ 1/2 d+1 i v i v i 20) resp. S = i I d v i v i and V = i I d λ 1/2 i V ) v i v i ). Proof. Let S and V be as above. Assume further that V Gl d) + and notice that then λv S V ) = λs 1/2 V 2 S 1/2 ). By Theorem 3.2 we get that, for every J I d with J =, λ i S) λ iv 2 ) = i J i J λ i S 1/2 S 1/2 V 2 S 1/2 )S 1/2 ) λ i S 1 ) λ i S 1/2 V 2 S 1/2 ). This shows that λ λ S) log λv S V ) or equivalently, that λs) λ log λv S V ). Moreover, the previous facts also show that if λv S V ) = λs) λ ) then S 1/2 is an UMM of S 1/2 V 2 S 1/2. By Theorem 4.2 we see that S 1/2 and S 1/2 V 2 S 1/2 commute, which in turn implies that S and V commute. Since S and V commute we conclude that there exists an o.n.b. {w i } i Id S = i I d w i w i and V = i I d λ σi) V ) w i w i of C d such that for some permutation σ S d. That is, in this case we have that ) ). λs) λ V 2 ) = λv SV ) = λs) λ σv 2 ) Notice that by replacing S and V by ts and tv for sufficiently large t > 0 we can always assume that S I Gl d) + and V I Gl d) +. Using the properties of the logaritm, we conclude that the vectors log λs) and log λv 2 ) R >0 ) are such that ) ) log λs) + log λ V 2 ) = log λs) + log λ σv 2 ). By [6, Proposition 6.6] see also [6, Remar 6.7]) we conclude that log λs) = log λ σ S). That is, if we set v i = w σ 1 i) for i I d then the o.n.b. {v i } i Id satisfies the conditions in Eq. 20). The general case, for V Gl d), follows by the reduction described at the end of the proof of Proposition 3.1. On the other hand, notice that a direct application of Theorem 3.2 shows that λ i V SV ) λ i V V ) = λ i V SV ) λ i V V ). 10

11 Hence, we conclude that λv SV ) log λs) λv V ) R d >0). Finally, in case that λv SV ) = λs) λv V ) we see that S is an UMM for S and therefore S and V commute. In this case it is straightforward to chec that there exists an o.n.b. {v i } i Id with the desired properties. Remar 4.4. Let S Gl d) + and let λ R d >0). Consider the set O S λ) = {V SV : V M d C), λv V ) = λ}. 21) Then, Theorem 4.3 shows that there exist log minimizers and maximizers in O S λ) and that their spectral and geometrical structure can be described explicitly. References [1] R. Bhatia, Matrix Analysis, Berlin-Heildelberg-New Yor, Springer [2] Horn, Johnson, matrix analysis... [3] A.A. Klyacho, Random wals on symmetric spaces and inequalities for matrix spectra. Special Issue: Worshop on Geometric and Combinatorial Methods in the Hermitian Sum Spectral Problem Coimbra, 1999). Linear Algebra Appl ), no. 1-3, [4] Li, Mathias Lidsii s multiplicative inequalities... [5] Massey, Ruiz, Stojanoff, Optimal duals and frame completions for majorization, acha... [6] Massey, Ruiz, Stojanoff, Optimal completions of a frame... 11