Convexity and Star-shapedness of Real Linear Images of Special Orthogonal Orbits

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1 Convexity and Star-shapedness of Real Linear Images of Special Orthogonal Orbits Pan-Shun Lau 1, Tuen-Wai Ng 1, and Nam-Kiu Tsing 1 1 Department of Mathematics, The University of Hong Kong, Pokfulam, Hong Kong August, 16 Abstract Let A R n n and SO n := {U R n n : UU t = I n, detu > } be the set of n n special orthogonal matrices. Dene the real) special orthogonal orbit of A by OA) := {UAV : U, V SO n }. In this paper, we show that the linear image of OA) is star-shaped with respect to the origin for arbitrary linear maps L : R n n R l if n l 1. In particular, for linear maps L : R n n R and when A has distinct singular values, we study B OA) such that LB) is a boundary point of LOA)). This gives an alternative proof of a result by Li and Tam on the convexity of LOA)) for linear maps L : R n n R. AMS Classication: 15A4, 15A18. Keywords: linear transformation, special orthogonal orbits, convexity, star-shapedness 1 Introduction Let O n := {U R n n : U t U = UU t = I n } and SO n := {U O n : detu > } be the sets of n n orthogonal matrices and n n special orthogonal matrices respectively. For any A R n n, we dene the special orthogonal orbit of A by OA) := {UAV : U, V SO n }. It is clear that every element in OA) has the same collection of singular values and the same sign of determinant. In [9], Thompson studied the set of diagonals of the matrices in OA), and in [8], Miranda and Thompson studied the panlau@hku.hk ntw@maths.hku.hk nktsing@hku.hk 1

2 characterizations of extreme values of LOA)) where L : R n n R is a linear map. A set S is said to be star-shaped with respect to c S if for all α 1 and x S, αx+1 α)c S. The c is called a star center of S. In this paper, we shall study the star-shapedness of images of OA) under arbitrary linear maps L : R n n R l. In fact the study of linear images of matrix orbits is a popular topic. If A, C are n n complex matrices and U n denotes the group of n n complex) unitary matrices, then the classical) numerical range of A, denoted by W A), and the C-numerical range of A, denoted by W C A), are simply the images of the unitary orbit of A, denoted by under the linear maps U n A) := {U AU : U U n }, X tre 1 X) and X trcx) respectively, where E 1 is the diagonal matrix with diagonal entries 1,,...,. It has been proved that W A) is always convex and W C A) is always star-shaped see [1], [], [1]). Many results on the convexity and the star-shapedness of other generalized numerical ranges, which can be expressed as some particular linear images of U n A), have been obtained e.g., see [1], [3], [4], [5], [6], [11], [1]). Our paper is organized as follows. In Section, we study an inclusion relation of LOA)) with L : R n n R l and n l 1. We then apply the inclusion relation to show that LOA)) is star-shaped for general A and L : R n n R l where n l 1. In particular, the star-shapedness holds for LOA)) with L : R n n R and n 3. Moreover, we shall extend our results to linear images of the following joint real) orthogonal orbits, O 1 A 1,..., A m ; G) := {A 1 V,..., A m V ) : V G}, O A 1,..., A m ; G) := {UA 1,..., UA m ) : U G}, O 3 A 1,..., A m ; G) := {UA 1 V,..., UA m V ) : U, V G}, where G = O n or SO n. In Section 3, we study boundary points of LOA)) with L : R n n R. When A R n n has distinct singular values, we shall discuss the conditions on U, V SO n under which LUAV ) will be a boundary point of LOA)). Then we show that the intersection of LOA)) and any of its supporting lines is path-connected. Combining the result in Section, convexity of LOA)) for L : R n n R then follows. This result was proved by Li and Tam [7] with a dierent approach. We shall also discuss the convexity of linear images of joint orthogonal orbits. Star-shapedness of linear image of OA) The following is the rst main theorem in this section.

3 Theorem.1. Let l 3. For any A R n n and any linear map L : R n n R l with n l 1, LOA)) is star-shaped with respect to the origin. We need some lemmas to prove Theorem.1. L : R n n R l can be expressed as Note that any linear map LX) = trp 1 X),..., trp l X) ) t for some P 1,..., P l R n n. For convenience, for M R n n and any P 1,..., P l R n n, we dene LP 1,..., P l ; M) := { trp 1 X),..., trp l X) ) t : X M }. For A, P 1,..., P l R n n, we let S A P 1,..., P l ) be the set containing P 1,..., P l ) where P 1,..., P l Rn n and LP 1,..., P l ; OA)) LP 1,..., P l ; OA)). This denition is motivated by Cheung and Tsing [1]. Below are some basic properties of S A P 1,..., P l ). Lemma.. Let A R n n. For any P 1,..., P l R n n, the followings hold: a) S XAY UP 1 V,..., UP l V ) = S A P 1,..., P l ) for any U, V, X, Y SO n ; b) UP 1 V,..., UP l V ) S A P 1,..., P l ), for any U, V SO n ; c) S A P 1,..., P l ) S AP 1,..., P l ) for any P 1,..., P l ) S AP 1,..., P l ); d) LP 1,..., P l ; OA))= { trp 1A),..., trp l A)) t :P 1,..., P l ) S AP 1,..., P l ) }. Proof. a), b) and c) are trivial. For d), follows from b) and follows from the denition of S A P 1,..., P l ). Lemma.3. The following statements are equivalent hence if one of these statements holds then the other three must also hold): a) LOA)) is star-shaped with respect to the origin for any A R n n and any linear map L : R n n R l ; b) S A P 1,..., P l ) is star-shaped with respect to n,..., n ) for any A R n n and any P 1,..., P l R n n, where n is the n n zero matrix; c) LSO n ) is star-shaped with respect to the origin for any linear map L : R n n R l ; d) S In P 1,..., P l ) is star-shaped with respect to n,..., n ) for any P 1,..., P l R n n. Proof. a) b)) For any P 1,..., P l ) S AP 1,..., P l ), U, V SO n and α 1, we have trαp 1UAV ),..., trαp luav )) t LP 1,..., P l; OA)) LP 1,..., P l ; OA)). 3

4 Hence αp 1,..., P l ) S AP 1,..., P l ). b) a)) Apply Lemma. b). a) c)) If we take A = I n, then OA) = SO n. c) a)) Let L : R n n R l be linear and A R n n. For any U SO n, dene linear map L UA : R n n R l by L UA X) = LUAX). For any U, V SO n and α 1, since L UA SO n ) is star-shaped with respect to the origin, there exists V SO n such that αluav ) = αl UA V ) = L UA V ) = LUAV ) LOA)). c) d)) Apply similar arguments as those in a) b). To prove Theorem.1, we apply Lemma.3 and show the star-shapedness of S In P 1,..., P l ) for any P 1,..., P l R n n with n l 1. For simplicity, we denote S In P 1,..., P l ) by SP 1,..., P l ). In fact, by the following lemma, we may focus only on the case of n = l 1. Lemma.4. If S ˆP 1,..., ˆP l ) is star-shaped with respect to the origin for all ˆP 1,..., ˆP l R n n, then for all m > n and for all P 1,..., P l R m m, SP 1,..., P l ) is star-shaped with respect to the origin. Proof. Let m = n + k where k is a positive integer. SP 1,..., P l ), we write P i P = i1 P i, P i3 P i4 For any P 1,..., P l ) where P i1 Rn n and P i4 Rk k. We shall show that P 1ɛ),..., P l ɛ)) SP 1,..., P l ) where P i ɛ) = ɛi n I k )P i and ɛ 1. For any U SO m, we write U1 U U =, U 3 U 4 where U 1 R n n and U 4 R k k. Then for ɛ 1, by the hypothesis of the lemma, there exists V SO n such that trp 1 ɛ)u),..., trp lɛ)u) ) t = ɛ trp 11U 1 + P 1U 3 ),..., trp l1u 1 + P lu 3 )) t + trp 13U + P 14U 4 ),..., trp l3u + P l4u 4 )) t = tr [ P 11U 1 + P 1U 3 )V ],..., tr [ P l1u 1 + P lu 3 )V ]) t ) t + trp 13U + P 14U 4 ),..., trp l3u + P l4u 4 ) = tr [ P 1UV I k ) ],..., tr [ P luv I k ) ]) t LP 1,..., P l; SO m ) LP 1,..., P l ; SO m ). 4

5 Since this holds for all U SO m, we have P 1ɛ),..., P l ɛ)) SP 1,..., P l ). Note that the preceding result also holds if we multiply arbitrary n rows of P i by ɛ 1. We re-apply the result by considering all n-combinations of rows to obtain ɛ N P 1,..., P l ) SP 1,..., P l ), where N = m!. For any α 1, n!k! we put ɛ = N α to obtain αp 1,..., P l ) SP 1,..., P l ). and We now consider the following recursively dened matrices. Let cos θ1 sin θ Rθ 1 ) = 1 sin θ 1 cos θ 1 [ ] cos θ Rθ 1,..., θ k ) = k I N sin θ k Rθ 1,..., θ k 1 ) sin θ k Rθ 1,..., θ k 1 ) t cos θ k I N where N = k 1. Note that Rθ 1,..., θ k ) SO k. Lemma.5. Let l and P 1,..., P l R N N where N = l 1. Then for any U, V SO N, the set EU, V ) := { tr Rθ 1,..., θ l 1 )UP 1 V ),..., tr Rθ 1,..., θ l 1 )UP l V )) t } :θ1,..., θ l 1 [, π] is an ellipsoid in R l centered at the origin and is a subset of LP 1,..., P l ; SO N ). Proof. We rst show that for any A R N N where N = l 1, cos θ l 1 tr Rθ 1,..., θ l 1 )A ) = [ a 1 ] sin θ l 1 cos θ l a l sin θ l 1 sin θ l cos θ l 3. sin θ l 1 sin θ l sin θ 1 for some a 1,..., a l R by induction on l. The case for l = is trivial. Now assume it is true for l m where m and consider A R M M where M = m 1. We write A1 A A = A 3 A 4 where A i R M M, i = 1,..., 4. Then tr Rθ 1,..., θ m )A ) = cos θ m tra 1 + A 4 ) + sin θ m tr Rθ 1,..., θ m 1 )A 3 A t ) ). By induction assumption on tr Rθ 1,..., θ m 1 )A 3 A t ) ), tr Rθ 1,..., θ m )A ) is in the desired form. Hence we have cos θ l 1 sin θ l 1 cos θ l EU, V ) = T sin θ l 1 sin θ l cos θ l 3 : θ 1,..., θ l 1 [, π],. sin θ l 1 sin θ l sin θ 1 5

6 for some T R l l and hence EU, V ) is an ellipsoid in R l centered at the origin. As Rθ 1,..., θ k ) is a special orthogonal matrix, EU, V ) LP 1,..., P l ; SO N ). Lemma.6. Let l 3. For any P 1,..., P l R N N where N = l 1, there exist U, V SO N such that EU, V ) dened in Lemma.5 degenerates i.e., EU, V ) is contained in an ane hyperplane in R l ). Proof. From the proof of Lemma.5, we see that if there exist U, V SO N such that P 1) UP 1 V = 1 P 1) P 1) 3 P 1) 4 where P 1) i R N N, i = 1,..., 4, trp 1) 1 + P 1) 4 ) = and P 1) = P 1) 3 =, then the rst coordinate of EU, V ) is always and hence EU, V ) degenerates. Let U, V SO N be such that U P 1 V = diagp 1,..., p N ). Then U = U, V = V 1 1 ) 1 1 will give the desired UP 1 V. Note that, by considering P 1 = 1 and P =, then for any 1 U, V SO, the ellipse EU, V ) dened in Lemma.5 is always non-degenerate. Hence Lemma.6 and Theorem.1 fail to hold for l =. We are now ready to prove our rst main result. Proof of Theorem.1. By Lemma.3 and Lemma.4, it suces to show that for any P 1,..., P l R N N with N = l 1, SP 1,..., P l ) is star-shaped with respect to N,..., N ). Let P 1,..., P l ) SP 1,..., P l ) and α 1. For any U SO N, we dene EI N, U) as in Lemma.5. If α trp 1U),..., trp 1U) ) t EI N, U), then we have α trp 1U),..., trp 1U) ) t LP 1,..., P l; SO N ) LP 1,..., P l ; SO N ). Assume now α trp 1U),..., trp 1U) ) t / EIN, U). As the center of EI N, U) is the origin, we have α trp 1U),..., trp 1U) ) t lies inside the ellipsoid EIN, U). As SO N SO N is path connected, consider a continuous function f : [, 1] SO N SO N with f) = I N, U) and f1) = U, V ) where U, V ) are de- ned in Lemma.6. Then by continuity of f, there exists s [, 1] such that α trp 1U),..., trp 1U) ) t Efs)) LP 1,..., P l ; SO N) LP 1,..., P l ; SO N ). As it is true for all U SO N, we have αp 1,..., P l) + 1 α) n,..., n ) = αp 1,..., P l) SP 1,..., P l ). 6

7 In fact for l =, we have the following theorem, the proof of which is given by Lemma.8 to Corollary.11. Theorem.7. Let A R n n and L : R n n R be a linear map with n 3. Then LOA)) is star-shaped with respect to the origin. Lemma.8. Let n. For any P, Q R n n, U SO n, the locus of the point trtθ P U), trt θ QU) ) t where Tθ = Rθ) I n forms an ellipse EU) in R when θ runs through [, π]. Proof. We write P = p 1) p ) P 3), Q = q 1) q ) Q 3) and U = [ u 1) u ) U 3) ] where p t 1), pt ), qt 1), qt ), u1), u ) R n and P t 3), Qt 3), U 3) R n n ). Direct computation shows trt θ P U) = cos θp 1) u 1) + p ) u ) ) + sin θp ) u 1) p 1) u ) ) + trp t 3) U 3) ). Similarly for trt θ QU). Hence trtθ P U) p1) u = 1) + p ) u ) p ) u 1) p 1) u ) cos θ trp3) U trt θ QU) q 1) u 1) + q ) u ) q ) u 1) q 1) u ) + 3) ) sin θ trq 3) U 3), ) the locus of which forms an ellipse possibly degenerate) when θ runs through [, π]. Lemma.9. For any P, Q R n n with n 3, there exists U SO n such that the ellipse EU ) dened in Lemma.8 degenerates. Proof. Note that EU) degenerates if we nd orthonormal vectors u 1), u ) R n such that the matrix p1) u 1) + p ) u ) p ) u 1) p 1) u ) q 1) u 1) + q ) u ) q ) u 1) q 1) u ) is singular. We will show that for any given p 1, p R n, there exist orthonormal vectors u 1, u such that p t 1u = p t u 1 = p t 1u 1 + p t u =. By scaling and rotating, we assume without loss of generality that p 1 = 1,,..., ) t and p = a, b,,..., ) t where a, b R and b 1. If a = or b =, we can take u 1 = b,, 1 b,,..., ) t and u =, 1,,..., ) t. Now, assume that a and < b 1. For θ [, π] consider unit vectors cos θ sin θ v θ =. and w θ = 1 b sin θ + a 7 b sin θ a sin θ a cos θ..

8 Clearly, p t 1v θ = p t w θ = vθ tw θ =. Dene fθ) = p 1 w θ + p v θ = b cos θ b sin θ which is a continuous function with f) = b and fπ) = b. b sin θ + a Hence there exists θ [, π] such that fθ ) =. Then we take u = v θ u 1 = w θ. and Lemma.1. For P, Q R n n, n 3 and ɛ 1 we dene [ [ ɛi ɛi P ɛ = P and Q In ] ɛ = Q. In ] Then P ɛ, Q ɛ ) SP, Q). Proof. For any U SO n, consider the ellipse EU) dened in Lemma.8. If trpɛ U), trq ɛ U) ) t EU), then we have trpɛ U), trq ɛ U) ) t LP, Q; SOn ). Now assume that trp ɛ U), trq ɛ U) ) t / EU). Then trpɛ U), trq ɛ U) ) t lies inside the ellipse EU). Since SO n is path-connected, consider a continuous function f : [, 1] SO n with f) = U and f1) = U where U is dened in Lemma.9. Since Ef1)) degenerates, by continuity of f, there exist s [, 1] such that trp ɛ U), trq ɛ U) ) t Efs)) LP, Q; SOn ). As it is true for all U SO n, we have LP ɛ, Q ɛ ; SO n ) LP, Q; SO n ) and hence P ɛ, Q ɛ ) SP, Q). Lemma.1 remains valid if we consider S A P, Q) instead of SP, Q). Corollary.11. Let A R n n and n 3. For any P, Q R n n and ɛ 1, we dene ɛi ɛi P ɛ = P and Q ɛ = Q. Then P ɛ, Q ɛ ) S A P, Q). I n I n Proof. For any U, V SO n, let P = P UAV, Q = QUAV, P ɛ = ɛi I n )P = P ɛ UAV and Q ɛ = ɛi I n )Q = Q ɛ UAV. By Lemma.1, because P ɛ, Q ɛ) SP, Q ), there exists W SO n such that trpɛ UAV ), trq ɛ UAV ) ) t = trp ɛ, trq ɛ) t = trp W ), trq W ) ) t = trp UAV W ), trquav W ) ) t LP, Q; OA)). As this is true for all U, V SO n, we have LP ɛ, Q ɛ ; OA)) LP, Q; OA)). Note that in Lemma.1 and Corollary.11, P ɛ, Q ɛ can be dened by picking arbitrary two rows of P and Q instead of the rst two rows. We are now ready to prove our second main theorem. 8

9 Proof of Theorem.7. By Lemma.3, it suces to show that for all P, Q R n n, SP, Q) is star-shaped with respect to n, n ). Let P, Q ) SP, Q) and α 1. We apply Lemma.1 repeatedly to every two rows of P, Q. Then we have ɛ N P, ɛ N Q ) SP, Q ) SP, Q) where N = n! n )!. Taking ɛ = N α, we have αp, Q ) = αp, Q ) + 1 α) n, n ) SP, Q). For the case of l = and l = 3, we know that n = 3 and n = 4 are respectively the smallest integers such that LOA)) is star-shaped for all A R n n and all linear maps L : R n n R l. However, for l 4, n = l 1 may not be the smallest integer to ensure star-shapedness of LOA)). One may ask the following question. Problem 1. For a given l 4, what is the smallest n such that LSO n ) is star-shaped for all linear maps L : R n n R l? The preceding results on star-shapedness of LOA)) can be easily generalized to the following joint orbits. We let R n n ) m := {A 1,..., A m ) : A 1,..., A m R n n }. Denition 1. For any A 1,..., A m R n n, we dene where G = O n or SO n. O 1 A 1,..., A m ; G) := {A 1 V,..., A m V ) : V G}, O A 1,..., A m ; G) := {UA 1,..., UA m ) : U G}, O 3 A 1,..., A m ; G) := {UA 1 V,..., UA m V ) : U, V G}, Theorem.1. Let L : R n n ) m R l be linear, A 1,..., A m ) R n n ) m and G = O n or SO n. If i) l = and n 3, or ii) l 3 and n l 1, then LO i A 1,..., A m ; G)), i = 1,, 3, are star-shaped with respect to the origin. Proof. The case of G = O n can be derived from the case G = SO n easily. Hence we consider the case G = SO n only and simply denote O i A 1,..., A m ; SO n ) by O i A 1,..., A m ). For any given L : R n n ) m R l, express it by LX 1,..., X m ) = tr m ) m )) t P 1) i X i,..., tr P l) i X i, 9

10 for some P j) i R n n, i = 1,..., m, j = 1,..., l. For O 1 A 1,..., A m ) we have LO 1 A 1,..., A m )) m ) m )) t = tr P 1) i A i U,..., tr P l) i A i U : U SO n m ) m = L P 1) i A i,..., P l) i A i ; SO n. Similarly for LO A 1,..., A m )). Hence the star-shapedness follows from Theorem.1 and Theorem.7. Now consider the case of O 3 A 1,..., A m ). For any U, V SO n, we have LUA 1 V,..., UA m V ) = L tr m By star-shapedness of L α 1 we have m m P 1) i αlua 1 V,..., UA m V ) L P 1) i UA i V P 1) i UA i,..., m ) m,..., tr m P l) i UA i V P l) i UA i ; SO N ). )) t UA i,..., m P l) i UA i ; SO N ), for any P 1) i UA i,..., LO 3 A 1,..., A m )). m ) t P 1) i UA i ; SO N 3 Convexity of linear image of OA) We rst give two non-convex examples, one is a linear image of OA) under L : R n n R l with l 3 and another is a linear image of O 3 A 1,..., A m ) under L : R n n ) m R l with l. Example 1. Consider OI n ) = SO n with n and the linear map L : R n n R l with l 3 dened by LX) = trp 1 X),..., trp l X)) t where P 1 = I n, P = I n 1, P 3 = I n 1, 1

11 and P j = n for j = 4,..., l. The [ mid-point ]) of points LI n ) = n, n 1 1, n,,..., ) t and L I n = n, n, n 1,,..., ) 1 t is in LP 1,..., P l ; SO n ) only if there exists U SO n having the form u11 u U = I n 1 u 1 u with u 11 = 1 = u 1. This is impossible as u 11 + u 1 = 1. Hence LSO n ) is non-convex. Example. For n 3, m, l, consider the matrices, 1 A 1 = n 3, A = 1 n 3, A j = n, j = 3,..., m, and the linear map L : R n n ) m R l dened by LX 1,..., X m ) := tra 1 X 1 + A X ), tra X 1 A 1 X ),,..., ) t. 1 1 By taking U = V = I n, and U = 1 I n 3, V = 1 I n respectively, we have,,,..., ) t,,,,..., ) t LO 3 A 1,..., A m )). We shall show that their mid-point which is 1, 1,,..., ) t / LO 3 A 1,..., A m )). For any U = [u ij ], V = [v ij ] SO n, by direct computation we have u 11 v 11 u 1 v 13 UA 1 V = u 1 v 1, UA V = u v. Hence 1, 1,,..., ) LO 3 A 1,..., A m )) only if u 11 v 11 + u v = 1 = u 1 v 1 u 1 v 13 for some U, V SO n. We shall show that such U, V do not exist. For X = x ij ), Y = y ij ) R n n, denote X Y := x ij y ij ) R n n. Since each absolute row column) sum of U V is not greater than one, we have 1, 1,,..., ) LO 3 A 1,..., A m )) only if there exist U, V SO n such that 1 U V = or U V = The possible choices of the leading principal submatices of U and V are 1 k1 ± k k 1 k where k 1, k = ±1. However, any two of them will not give the U V as required. 11

12 From the above two examples we know that LOA)) is not convex in general. However if the codomain of L is R then LOA)) is always convex. This result was obtained by Li and Tam [7] by using techniques in Lie algebra. In the following, we shall give an alternative proof on this result by showing that LOA)) has convex boundary for all A R n n and linear L : R n n R, i.e., the intersection of LOA)) with any of its supporting lines is path connected. Combining with the star-shapedness property of LOA)), the convexity of LOA)) follows. We rst need some notations. Denition. For A = a ij ) R n n, we denote its diagonal as da) = a 11, a,..., a nn ) t R n. We further denote the sum of the rst k diagonal elements of A by t k A). Moreover for P R n n, we denote rp, A) = max{trp UAV ) : U, V SO n } and G P A) = {B OA) : trp B) = rp, A)}. We shall characterize the set G P A) when A has distinct singular values and then show that it is path connected. Note that for any U, V SO n, G P UAV ) = G P A) and G UP V A) = {V t BU t : B G P A)}. Hence we may assume that A, P are diagonal matrices. Lemma 3.1. Let A = diaga 1,..., a n 1, a n ) where a 1 > a > > a n 1 > a n and B OA). If t k B) = t k A) then W W t B = A, where W SO k, X 1, X SO n k. Proof. Let B = UAV where U, V SO n and write U11 U U = u ij ) = 1 V11 V, V = v U 1 U ij ) = 1, V 1 V where U 11, V 11 R k k, U, V R n k) n k). Denote V11 U11 U 1 = u 1 u n, = V 1 X 1 X v 1. v n where u t j = u 1j,..., u kj ), v j = v j1,..., v jk ), j = 1,..., n. Then t k UAV ) = tru 11 A 11 V 11 + U 1 A V 1 ) = tra 11 V 11 U 11 + A V 1 U 1 ) = n a iv i u i. Since v i u i 1, n v i u i k and a 1 > > a k > > a n, we have n a iv i u i k a ii with equality holds if and only if v i u i = 1 for i k and v i u i = for i > k. Hence we have v i = u t i and u iu t i = 1. Now U = W X 1 and V = W t X 1 where W O k, X 1, X O n k and detw = detx 1 = detx. If detw = detx 1 = detx = 1, then we have B = W D 1 ) X 1 D )) A W D 1 ) t D X )) where D 1 = I k 1 1 and D = 1 I n k 1., 1

13 Thompson [9] gave the following result on characterizing the diagonal elements of OA). Proposition 3.. [9] A vector d = d 1,..., d n ) is the diagonal of a matrix A R n n with singular values s 1 s 1 s n if and only if d lies in the convex hull of those vectors ±s σ1),..., ±s σn) ) with an even number possibly zero) of negative signs and arbitrary permutation σ. For matrices A, B R n n, the following result by Miranda and Thompson [8] can be regarded as a characterization of the extreme values of OA) under the linear map X trbx). Proposition 3.3. [8] Let A, B R n n have singular values s 1 A) s n A) and s 1 B) s n B) respectively. Then n 1 max trbuav ) = s i A)s i B) + sign detab))s n A)s n B). U,V SO n Theorem 3.4. Let A = diaga 1,..., a n 1, ±a n ) where a 1 > > a n and P = p 1 I n1 p k I nk where p 1 > > p k and n n k = n. Then i) if p k >, 1 G P A) = U... U k A U t 1... : U i SO ni, i = 1,..., k ; Uk t ii) if p k =, U 1 U1 t... G P A) = U k 1 A... Uk 1 t : U V U i SO ni, i = 1,..., k 1,. U, V SO nk In both cases, G P A) is path connected. Proof. ) Obvious. ). We assume that A = A 1 A k where A i R ni ni. We have rp, A) = dp ) t da) = k p itra i. Let U, V SO n such that trp UAV ) = rp, A) = dp ) t duav ). Write B 11 B 1 B 1k B 1 B B k UAV = B =..... B k1 B k B kk where B ij R ni nj. We have trp UAV ) = trp B) = k p itrb ii. We shall show that trb ii = tra i for all i whenever p i >. By Proposition 3., 13

14 db) = α i s i where α i >, α i = 1 and s i are vector of ±a σ1),..., ±a σn) ), σ is a permutation on {1,..., n} and the number of negative signs is even odd, respectively) if deta, respectively). If k = 1, then P = p 1 I, and the proof is trivial. Now consider k > 1, hence p 1 >. We rst show that trb 11 = tra 1. Note that trb 11 < tra 1 holds if and only if at least one of the following cases hold: 1) there exists i 1 such that the rst n 1 elements of s i1 contain a j where j n 1 ; ) there exists i 1 such that the rst n 1 elements of s i1 contain ±a j where j > n 1. In case 1), we construct s i 1 from s by multiplying 1 to a j and arbitrary a q for some q > n 1. If in case ), then there exists i < n 1 such that ±a i will not be the rst n 1 elements of s i1. In this case, we construct s i 1 from s i1 by interchanging ±a j and ±a i and multiplying 1 to both if necessary to have a i instead of a i. Replace s i1 in α i s i by s i 1 to form s. By Proposition 3., there exists B OA) such that db ) = s. We shall have dp ) t db) = dp ) t α i s i ) = dp ) t s + dp ) t s i1 s i 1 ) < dp ) t s, which contradicts the assumption on B. Therefore, we have trb 11 = tra 1. By Lemma 3.1, we have U = U 1 U and V = V1 t V where U 1, V 1 SO n1, V, U SO n n1 and V1 t = U 1. Apply similar approach for B ii where p i >. Hence, if p k >, we have U = U 1 U k and V = U t where U i SO ni, i = 1,..., k; otherwise if p k =, U = U 1 U k 1 U and V = U1 t Uk 1 t V where U i SO ni, i = 1,..., k 1, U, V SO nk. The path connectedness of G P A) follows from the path connectedness of SO ni for all i. Corollary 3.5. If A R n n has n distinct singular values, then LOA)) has convex boundary for all linear maps L : R n n R. Proof. Let P, Q R n n be such that LP, Q; OA)) = LOA)). Then LOA)) has convex boundary if for any θ [, π], the set { sin θx + cos θy : x, y) LP, Q; OA)), cos θx + sin θy = r θ }, where r θ = max{cos θx + sin θy : x, y) LP, Q; OA))}, is path connected. For any θ [, π], we dene P θ = sin θp + cos θq and Q θ = cos θp + sin θq, then we have { sin θx + cos θy : x, y) LP, Q; OA)), cos θx + sin θy = r θ } = {tr P θuav ) : U, V SO n, tr Q θuav ) = r θ } = {trp θx) : X G Q θ A)} Hence by Theorem 3.4, it is path connected. Note that a set M R is convex if and only if it is star-shaped and has convex boundary. Hence by Theorem.1 and Corollary 3.5, the following result is clear. 14

15 Theorem 3.6. Let n 3. If A R n n has n distinct singular values, then LOA)) is convex for all linear maps L : R n n R. In fact, the condition of distinct singular values in Theorem 3.6 can be removed by applying the following lemma. Lemma 3.7. Let L : R n n R l be a linear map. Suppose LOA)) is convex for all A in a dense set S of R n n. Then LOA)) is convex for all A R n n. Proof. Suppose that A R n n such that LOA )) is not convex. Then there exist x 1, x LOA )) such that y = 1 x 1 +x ) / LOA )). Since LOA )) is compact, there exists ɛ > such that By, ɛ) := {x R l : x y < ɛ} has empty intersection with LOA )). Since S is dense in R n n, there exists A ɛ S such that for all U, V SO n, LUA V ) LUA ɛ V ) < ɛ. Hence there exist x 1, x LOA ɛ )) such that x 1 x 1 < ɛ and x x < ɛ. By convexity of LOA ɛ)), y = 1 x 1 + x ) LOA ɛ ). We have y y = 1 x 1 + x ) 1 x 1 + x ) < 1 ɛ + ɛ ) = ɛ. By assumption of A ɛ, there exists z LOA )) such that z y < ɛ. Then z y = z y ) + y y) < z y ) + y y) < ɛ + ɛ = ɛ, contradicting the fact that By, ɛ) LOA )) =. Since the set of n n matrices with n distinct singular values is dense in R n n, by Lemma 3.7 we have the following result. Theorem 3.8. Let n 3. LOA)) is convex for all linear maps L : R n n R and A R n n. From the proof of Corollary.1, the convexity of LOA)) can be extended to LO i A 1,..., A m )), i = 1,. Corollary 3.9. Let n 3. LO i A 1,..., A m )), i = 1,, is convex for all linear maps L : R n n ) m R and A 1,..., A m R n n. References [1] W.S. Cheung, N.K. Tsing, The C-numerical Range of Matrices is Starshaped, Linear and Multilinear Algebra ), [] F. Hausdor, Das Wertvorrat einer Bilinearform, Math. Zeit ), [3] A. Horn, C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge,

16 [4] C.K. Li, C-numerical Ranges and C-numerical Radii, Linear and Multilinear Algebra, ), [5] C.K. Li, Y.T. Poon, Convexity of the Joint Numerical Range, SIAM J. Matrix Analysis Appl ), [6] C.K. Li, Y.T. Poon, Generalized Numerical Ranges and Quantum Error Correction, J. Operator Theory ), [7] C.K. Li, T.Y. Tam, Numerical Ranges Arising from Simple Lie Algebras, Canad. J. Math. 5 ), [8] H. Miranda, R.C. Thompson, Group Majorization, the Convex Hulls of Sets of Matrices, and the Diagonal Element-Singular Value Inequalities, Linear Algebra Appl ), [9] R.C. Thompson, Singular Values, Diagonal Elements, and Convexity, SIAM J. Appl. Math ), [1] O. Toeplitz, Das algebraische Analogon zu einem Satze von Fejer, Math. Zeit. 1918), [11] N.K. Tsing, On the Shape of the Generalized Numerical Ranges, Linear and Multilinear Algebra, ), [1] R. Westwick, A Theorem on Numerical Range, Linear and Multilinear Algebra, 1975),