On Numerical Range of Sp(N, C)

Spec. Matrices 2016; 4:328–332

Research Article Open Access Special Issue: 5th International Conference on Matrix Analysis and Applications

Wen Yan*, Jicheng Tao, and Zhao Lu On numerical range of sp(2n, C)

DOI 10.1515/spma-2016-0032 Received March 20, 2016; accepted November 16, 2016

Abstract: In this paper we studied the classical numerical range of matrices in sp(2n, C). We obtained some ! A1 A2 result on the relationship between the numerical range of a matrix in A = > ∈ sp(2n, C) and that 0 −A1 of its diagonal block, the singular values of its o-diagonal block A2. Keywords: Numerical range, elliptical disk.

MSC: 15A45, 15A18

1 Introduction

Let A ∈ Cn×n be a complex matrix. The numerical range of A is the set of all complex numbers, denoted by W(A), of the following form: n W(A) := {x*Ax : x ∈ C and kxk = 1}. By the well-known Toeplitz-Hausdor theorem W(A) is convex [1, 5, 6]. When n = 2, the numerical range W(A) p * 2 2 is the closed elliptical disk with foci the eigenvalues λ1 and λ2 and the minor axis length tr A A − |λ1| − |λ2| [4], where tr A*A is the trace of the matrix A*A. There are many generalizations of the classical numerical range. Among all of these generalizations, Tam[7] studied the numerical range that is associated with Lie algebras. In this paper, we focused on the classical numerical range of matrices in sp(2n, C), a Lie algebra of type C.

2 Numerical range of matrices in sp(2n, C)

Consider the complex symplectic Lie algebra [3, p.3] which is simple for n ≥ 1:

sp(2n, C) := sp(2n) ⊕ isp(2n) ( ! ) A1 A2 > > = > : A1, A2, A3 ∈ Cn×n , A2 = A2, A3 = A3 . A3 −A1

The compact group K = Sp(2n, C) ∩ U(2n) [3] consists of the matrices ! U −V ∈ U(2n). V U

*Corresponding Author: Wen Yan: Department of Mathematics, Tuskegee University, AL, 36088, E-mail: [email protected] Jicheng Tao: Department of Mathematics, China Jiliang University, Hangzhou 310000, China, E-mail: [email protected] Zhao Lu: Department of Electrical Engineering, Tuskegee University, AL 36088, USA, E-mail: [email protected]

It is known that for any B ∈ sp(2n, C), there is U ∈ K such that UBU* ∈ b ⊂ sp(2n, C), where ( ! ) A1 A2 > b := > , A1 ∈ Cn×n is upper triangular, A2 = A2 (2.1) 0 −A1 is a Borel subalgebra of sp(2n, C). The eigenvalues of A ∈ sp(2n, C) occur in pairs but opposite in sign as we can see it from (2.1). By Toeplitz-Hausdor theorem, the numerical range W(A) of a matrix A ∈ sp(2n, C) is n W A λ 1 |a | λ convex. When = 1, ( ) is the elliptical disk with foci ± and the minor axis length 2 2 , where ± are the eigenvalues of A.

Lemma 2.1. If A ∈ sp(2n, C), then 0 ∈ W(A).

Proof. Let A ∈ sp(2n, C). The eigenvalues ±λ1, ... , ±λn of A are in the numerical range W(A). Since the set W(A) is convex, 1 1 0 = λ + (−λ ) ∈ W(A). 2 1 2 1

Now we explore the relationship between the numerical range of A and that of A1, the singular values of A2. ! A1 A2 Theorem 2.2. Let A = > ∈ b. If the largest singular value of A2 is s , then the circular disk Ds/2 0 −A1 r s W A D ⊂ W A centered at the origin and with the radius = 2 is a subset of the numerical range ( ), i.e., s/2 ( ).

> Proof. By Takagi’s factorization of complex symmetric matrices, A2 = Udiag (s1, ... , sn)U with U ∈ U(n) and s1 ≥ · · · ≥ sn the singular values of A2. Then

* U A2U = diag (s1, ... , sn). ! X Let X be the rst column of U and X = √1 1 . Direct computation shows that 1 2 X1 ! ! * 1 * > A1 A2 X1 W(A) 3 X AX = (X1, X1 ) > 2 0 −A1 X1 1 > > = (X* A X − X A X + X* A X ) 2 1 1 1 1 1 1 1 2 1 1 > = [X* A X − (X* A X ) + X* A X ] 2 1 1 1 1 1 1 1 2 1 1 = X* A X 2 1 2 1 1 = s . 2 1 X e−iθ/2X θ ∈ π 1 s eiθ ∈ W A { 1 eiθ s Replace 1 with 1, [0, 2 ] in above equation, we can see that 2 1 ( ). So the circle 2 1 : θ ∈ [0, 2π]} ⊂ W(A). Since W(A) is convex, Ds/2 ⊂ W(A). ! a b In Theorem 2.2, if n = 1, A = . The singular value of the (2, 1) block is |b| and the numerical range 0 −a

W(A) is the elliptical disk with foci a, −a and minor axis length |b|. Clearly the circular disk D|b|/2 is a subset of W(A). ! A1 A2 Lemma 2.3. Let A = > ∈ sp(2n, C). The numerical range of A is the union of all elliptical disks 0 −A1 * * * n with foci X A1X ∈ W(A1) and −Y A1Y ∈ −W(A1) and minor axis length |X A2Y|, where X, Y ∈ C are unit vectors. 330 Ë Wen Yan, Jicheng Tao, and Zhao Lu

iα ! n e sin θX n Proof. For any unit vector Z ∈ C2 , let Z = , where X, Y are unit vectors in C . eiβ cos θY

* * 2 −i(α−β) * > > 2 Z AZ = X A1X sin θ + e X A2Y sin θ cos θ − Y A1 Y cos θ * iα ! * * ! iα ! e sin θ X A1XX A2Y e sin θ = iβ * > iβ e cos θ 0 −(Y A1Y) e cos θ which generates the required ellipse containing Z*AZ if we let θ, α, β run over all the values in [0, 2π]. By the convexity property of numerical range, the union of all these closed elliptical disks is the numerical range of A. ! A1 A2 Actually, Lemma 2.3 holds for any matrix A = , where A1 and A3 are square matrix. 0 A3 For any square matrix A and s > 0, dene the set

iθ W(A, s) := {a + re : a ∈ W(A), 0 ≤ θ ≤ 2π, and 0 ≤ r ≤ s}.

Clearly W(A) = W(A, 0) ⊂ W(A, s). The set W(A, s) is the region obtained by expanding the region W(A) by s units in all directions. The following is the main result of this paper. ! A1 A2 > Theorem 2.4. For any A = > ∈ b, dene B = A1 ⊕ (−A1 ). Then 0 −A1

W A ⊆ W B 1 s s A 1. ( ) ( , 2 ), where is the largest singular values of 2. A A sI W A W B 1 s 2. If 1 is skew symmetric and 2 = n, then ( ) = ( , 2 ). W A W B 1 s W A W A x ∂W A 3. If ( ) = ( , 2 ), then ( 1) = (− 1), furthermore for any point on the boundary ( 1) that n * is not convex combination of any other two points in W(A1), there is X ∈ C such that x = X A1X = * > −X A1 X.

! X1 2n > > Proof. (1) Let X = ∈ C be any unit vector with X1 = (x1, ... , xn) and X2 = (xn+1, ... , x2n) . Then X2

* * * X AX = X BX + X1A2X2 * * = X BX + X1A2X2 * * > = X BX + (UX1) UA2U (UX2) for any U ∈ U(n) (2.2)

> Since A2 is symmetric, there exists a U ∈ U(n) such that UA2U = diag (s1, ... , sn), where s1 ≥ · · · ≥ sn are singular values of A2. From now on, we always assume that A2 = diag (s1, ... , sn) unless otherwise specied. > > Let Y1 = (y1, ... , yn) = UX1 and Y2 = (yn+1, ... , y2n) = UX2. In (2.2),

* > * |(UX1) UA2U UX2| = |Y1diag (s1, ... , sn)Y2| n X ≤ |yk| sk |yn+k| k=1 n X 1 2 2 ≤ sk(|yk| + |yn k| ) 2 + k=1 n 1 X 2 2 ≤ s (|yk| + |yn k| ) (2.3) 2 1 + k=1 n 1 X 2 2 1 = s (|xk| + |xn k| ) = s. 2 1 + 2 k=1 On numerical range of sp(2n, C) Ë 331

X*AX ∈ W B 1 s W A ⊆ W B 1 s Thus ( , 2 ). Hence ( ) ( , 2 ). The equality in (2.3) holds if and only if iβ Y2 = e Y1 for some β and sj = s for all 1 ≤ j ≤ n where yj ≠ 0. (2) A A sI W B 1 s Assume that 1 is skew symmetric and 2 = n, we will show that the boundary of ( , 2 ) is a sub- W A p Z*BZ 1 seiθ W B s Z αX> αX> > ∈ set of ( ). Let = + 2 be any point on the boundary of ( , ), where = (sin 1 , cos 2 ) 2n n C for some unit vectors X1, X2 ∈ C and α ∈ [0, 2π]. 1 iθ p = sin2 αX* A X + cos2 αX* A X + se . 1 1 1 2 1 2 2 Since the numerical range W(A ) is convex, the convex combination sin2 αX* A X + cos2 αX* A X = 1 ! 1 1 1 2 1 2 e−iθ/2X X* A X ∈ W(A ) for some unit vector X ∈ n. Let Z = √1 3 . Then 3 1 3 1 3 C 1 2 iθ/2 e X3

* 1 * 1 * 1 iθ * W(A) 3 Z AZ = X A X + X A X + e X sIn X = p 1 1 2 3 1 3 2 3 1 3 2 3 3 W B 1 s W A W B 1 s ⊂ W A W A So the boundary of ( , 2 ) is a subset of ( ). Thus ( , 2 ) ( ) since ( ) is convex. Therefore W A W B 1 s ( ) = ( , 2 ). (3) W A W B 1 s W A W A W A W A Suppose that ( ) = ( , 2 ). Assume that ( 1) ≠ (− 1). Since ( 1) and (− 1) are convex, * we can pick a point p = X1A1X1, kX1k = 1 on the boundary of W(A1) that satises the following: the sup- porting line ` of W(A1) through p does not intersects W(−A1) and both W(A) and W(−A1) are on the same ` pq 1 seiθ ` q ∈ W A q p 1 seiθ side of . Choose a vector = 2 that is perpendicular to and ̸ ( 1). So = + 2 . Because W(B) = conv {W(A1) ∪ W(−A1)}, the convex hull containing W(A1) and W(−A1), q is on the boundary of W(B, 1 s) and the distance between q and W(B) is 1 s. Since W(A) = W(B, 1 s), there exists a unit vector 2 ! 2 2 sin αY1 n Z = ∈ C2 such that cos αY2

> 1 q = Z*AZ = sin2 αY* A Y + cos2 α(−Y* A Y ) + sin(2α)Y* A Y ∈ W(A) (2.4) 1 1 1 2 1 2 2 1 2 2 2 * 2 * > * * > Because sin αY1A1Y1 + cos α(−Y2A1 Y2) is a convex combination of Y1A1Y1 ∈ W(A1) and −Y2A1 Y2 ∈ * W(−A1), hence it is in W(B). Thus the distance between q = Z AZ and W(B), 1 d(q, W(B)) ≤ | sin(2α)Y* A Y | 2 1 2 2 1 ≤ s sin(2α) by inequalities (2.2) and (2.3) 2 iβ 2 * 2 * > For that the equality holds, sin 2α = 1, Y2 = e Y1 and sin αY1A1Y1 +cos α(−Y2A1 Y2) = p. But if this is the * * > case, then Y1A1Y1 = −Y2A1 Y2 = p, that means p is also in W(−A1), which is not true. So W(A1) = W(−A1). Furthermore, if x = p is a point on the boundary of W(A1) = W(−A1) that is not a convex combination of any W A q d q W B 1 s Y eiβ Y other points in ( 1), let be the point in Equation (2.4). Since the distance ( , ( )) = 2 , 1 = 2 * * > * * > and Y1A1Y1 = −Y2A1 Y2 = p. Let X = Y1, then X A1X = −X A1 X = p = x. ! Bm×m Cm×n Theorem 2.4 holds for more general matrices. Let A = be any m + n square matrix with 0 Dn×n * m ≤ n. Let C = U1ΣU2 be the singular value decomposition of C. So we can assume that the block C = Σ. If W(A) = W(B, 1 s), then W(B) = W(D) and for each x on the boundary of W(B) that is not a convex combination 2 ! m X n of another two points of W(B), then there exists X ∈ C and Y = ∈ C such that x = X*BX = Y*DY. 0 B D C sI W A W B 1 s W A W B 1 s A Clearly if = and = n, then ( ) = ( , 2 ). We conjecture that ( ) = ( , 2 ) if and only if is unitarily similar to a matrix of the following form   B1 0 sIk 0    0 B2 0 Σ1    ,  0 0 B1 0  0 0 0 D2 where W(B2) ∪ W(D2) ⊂ W(B1) and the singular values of Σ1 are less than or equal to s. 332 Ë Wen Yan, Jicheng Tao, and Zhao Lu

Acknowledgement: The work of Jicheng Tao was supported by the provincial Natural Science Foundation of Zhejiang (No. LY16A010009).

References

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