Numerical Range and Positive Block Matrices

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A X A + B + ωI . X∗ B ≤k k

Here I stands for the identity matrix and the width of W (X) is the smallest distance between two parallel straight lines such that the strip between these two lines contains W (X). Hence the partial trace A + B may be used to give an upper bound for the norms of the full block-matrix. This note will provide a lower bound, stated in Section 2, and several consequences. Theorem 1.1 is the first inequality involving the width of the numerical range; classical results rather deal with the numerical radius, w(X) = max z : z W (X) . Our new {| | ∈ } lower bound will also have an unusual feature as it involves the distance from 0 to the numerical range, dist(0, W (X)) = min z : z W (X) . For a background on the {| | ∈ } numerical range we refer to [12], where the term of Field of values is used. Some very interesting inequalities for the numerical radius can be found in [11], [13], and in the recent article [8]. In case of Hermitian off-diagonal blocks, Theorem 1.1 holds with w = 0. More generally, if X = X = aI + bH for some a, b C and some Hermitian matrix H, we ∈ have ω = 0 as W (X) is a line segment. This special case of the theorem was first shown by Mhanna [14]. In particular, if the off-diagonal blocks are normal two-by-two matrices, then we can take ω = 0. This does not hold any longer for three-by-three normal matrices, a detailed study of this phenomenon is given in [10] and [9]. For Hermitian off-diagonal blocks, a stronger statement than Theorem 1.1 with w =0 holds. The following decomposition was shown in [6, Theorem 2.2].

A X Theorem 1.2. Let be a positive matrix partitioned into four Hermitian blocks X B in M . Then, for some pair of unitary matrices U, V M , n ∈ 2n A X 1 A + B 0 0 0 = U U ∗ + V V ∗ . X B 2 0 0 0 A + B For decompositions of positive matrices partitioned into a larger number of blocks, see [5]. We close this section by recalling some facts on symmetric norms, classical text books such as [2], [12] and [15] are good references. M M+ A symmetric norm on n, can be deﬁned by its restriction to the positive cone n . M+ Symmetric norms on n are characterized by three properties: (i) λA = λ A for all A M+ and all λ 0, k k k k ∈ n ≥ (ii) UAU ∗ for all A M+ and all unitaries U M , k k ∈ n ∈ n (iii) A A + B A + B for all A, B M+. k k≤k k≤k k k k ∈ n 2 Let λ↓(A) λ↓ (A) stand for the eigenvalues of A M+ arranged in non-increasing 1 ≥···≥ n ∈ n order. Then, the Ky Fan k-norms,

k A = λ↓(A) k k(k) j j=1 X are symmetric norms, k =1,...,n. Thus A (1) is the operator norm, usually denoted k k + by A ∞ while A is the trace norm, usually written A . For A, B M , the k k k k(n) k k1 ∈ n following conditions are equivalent: (a) A B for all k =1,...,n, k k(k) ≤k k(k) (b) A B for all symmetric norms, k k≤k k (c) The vector of the eigenvalues of A is dominated by a convex combination of permutations of the vector of the eigenvalues of B, equivalently,

n+1 A α U BU ∗ ≤ i i i i=1 X for some unitary matrices U and some scalars α 0 such that n+1 α = 1. i i ≥ i=1 i When these conditions hold (especially when explicitly stated as (a))P one says that A is weakly majorized by B and one writes A B. If furthemore in (a) one has the ≺w equality A = B , that it is A and B have the same trace, then A is majorized k k(n) k k(n) by B, written A B. Thus A B means that (c) holds with the equality sign: A is in ≺ ≺ the convex hull of the unitary orbit of B. Theorem 1.2 is a special majorization. A linear map Φ : M M is called doubly stochastic if Φ preserves positivity, n → n identity, and trace. For all A M+, we then have Φ(A) A, see the last section of ∈ n ≺ Ando’s survey [1].

2 The distance from 0 to the numerical range

We state our main result and infer several corollaries. The proof of the theorem is postponed to Section 3. A X Theorem 2.1. Let be a positive matrix partitioned into four blocks in M X∗ B n and let d = dist(0, W(X)). Then, for all symmetric norms,

A X A + B A + B + dI dI . X∗ B ≥ 2 ⊕ 2 −

Here, the direct sum is a standard notation for block-diagonal mat rices X 0 X Y = . ⊕ 0 Y 3 Since we have equality for the trace, Theorem 2.1 is a majorization relation. We have (A + B)/2 dI, otherwise, the trace norm of the left-hand side would be strictly ≥ smaller than the right-hand side one, a contradiction with the theorem. By a basic principle of majorization, Theorem 2.1 is equivalent to some trace inequalities. A X Corollary 2.2. Let be a positive matrix partitioned into four blocks in M X∗ B n and let d = dist(0, W(X)). Then, for every convex function g : [0, ) ( , ), ∞ → −∞ ∞ A + B A + B A X Tr g + dI + Tr g dI Tr g . 2 2 − ≤ X∗ B Symmetric norms on M+ are the homogeneous, unitarily invariant, convex func- k·k n tionals. The concave counterpart, the symmetric anti-norms , have been introduced k·k! and studied in papers [3] and [4, Section 4]. We recall their basic properties, parallel to M+ those of symmetric norms given at the end of Section 1. Symmetric anti-norms on n are continuous functionals characterized by three properties:

(i) λA = λ A for all A M+ and all λ 0, k k! k k! ∈ n ≥ (ii) UAU ∗ for all A M+ and all unitaries U M , k k! ∈ n ∈ n (iii) A + B A + B for all A, B M+. k k! ≥k k! k k! ∈ n Let λ↑(A) λ↑ (A) stand for the eigenvalues of A M+ arranged in non-decreasing 1 ≤···≤ n ∈ n order. Then, the Ky Fan k-anti-norms,

k A = λ↑(A) k k(k)! j j=1 X are symmetric anti-norms, k =1,...,n. The following conditions are equivalent:

(a) A B for all k =1,...,n, k k(k)! ≥k k(k)! (b) A B for all symmetric anti-norms, k k! ≥k k! (c) The vector of the eigenvalues of A is dominated by some convex combination of permutations of the vector of the eigenvalues of B, equivalently,

n+1 A α U BU ∗ ≥ i i i i=1 X for some unitary matrices U and some scalars α 0 such that n+1 α = 1. i i ≥ i=1 i P

4 The continuity assumption is not essential, but deleting it would lead to rather strange functionals which are not continuous on the boundary of M+, such as A := Tr A if A n k k! is invertible and A := 0 if A is not invertible. k k! Note that the trace norm is both a symmetric norm and a symmetric anti-norm and that the majorization A B in M+ also entails that A B for all symmetric ≺ n k k! ≥ k k! anti-norms. Thus Theorem 2.1 is equivalent to the following statement: A X Corollary 2.3. Let be a positive matrix partitioned into four blocks in M , X∗ B n let d = dist(0, W (X)). Then, for all symmetric anti-norms,

A + B A + B A X + dI dI . 2 ⊕ 2 − ≥ X∗ B ! !

A X Corollary 2.4. Let be a positive matrix partitioned into four blocks in M X∗ B n and let d = dist(0, W(X)). Then,

A X A + B λ↓ λ↓ d 1 X∗ B − 1 2 ≥ and A + B A X λ↑ λ↑ d. 1 2 − 1 X∗ B ≥ Proof. The ﬁrst inequality follows from Therorem 2.1 applied to the symmetric norm ↓ A λ1(A) (the operator norm on the positive cone), while the second inequality follows 7→ ↑ from Corollary 2.3 applied to the anti-norm A λ (A) 7→ 1 By adding these two inequalities we get an estimate for the spread of the matrices, i.e., for the diameter of the numerical ranges.

Corollary 2.5. For every positive matrix partitioned into four blocks of same size,

A X A + B diam W diam W 2d, X∗ B − 2 ≥ where d is the distance from 0 to W (X).

Of course A X A 0 A + B diam W diam W diam W , X∗ B ≥ 0 B ≥ 2 however the ratio 1 A X A 0 ρ = diam W diam W 2d X∗ B − 0 B 5 can be arbitrarily small as shown by the following example where the blocks are in M2, α 0 1 0 A X 0 α−1 0 1 = , X∗ B  1 0 α−1 0   0 1 0 α      and by noting that ρ then takes the value 2/α which tends to 0 as α . →∞ The Minkowki inequality for positive m-by-m matrices,

det1/m(A + B) det1/m(A) + det1/m(B), ≥ shows that the functional A det1/m(A) is a symmetric anti-norm on M+ . For this 7→ m anti-norm Theorem 2.1 reads as: A X Corollary 2.6. Let be a positive matrix partitioned into four blocks in M X∗ B n and let d = dist(0, W(X)). Then,

A + B 2 A X det d2I det . 2 − ≥ X∗ B ( ) Letting X = 0, we recapture a basic property: the determinant is a log-concave map on the positive cone of Mn. Hence Corollary 2.6 reﬁnes this property. By a basic principle of majorization, Corollary 2.3 is equivalent to the following seemingly more general statement. A X Corollary 2.7. Let be a positive matrix partitioned into four blocks in M , X∗ B n let d = dist(0, W (X)), and letf(t) be a nonnegative concave function on [0, ). Then, ∞ A + B A + B A X f + dI f dI f 2 ⊕ 2 − ≥ X∗ B ! ! for all symmetric anti-norms.

3 Proof of Theorem 2.1

M+ We want to show the majorization in 2n A+B + dI 0 A X 2 (3.1) 0 A+B dI ≺ X∗ B 2 − where d = dist(0, W (X). We use two lemmas, the ﬁrst one might belong to folklore. Lemma 3.1. Let A m and B m be two families of r-by-r positive matrices such { k}k=1 { k}k=1 that A B for each k. Then, k ≺ k m A m B . ⊕k=1 k ≺ ⊕k=1 k 6 Proof. Let p denote any integer such that 0 p m, k = 1,...,m. With this k ≤ k ≤ notation, we then have, for each integer p =1,...,mr,

p m pk ↓ m λj ( k=1Ak) = max (Ak) ⊕ p1+p2+···+pm=p j=1 j=1 X Xk=1 X m pk

max (Bk) ≤ p1+p2+···+pm=p k=1 j=1 p X X = λ↓ ( m B ) j ⊕k=1 k j=1 X with equality for p = mr.

Lemma 3.2. Let X,Y M+ and let δ > 0 be such that X Y δI. Then, ∈ n ≥ ≥ X + δI 0 X + Y 0 . 0 X δI ≺ 0 X Y − −

Proof. Let e n be an orthonormal basis of Cn and deﬁne two n-by-n diagonal posi- { k}k=1 tive matrices D = diag( e , (X + Y )e ,..., e , (X + Y )e ) + h 1 1i h n ni and D− = diag( e , (X Y )e ,..., e , (X Y )e ). h 1 − 1i h n − ni Since extracting a diagonal is a doubly stochastic map (a pinching), we have

D 0 X + Y 0 + . (3.2) 0 D− ≺ 0 X Y − n ↓ Now, choose the basis ek k=1 as a basis of eigenvectors for X, λk(X)= ek, Xek , and { } M+ h i observe that the majorization in 2 ,

↓ λk(X)+ δ 0 ek, (X + Y )ek 0 ↓ h i , 0 λ (X) δ ≺ 0 ek, (X Y )ek k − h − i holds for every k. Applying Lemma 3.1 then shows that

n ↓ n λk(X)+ δ 0 ek, (X + Y )ek 0 ↓ h i . 0 λk(X) δ ≺ 0 ek, (X Y )ek Mk=1 − Mk=1 h − i This means that X + δI 0 D+ 0 0 X δI ≺ 0 D− − and we may combine this majorization with (3.2) to complete the proof.

7 We turn to the proof of (3.1). Proof. Suppose ﬁrst that d = 0, that is 0 W (X). Note that ∈ A 0 A X (3.3) 0 B ≺ X∗ B as the operation of taking the block diagonal is doubly stochastic. Using the unitary congruence with 1 I I J = − (3.4) √ I I 2 we observe that A 0 A+B A−B J J ∗ = 2 2 0 B A−B A+B 2 2 Hence we have A+B 2 0 A 0 0 A+B ≺ 0 B 2 and combining with (3.3) establishes (3.1) for the case d = 0. Now assume that d > 0, that is 0 / W (X). Using the unitary congruence imple- ∈ mented by I 0 0 e−iθI we may replace the right hand side of (3.1) with A eiθX e−iθX∗ B Thanks to the rotation property W (eiθX) = eiθW (X), by choosing the adequate θ, we may then and do assume that W (X) lies the half-plane of C consiting of complex numbers with real parts greater or equal than d, W (X) z = x + iy : x d . ⊂{ ≥ } The projection property for the real part of the numerical range, Re W (X)= W (Re X) with Re X =(X + X∗)/2, then ensures that Re X dI. ≥ Now, using again a unitary congruence with (3.4), wet get A X A+B + ReX J J ∗ = 2 ∗ X∗ B A+B ReX ∗ 2 − where stands for unspeciﬁed entries. Hence ∗ A+B 2 + ReX 0 A X 0 A+B ReX ≺ X∗ B 2 − and applying Lemma 3.2 then yields (3.1).

8 References

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Jean-Christophe Bourin Laboratoire de mathématiques de Besan¸con, UMR no 6623, CNRS, Universitéde Bourgogne Franche-Comté Email: [email protected]

9 Eun-Young Lee Department of mathematics, KNU-Center for Nonlinear Dynamics, Kyungpook National University, Daegu 702-701, Korea. Email: [email protected]