18 Chapter 18 Numerical Range

18
Numerical Range
Chi-Kwong Li
College of William and Mary
18.1 Basic Properties and Examples . . . . . . . . . . . . . . . . . . . . . .
18.2 The Spectrum and Special Boundary Points . . . . . . . . .
18.3 Location of the Numerical Range . . . . . . . . . . . . . . . . . . .
18.4 Numerical Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.5 Products of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.6 Dilations and Norm Estimation . . . . . . . . . . . . . . . . . . . .
18.7 Mappings on Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18-1
18-3
18-4
18-6
18-8
18-9
18-11
18-11
The numerical range W(A) of an n × n complex matrix A is the collection of complex numbers of the form
x∗ Ax, where x ∈ Cn is a unit vector. It can be viewed as a “picture” of A containing useful information of A.
Even if the matrix A is not known explicitly, the “picture” W(A) would allow one to “see” many properties
of the matrix. For example, the numerical range can be used to locate eigenvalues, deduce algebraic and
analytic properties, obtain norm bounds, help find dilations with simple structure, etc. Related to the
numerical range are the numerical radius of A defined by w (A) = maxµ∈W(A) |µ| and the distance of
(A) = minµ∈W(A) |µ|. The quantities w (A) and w
(A) are useful in
W(A) to the origin denoted by w
studying perturbation, convergence, stability, and approximation problems.
Note that the spectrum σ (A) can be viewed as another useful “picture” of the matrix A ∈ Mn . There
are interesting relations between σ (A) and W(A).
18.1
Basic Properties and Examples
Definitions and Notation:
Let A ∈ Cn×n . The numerical range (also known as the field of values) of A is defined by
W(A) = {x∗ Ax : x ∈ Cn , x∗ x = 1}.
The numerical radius of A and the distance of W(A) to the origin are the quantities
w (A) = max{|µ| : µ ∈ W(A)}
and
(A) = min{|µ| : µ ∈ W(A)}.
w
Furthermore, let
W(A) = {a : a ∈ W(A)}.
18-1
18-2
Handbook of Linear Algebra
Facts:
The following basic facts can be found in most references on numerical ranges such as [GR96], [Hal82],
and [HJ91].
1. Let A ∈ Cn×n , a, b ∈ C. Then W(a A + b I ) = aW(A) + b.
2. Let A ∈ Cn×n . Then W(U ∗ AU) = W(A) for any unitary U ∈ Cn×n .
3. Let A ∈ Cn×n . Suppose k ∈ {1, . . . , n − 1} and X ∈ Cn×k satisfies X ∗ X = Ik . Then
W(X ∗ AX) ⊆ W(A).
4.
5.
6.
7.
In particular, for any k × k principal submatrix B of A, we have W(B) ⊆ W(A).
Let A ∈ Cn×n . Then W(A) is a compact convex set in C.
If A1 ⊕ A2 ∈ Mn , then W(A) = conv {W(A1 ) ∪ W(A2 )}.
Let A ∈ Mn . Then W(A) = W(AT ) and W(A∗ ) = W(A).
If A ∈ C2×2 has eigenvalues λ1 , λ2 , then W(A) is an elliptical disk with foci λ1 , λ2 , and minor axis
∗
with length {tr (A A) − |λ1 | − |λ2 | }
2
2 1/2
λ
. Consequently, if A = 1
0
b
, then the minor axis of
λ2
the elliptical disk W(A) has length |b|.
8. Let A ∈ Cn×n . Then W(A) is a subset of a straight line if and only if there are a, b ∈ C with a = 0
such that a A + b I is Hermitian. In particular, we have the following:
(a) A = a I if and only if W(A) = {a}.
(b) A = A∗ if and only if W(A) ⊆ R.
(c) A = A∗ is positive definite if and only if W(A) ⊆ (0, ∞).
(d) A = A∗ is positive semidefinite if and only if W(A) ⊆ [0, ∞).
9. If A ∈ Cn×n is normal, then W(A) = conv σ (A) is a convex polygon. The converse is true if n ≤ 4.
10. Let A ∈ Cn×n . The following conditions are equivalent.
(a) W(A) = conv σ (A).
(b) W(A) is a convex polygon with vertices µ1 , . . . , µk .
(c) A is unitarily similar to diag (µ1 , . . . , µk ) ⊕ B such that W(B) ⊆ conv {µ1 , . . . , µk }.
11. Let A ∈ Cn×n . Then A is unitary if and only if all eigenvalues of A have modulus one and
W(A) = conv σ (A).
12. Suppose A = (Aij )1≤i, j ≤m ∈ Mn is a block matrix such that A11 , . . . , Amm are square matrices and
/ {(1, 2), . . . , (m − 1, m), (m, 1)}. Then W(A) = c W(A) for any c ∈ C
Aij = 0 whenever (i, j ) ∈
satisfying c m = 1. If Am,1 is also zero, then W(A) is a circular disk centered at 0 with the radius
equal to the largest eigenvalue of (A + A∗ )/2.
Examples:
1. Let A = diag (1, 0). Then W(A) = [0, 1].
0
2. Let A =
0
2
. Then W(A) is the closed unit disk D = {a ∈ C : |a| ≤ 1}.
0
2
3. Let A =
0
2
. Then by Fact 7 above, W(A) is the convex set whose boundary is the ellipse with
1
foci 1 and 2 and minor axis 2, as shown in Figure 18.1.
0
4. Let A = diag (1, i, −1, −i ) ⊕
0
vertices 1, i, −1, −i .
1
. By Facts 5 and 7, the boundary of W(A) is the square with
0
18-3
Numerical Range
1
0.5
0.5
1
1.5
2
–0.5
–1
FIGURE 18.1 Numerical range of the matrix A in Example 3.
Applications:
1. By Fact 6, if A is real, then W(A) is symmetric about the real axis, i.e., W(A) = W(A).
2. Suppose A ∈ Cn×n , and there are a, b ∈ C such that (A − a I )(A − b I ) = 0n . Then A is unitarily
similar to a matrix of the form
a
a Ir ⊕ b Is ⊕
0
d1
a
⊕ ··· ⊕
b
0
dt
b
with d1 ≥ · · · ≥ dt > 0, where r + s + 2t = n. By Facts 1, 5, and 7, the set W(A) is the elliptical
disk with foci a, b and minor axis of length d, where
d = d1 =
A
22 − |a|2 A
22 − |b|2
1/2
/
A
2
if t ≥ 1, and d = 0 otherwise.
3. By Fact 12, if A ∈ Cn×n is the basic circulant matrix E 12 + E 23 + · · · + E n−1,n + E n1 , then
W(A) = conv {c ∈ C : c n = 1}; if A ∈ Mn is the Jordan block of zero J n (0), then W(A) =
{c ∈ C : |c | ≤ cos(π/(n + 1))}.
4. Suppose A ∈ Cn×n is a primitive nonnegative matrix. Then A is permutationally similar to a
block matrix (Aij ) as described in Fact 12 and, thus, W(A) = c W(A) for any c ∈ C satisfying
c m = 1.
18.2
The Spectrum and Special Boundary Points
Definitions and Notation:
Let ∂S and int (S) be the boundary and the interior of a convex compact subset S of C.
A support line of S is a line that intersects ∂S such that S lies entirely within one of the closed
half-planes determined by .
A boundary point µ of S is nondifferentiable if there is more than one support line of S passing
through µ.
An eigenvalue λ of A ∈ Cn×n is a reducing eigenvalue if A is unitarily similar to [λ] ⊕ A2 .
Facts:
The following facts can be found in [GR96],[Hal82], and [HJ91].
1. Let A ∈ Cn×n . Then
σ (A) ⊆ W(A) ⊆ {a ∈ C : |a| ≤ A
2 }.
18-4
Handbook of Linear Algebra
2. Let A, E ∈ Cn×n . We have
σ (A + E ) ⊆ W(A + E ) ⊆ W(A) + W(E )
⊆ {a + b ∈ C : a ∈ W(A),
b ∈ C with
|b| ≤ E 2 }.
3. Let A ∈ Cn×n and a ∈ C. Then a ∈ σ (A) ∩ ∂ W(A) if and only if A is unitarily similar to a Ik ⊕ B
such that a ∈
/ σ (B) ∪ int (W(B)).
4. Let A ∈ Cn×n and a ∈ C. Then a is a nondifferentiable boundary point of W(A) if and only if A
/ W(B). In particular, a is a reducing eigenvalue of A.
is unitarily similar to a Ik ⊕ B such that a ∈
5. Let A ∈ Cn×n . If W(A) has at least n − 1 nondifferentiable boundary points or if at least n − 1
eigenvalues of A (counting multiplicities) lie in ∂ W(A), then A is normal.
Examples:
0
1. Let A = [1] ⊕
0
2
. Then W(A) is the unit disk centered at the origin, and 1 is a reducing
0
eigenvalue of A lying on the boundary of W(A).
0
2. Let A = [2] ⊕
0
2
. Then W(A) is the convex hull of unit disk centered at the origin of the
0
number 2, and 2 is a nondifferentiable boundary point of W(A).
Applications:
1. By Fact 1, if A ∈ Cn×n and 0 ∈
/ W(A), then 0 ∈
/ σ (A) and, thus, A is invertible.
2. By Fact 4, if A ∈ Cn×n , then W(A) has at most n nondifferentiable boundary points.
3. While W(A) does not give a very tight containment region for σ (A) as shown in the examples
in the last section. Fact 2 shows that the numerical range can be used to estimated the spectrum
of the resulting matrix when A is under a perturbation E . In contrast, σ (A) and σ (E ) usually
0
do not carry much information about σ (A + E ) in general. For example, let A =
0
M
and
0
√
0
. Then σ (A) = σ (E ) = {0}, σ (A + E ) = {± Mε} ⊆ W(A + E ), which is the
0
√
elliptical disk with foci ± Mε and length of minor axis equal to | |M| − |ε| |.
0
E =
ε
18.3
Location of the Numerical Range
Facts:
The following facts can be found in [HJ91].
1. Let A ∈ Cn×n and t ∈ [0, 2π). Suppose xt ∈ Cn is a unit eigenvector corresponding to the largest
eigenvalue λ1 (t) of e i t A + e −i t A∗ , and
Pt = {a ∈ C : e it a + e −it ā ≤ λ1 (t)}.
Then
e it W(A) ⊆ Pt ,
λt = x∗t Axt ∈ ∂ W(A) ∩ ∂Pt
18-5
Numerical Range
and
W(A) = ∩r ∈[0,2π) e −ir Pr = conv {λr : r ∈ [0, 2π )}.
If T = {t1 , . . . , tk } with 0 ≤ t1 < · · · < tk < 2π and k > 2 such that tk − t1 > π, then
PTO (A) = ∩r ∈T e −ir Pr
and
PTI (A) = conv {λr : r ∈ T }
are two polygons in C such that
PTI (A) ⊆ W(A) ⊆ PTO (A).
Moreover, both the area W(A)\ PTI (A) and the area of PTO (A)\W(A) converge to 0 as max{t j −t j −1 :
1 ≤ j ≤ k + 1} converges to 0, where t0 = 0, tk+1 = 2π.
2. Let A = (aij ) ∈ Cn×n . For each j = 1, . . . , n, let
gj =
(|aij | + |a j i |)/2
and
G j (A) = {a ∈ C : |a − a j j | ≤ g j }.
i = j
Then
W(A) ⊆ conv ∪nj=1 G j (A).
Examples:
2 2
1. Let A =
. Then W(A) is the circular disk centered at 2 with radius 1 In Figure 18.2, W(A)
0 2
is approximated by PTO (A) with T = {2kπ/100 : 0 ≤ k ≤ 99}. If T = {0, π/2, π, 3π/2}, then
the polygon PTO (A) in Fact 1 is bounded by the four lines {3 + bi : b ∈ R}, {a + i : a ∈ R},
{1 + bi : b ∈ R}, {a −i : a ∈ R}, and the polygon PTI (A) equals the convex hull of {2, 1 +i, 0, 1 −i }.
⎡
5i
⎢
2. Let A = ⎣ 4
1
2
−3i
3
⎤
3
⎥
−2⎦. In Figure 18.3, W(A) is approximated by PTO (A) with T = {2kπ/100 :
9
0 ≤ k ≤ 99}. By Fact 2, W(A) lies in the convex hull of the circles G 1 = {a ∈ C : |a − 5i | ≤ 5},
G 2 = {a ∈ C : |a + 3i | ≤ 5.5}, G 3 = {a ∈ C : |a − 9| ≤ 4.5}.
1.5
imaginary axis
1
0.5
0
−0.5
−1
−1.5
0.5
1
1.5
2
real axis
2.5
3
3.5
FIGURE 18.2 Numerical range of the matrix A in Example 1.
18-6
Handbook of Linear Algebra
10
8
6
imaginary axis
4
2
0
−2
−4
−6
−8
−6
−4
−2
0
2
4
real axis
6
8
10
12
FIGURE 18.3 Numerical range of the matrix A in Example 2.
Applications:
1. Let A = H + i G , where H, G ∈ Cn×n are Hermitian. Then
W(A) ⊆ W(H) + i W(G ) = {a + i b : a ∈ W(H), b ∈ W(G )},
which is PTO (A) for T = {0, π/2, π, 3π/2}.
2. Let A = H + i G , where H, G ∈ Cn×n are Hermitian. Denote by λ1 (X) ≥ · · · ≥ λn (X) for a
Hermitian matrix X ∈ Cn×n . By Fact 1,
w (A) = max{λ1 (cos t H + sin tG ) : t ∈ [0, 2π )}.
If 0 ∈
/ W(A), then
(A) = max{{λn (cos t H + sin tG ) : t ∈ [0, 2π)} ∪ {0}}.
w
3. By Fact 2, if A = (aij ) ∈ Cn×n , then
w (A) ≤ max{|a j j | + g j : 1 ≤ j ≤ n}.
In particular, if A is nonnegative, then w (A) = λ1 (A + AT )/2.
18.4
Numerical Radius
Definitions:
Let N be a vector norm on Cn×n . It is submultiplicative if
N(AB) ≤ N(A)N(B)
for all
A, B ∈ Cn×n .
It is unitarily invariant if
N(UAV) = N(A)
for all
A ∈ Cn×n and unitary U, V ∈ Cn×n .
It is unitary similarity invariant (also known as weakly unitarily invariant) if
N(U ∗ AU) = N(A)
for all
A ∈ Cn×n and unitary U ∈ Cn×n .
18-7
Numerical Range
Facts:
The following facts can be found in [GR96] and [HJ91].
1. The numerical radius w (·) is a unitary similarity invariant vector norm on Cn×n , and it is not
unitarily invariant.
2. For any A ∈ Cn×n , we have
ρ(A) ≤ w (A) ≤ A
2 ≤ 2w (A).
3. Suppose A ∈ Cn×n is nonzero and the minimal polynomial of A has degree m. The following
conditions are equivalent.
(a) ρ(A) = w (A).
(b) There exists k ≥ 1 such that A is unitarily similar to γ U ⊕ B for a unitary U ∈ Ck×k and
B ∈ C(n−k)×(n−k) with w (B) ≤ w (A) = γ .
(c) There exists s ≥ m such that w (As ) = w (A)s .
4. Suppose A ∈ Cn×n is nonzero and the minimal polynomial of A has degree m. The following
conditions are equivalent.
(a) ρ(A) = A
2 .
(b) w (A) = A
2 .
(c) There exists k ≥ 1 such that A is unitarily similar to γ U ⊕ B for a unitary U ∈ Ck×k and a
B ∈ C(n−k)×(n−k) with B
2 ≤ A
2 = γ .
(d) There exists s ≥ m such that As 2 = A
s2 .
5. Suppose A ∈ Cn×n is nonzero. The following conditions are equivalent.
(a) A
2 = 2w (A).
(b) W(A) is a circular disk centered at origin with radius A
2 /2.
0
(c) A/
A
2 is unitarily similar to A1 ⊕ A2 such that A1 =
0
2
and w (A2 ) ≤ 1.
0
6. The vector norm 4w on Cn×n is submultiplicative, i.e.,
4w (AB) ≤ (4w (A))(4w (B)) for all
A, B ∈ Cn×n .
The equality holds if
0
X=Y =
0
T
2
.
0
7. Let A ∈ Cn×n and k be a positive integer. Then
w (Ak ) ≤ w (A)k .
8. Let N be a unitary similarity invariant vector norm on Cn×n such that N(Ak ) ≤ N(A)k for any
A ∈ Cn×n and positive integer k. Then
w (A) ≤ N(A)
for all
A ∈ Cn×n .
9. Suppose N is a unitarily invariant vector norm on Cn×n . Let
D=
2Ik ⊕ 0k
2Ik ⊕ I1 ⊕ 0k
if n = 2k,
if n = 2k + 1.
18-8
Handbook of Linear Algebra
Then a = N(E 11 ) and b = N(D) are the best (largest and smallest) constants such that
aw (A) ≤ N(A) ≤ bw (A)
A ∈ Cn×n .
for all
10. Let A ∈ Cn×n . The following are equivalent:
(a) w (A) ≤ 1.
(b) λ1 (e i t A + e −i t A∗ )/2 ≤ 1 for all t ∈ [0, 2π).
(c) There is Z ∈ C
n×n
I +Z
such that n ∗
A
A
is positive semidefinite.
In − Z
(d) There exists X ∈ C2n×n satisfying X ∗ X = In and
A=X
18.5
∗
0n
2In
0n
0n
X.
Products of Matrices
Facts:
The following facts can be found in [GR96] and [HJ91].
(A) > 0. Then
1. Let A, B ∈ Cn×n be such that w
σ (A−1 B) ⊆ {b/a : a ∈ W(A), b ∈ W(B)}.
2. Let 0 ≤ t1 < t2 < t1 + π and S = {rei t : r > 0, t ∈ [t1 , t2 ]}. Then σ (A) ⊆ S if and only if there is
a positive definite B ∈ Cn×n such that W(AB) ⊆ S.
3. Let A, B ∈ Cn×n .
(a)
(b)
(c)
(d)
If AB = BA, then w (AB) ≤ 2w (A)w (B).
If A or B is normal such that AB = BA, then w (AB) ≤ w (A)w (B).
If A2 = a I and AB = B A, then w (AB) ≤ A
2 w (B).
If AB = BA and AB∗ = B ∗ A, then w (AB) ≤ min{w (A)
B
2 , A
2 w (B)}.
4. Let A and B be square matrices such that A or B is normal. Then
W(A ◦ B) ⊆ W(A ⊗ B) = conv {W(A)W(B)}.
Consequently,
w (A ◦ B) ≤ w (A ⊗ B) = w (A)w (B).
(See Chapter 8.5 and 10.4 for the definitions of t A ◦ B and A ⊗ B.)
5. Let A and B be square matrices. Then
w (A ◦ B) ≤ w (A ⊗ B) ≤ min{w (A)
B
2 , A
2 w (B)} ≤ 2w (A)w (B).
6. Let A ∈ Cn×n . Then
w (A ◦ X) ≤ w (X)
for all
X ∈ Cn×n
if and only if A = B ∗ WB such that W satisfies W
≤ 1 and all diagonal entries of B ∗ B are
bounded by 1.
18-9
Numerical Range
Examples:
1. Let A ∈ C9×9 be the Jordan block of zero J 9 (0), and B = A3 + A7 . Then w (A) = w (B) =
cos(π/10) < 1 and w (AB) = 1 > A
2 w (B). So, even if AB = BA, we may not have w (AB) ≤
min{w (A)
B
2 , A
2 w (B)}.
1
2. Let A =
0
1
. Then W(A) = {a ∈ C : |a − 1| ≤ 1/2}
1
and
W(A2 ) = {a ∈ C : |a − 1| ≤ 1},
whereas
conv W(A)2 ⊆ {sei t ∈ C : s ∈ [0.25, 2.25], t ∈ [−π/3, π/3]}.
So, W(A2 ) ⊆ conv W(A)2 .
1
3. Let A =
0
0
−1
and
0
B=
1
1
. Then σ (AB) = {i, −i }, W(AB) = i [−1, 1], and W(A) =
0
W(B) = W(A)W(B) = [−1, 1]. So, σ (AB) ⊆ conv W(A)W(B).
Applications:
1. If C ∈ Cn×n is positive definite, then W(C −1 ) = W(C )−1 = {c −1 : c ∈ W(C )}. Applying Fact 1
with A = C −1 ,
σ (CB) ⊆ W(C )W(B).
(C ) > 0, then for every unit vector x ∈ Cn x∗ C −1 x = y∗ C ∗ C −1 C y with
2. If C ∈ Cn×n satisfies w
−1
y = C x and, hence,
W(C −1 ) ⊆ {r b : r ≥ 0,
b ∈ W(C ∗ )} = {r b : r ≥ 0,
b ∈ W(C )}.
Applying this observation and Fact 1 with A = C −1 , we have
σ (AB) ⊆ {r ab : r ≥ 0,
18.6
a ∈ W(A),
b ∈ W(B)}.
Dilations and Norm Estimation
Definitions:
A matrix A ∈ Cn×n has a dilation B ∈ Cm×m if there is X ∈ Cm×n such that X ∗ X = In and X ∗ B X = A.
A matrix A ∈ Cn×n is a contraction if A
2 ≤ 1.
Facts:
The following facts can be found in [CL00],[CL01] and their references.
1. A has a dilation B if and only if B is unitarily similar to a matrix of the form
A ∗
.
∗ ∗
2. Suppose B ∈ C3×3 has a reducing eigenvalue, or B ∈ C2×2 . If W(A) ⊆ W(B), then A has a dilation
of the form B ⊗ Im .
18-10
Handbook of Linear Algebra
3. Let r ∈ [−1, 1]. Suppose A ∈ Cn×n is a contraction with
W(A) ⊆ S = {a ∈ C : a + ā ≤ 2r }.
Then A has a unitary dilation U ∈ C2n×2n such that W(U ) ⊆ S.
4. Let A ∈ Cn×n . Then
W(A) = ∩{W(B) : B ∈ C2n×2n is a normal dilation of A}.
If A is a contraction, then
W(A) = ∩{W(U ) : U ∈ C2n×2n is a unitary dilation of A}.
5. Let A ∈ Cn×n .
(a) If W(A) lies in an triangle with vertices z 1 , z 2 , z 3 , then
A
2 ≤ max{|z 1 |, |z 2 |, |z 3 |}.
(b) If W(A) lies in an ellipse E with foci λ1 , λ2 , and minor axis of length b, then
A
2 ≤ { (|λ1 | + |λ2 |)2 + b 2 +
(|λ1 | − |λ2 |)2 + b 2 }/2.
More generally, if W(A) lies in the convex hull of the ellipse E and the point z 0 , then
A
2 ≤ max |z 0 |, { (|λ1 | + |λ2 |)2 + b 2 +
(|λ1 | − |λ2 |)2 + b 2 }/2 .
6. Let A ∈ Cn×n . Suppose there is t ∈ [0, 2π) such that e i t W(A) lies in a rectangle R centered at
z 0 ∈ C with vertices z 0 ± α ± iβ and z 0 ± α ∓ iβ, where α, β > 0, so that z 1 = z 0 + α + iβ has
the largest magnitude. Then
A
2 ≤
|z 1 |
α+β
if R ⊆ conv {z 1 , z̄ 1 , −z̄ 1 },
otherwise.
The bound in each case is attainable.
Examples:
0
1. Let A =
0
√ 2
. Suppose
0
⎡
1
0
B =⎢
⎣0
0
0
1⎥
⎦
0
0
⎢
⎡
⎤
0
⎥
0
0
⎢0 i
B =⎢
⎢0 0
⎣
0
⎢
or
1
0
0
−1
0
0
⎤
⎥
⎥.
0⎥
⎦
0⎥
−i
Then W(A) ⊆ W(B). However, A does not have a dilation of the form B ⊗ Im for either of the
matrices because
√
A
2 = 2 > 1 = B
2 = B ⊗ Im 2 .
So, there is no hope to further extend Fact 1 in this section to arbitrary B ∈ C3×3 or normal matrix
B ∈ C4×4 .
18-11
Numerical Range
18.7
Mappings on Matrices
Definitions:
Let φ : Cn×n → Cm×m be a linear map. It is unital if φ(In ) = Im ; it is positive if φ(A) is positive
semidefinite whenever A is positive semidefinite.
Facts:
The following facts can be found in [GR96] unless another reference is given.
1. [HJ91] Let P(C) be the set of subsets of C. Suppose a function F : Cn×n → P(C) satisfies the
following three conditions.
(a) F (A) is compact and convex for every A ∈ Cn×n .
(b) F (a A + b I ) = a F (A) + b for any a, b ∈ C and A ∈ Cn×n .
(c) F (A) ⊆ {a ∈ C : a + ā ≥ 0} if and only if A + A∗ is positive semidefinite.
Then F (A) = W(A) for all A ∈ Cn×n .
2. Use the usual topology on Cn×n and the Hausdorff metric on two compact sets A, B of C defined by
d(A, B) = max max
a∈A
min |a − b|, max
b∈B
b∈B
min |a − b|
a∈A
The mapping A → W(A) is continuous.
3. Suppose f (x + i y) = (ax + by + c ) + i (d x + e y + f ) for some real numbers a, b, c , d, e, f . Define
f (H + i G ) = (a H + bG + c I ) + i (d H + eG + f I ) for any two Hermitian matrices H, G ∈ Cn×n .
We have
W( f (H + i G )) = f (W(A)) = { f (x + i y) : x + i y ∈ W(A)}.
4. Let D = {a ∈ C : |a| ≤ 1}. Suppose f : D → C is analytic in the interior of D and continuous on
the boundary of D.
(a) If f (D) ⊆ D and f (0) = 0, then W( f (A)) ⊆ D whenever W(A) ⊆ D.
(b) If f (D) ⊆ C+ = {a ∈ C : a + ā ≥ 0}, then W( f (A)) ⊆ C+ \ {( f (0) + f (0))/2} whenever
W(A) ⊆ D.
5. Suppose φ : Cn×n → Cn×n is a unital positive linear map. Then W(φ(A)) ⊆ W(A) for all A ∈ Cn×n .
6. [Pel75] Let φ : Cn×n → Cn×n be linear. Then
W(A) = W(φ(A))
for all
A ∈ Cn×n
if and only if there is a unitary U ∈ Cn×n such that φ has the form
X → U ∗ XU
or
X → U ∗ X T U.
7. [Li87] Let φ : Cn×n → Cn×n be linear. Then w (A) = w (φ(A)) for all A ∈ Cn×n if and only if there
exist a unitary U ∈ Cn×n and a complex unit µ such that φ has the form
X → µU ∗ XU
or
X → µU ∗ X T U.
References
[CL00] M.D. Choi and C.K. Li, Numerical ranges and dilations, Lin. Multilin. Alg. 47 (2000), 35–48.
[CL01] M.D. Choi and C.K. Li, Constrained unitary dilations and numerical ranges, J. Oper. Theory 46
(2001), 435–447.
18-12
Handbook of Linear Algebra
[GR96] K.E. Gustafson and D.K.M. Rao, Numerical Range: the Field of Values of Linear Operators and
Matrices, Springer, New York, 1996.
[Hal82] P.R. Halmos, A Hilbert Space Problem Book, 2nd ed., Springer-Verlag, New York, 1982.
[HJ91]R.A. Horn and C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, New York,
1991.
[Li87]C.K. Li, Linear operators preserving the numerical radius of matrices, Proc. Amer. Math. Soc. 99
(1987), 105–118.
[Pel75] V. Pellegrini, Numerical range preserving operators on matrix algebras, Studia Math. 54 (1975),
143–147.