Numerical range of a pair of strictly upper triangular matrices
Proyecciones Journal of Mathematics
Vol. 30, No 1, pp. 77-90, May 2011.
Universidad Católica del Norte
Antofagasta - Chile
Numerical range of a pair of strictly upper
triangular matrices
Wen Yan
Tuskegee University, U. S. A.
Received : January 2011. Accepted : March 2011
Abstract
Given two strictly upper triangular matrices X, Y ∈ Cm×m , we
study the range WY (X) = {tr nXn−1 Y ∗ : n ∈ N }, where N is the
group of unit upper triangular matrices in Cm×m . We prove that it
is either a point or the whole complex plane. We characterize when it
is a point.
We also obtain some convexity result for a similar range, where
N is replaced by any ball of C k (k = m(m − 1)/2) embedded in N ,
m ≤ 4.
2000 Mathematics Subject Classification : Primary 15A60.
Key Words and Phrases : Numerical range, unit upper triangular matrices, strictly upper triangular matrices.
78
Wen Yan
1. Introduction
Let Cm×m be the space of all m × m complex matrices. The classical
numerical range of A ∈ Cm×m is defined as
W (A) := {x∗ Ax : x∗ x = 1, x ∈ C m } ⊂ C.
The celebrated Toeplitz-Hausdorff theorem [9] asserts that W (A) is a compact convex subset of C. There are numerous generalizations [5, 1, 4, 8,
7, 10, 11, 12, 14] and our references are far from complete. One important
view is to deem the numerical range as the image of an orbit under the
linear functional [2] determined by A, that is,
W (A) = {tr Axx∗ : x ∈ C m , x∗ x = 1}.
The set
{xx∗ : x ∈ C m , x∗ x = 1} = O(E11 ) := {U E11 U ∗ : U ∈ U (m)}
is viewed as an orbit of the matrix E11 := diag (1, 0, . . . , 0) under the conjugation action of U (m), where U (m) denotes the unitary group in Cm×m .
In general, if C ∈ Cm×m , then denote by
O(C) := {U CU ∗ : U ∈ U (m)}
the orbit of C under the conjugation action of U (n). The C-numerical
range of A [13, 3] is defined to be the set
WC (A) := {tr AY : Y ∈ O(C)}.
If C = diag (1, . . . , 1, 0, . . . , 0), (k 1’s), it becomes Halmos’s k-numerical
range [7] of A
Wk (A) = {
k
X
j=1
x∗j Axj : x1 , . . . , xk ∈ C m
are orthonormal }.
If C = diag (c1 , . . . , cm ) (c’s are real), the C-numerical range of A becomes
Westwick’s c-numerical range [14] of A
Wc (A) = {
m
X
j=1
cj x∗j Axj : x1 , . . . , xm ∈ C m
are orthonormal }.
Numerical range of a pair of strictly upper triangular matrices
79
Westwick’s theorem [14] asserts that the c-numerical range of A is convex.
The orbital point of view leads to several generalizations of the numerical
range. Moreover the convexity result has been successfully extended in
the context of compact Lie groups [11] and most real classical semisimple
Lie algebras [8, 4, 12]. Usually the groups involved in the relevant orbital
generalizations are compact (for example U (m) is compact in the setting
of the c-numerical range).
In this note we consider the group of m×m unit upper triangular matrices which is non-semisimple and noncompact. By a unit upper triangular
matrix, we mean an upper triangular with diagonal entries all ones. Let N
be the group of unit upper triangular matrices in Cm×m . It is a unipotent
(noncompact) Lie group whose Lie algebra n is the set of strictly upper
triangular matrices in Cm×m . Given X ∈ n, denote by
O(X) := {nXn−1 : n ∈ N } ⊂ n
the orbit of X under the conjugation action of the group N . Let X, Y ∈ n.
The numerical range of the pair (X, Y ) is defined as
WY (X) := {tr nXn−1 Y ∗ : n ∈ N }.
It may be interpreted as the image of the orbit O(X) under the linear
functional determined by Y . In Section 2 we prove that WY (X) is either a
point (not necessarily the origin) or C. In Section 3, given r > 0, cij ∈ C,
1 ≤ i < j ≤ m, we consider a compact subset of N:
N1 := {n := (nij ) ∈ N :
X
1≤i<j≤m
|nij − cij |2 = r2 }.
In other words, the ball of radius r (with respect to the 2-norm) centered
at c of C s is embedded as N1 ⊂ N , where s = m(m − 1)/2. We consider
the restricted range:
WY1 (X) := {tr nXn−1 Y ∗ : n ∈ N1 }.
When m = 2, 3, 4 we prove that WY1 (X) is a convex set. When m > 4
convexity of WY1 (X) is unknown.
2. The shape of WY (X)
Theorem 1. Let X, Y ∈ n. When m = 2, WY (X) = {tr nXn−1 Y ∗ : n ∈
N } is a singleton set {xȳ} if
X=
Ã
0 x
0 0
!
, Y =
Ã
0 y
0 0
!
.
80
Wen Yan
When m > 2, WY (X) is either a point or the whole complex plane C.
P
If WY (X) is a point, then the point is 1≤i< ≤m xi ȳi . More precisely,
WY (X) = C if and only if one of the following is true.
(i) xjk ȳi 6= 0 for some i, j, k and
such that
(a) 1 ≤ i < j < k < ≤ m, or
(b) 1 ≤ i = j < k < − 1 ≤ m − 1, or
(c) 2 ≤ i + 1 < j < k = ≤ m.
(ii) xjk ȳi = 0 for all 1 ≤ i < j < k < ≤ m, but there exist i, such that
i < − 1, xi, −1 ȳi 6= 0 and xi, −1 ȳi 6= x t ȳ −1,t for all < t ≤ m, or
xi+1, ȳi 6= 0 and xi+1, ȳi 6= xti ȳt,i+1 for all 1 ≤ t < i.
Proof.
The case m = 2 is trivial. Suppose m > 2. Let n = (nij ) ∈ N .
Clearly M := n−1 is upper triangular. Because of the upper triangular
form of n, X, Y, M , we have
tr nXn−1 Y ∗ =
X
nij xjk Mk ȳi .
1≤i≤j<k≤ ≤m
Notice that the (k, ) entry of M is
⎧
⎪
1
⎪
⎪
⎪
⎪
0
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
if k = l,
if k >⎛1
nk,k+1 nk,k+2
nk,k+3
⎜
1
nk+1,k+2 nk+1,k+3
⎜
Mkl =
⎜
k+l
⎪
det
0
1
nk+2,k+3
(−1)
⎜
⎪
⎪
⎜
⎪
⎪
⎝
·
·
·
·
·
·
···
⎪
⎪
⎪
⎪
⎪
0
0
0
⎪
⎪
⎩
if k < l
Notice that Mk is a polynomial in the variables nst , k ≤ s < t ≤ .
Moreover the exponent of each nst in the expression (2.1) of Mk is either
0 or 1.
Evidently tr nXn−1 Y ∗ is a polynomial of nij , 1 ≤ i < j ≤ m. Since nij
does not appear in the polynomial Mk for i ≤ j < k ≤ , the exponent of
any nij (i < j) in tr nXn−1 Y ∗ is either 0 or 1. We use n1 , ..., nr to denote
those nij (i < j) which appear in the polynomial tr nXn−1 Y ∗ . Let
f0 (n1 , n2 , ..., nr ) := tr nXn−1 Y ∗ .
⎞
· · · nk,l−1
nk,l
· · · nk+1,l−1 nk+1,l ⎟
⎟
⎟
· · · nk+2,l−1 nk+2l ⎟
⎟
···
···
··· ⎠
···
1
nl−1,l
Numerical range of a pair of strictly upper triangular matrices
81
1. If f0 is a constant polynomial. Then {tr nXn−1 Y ∗ : n ∈ N } is a
point.
2. Otherwise, we can rewrite f0 as
f0 (n1 , ..., nr ) = n1 f1 (n2 , ..., nr ) + f2 (n2 , ..., nr ),
where f1 is either a nonconstant polynomial in n2 , n3 ..., nr or a nonzero
constant number c. In either case we can choose complex numbers
c2 , ..., cr for n2 , ..., nr such that f1 (c2 , ..., cr ) 6= 0. By the fundamental theorem of algebra {f0 (n1 , c2 , ..., cr ) : n1 ∈ C} = C. Hence
WY (X) = C.
So WY (X) is either a point or C.
We are going to show that WY (X) = C if either (i) or (ii) holds. Suppose (i)(a) is true, that is, there exists xj0 k0 ȳi0 0 6= 0 for some 1 ≤ i0 <
j0 < k0 < 0 ≤ m. Define
n(s) := (nij ) = Im + sEi0 ,j0 + sEk0 , 0 ∈ N,
s ∈ C,
and Eij is the matrix with 1 as the (i, j) entry and zeros elsewhere. So
M := n(s)−1 = Im − sEi0 ,j0 − sEk0 , 0 . Then
f (s) := tr n(s)Xn(s)−1 Y ∗ =
X
nij xjk Mkl ȳi
1≤i≤j<k≤ ≤m
is a quadratic polynomial in s, and the leading term of f (s) is ni0 j0 xj0 k0 Mk0 0 ȳi0 0 =
−xj0 k0 ȳi0 0 s2 . Therefore
C = {f (s) : s ∈ C} ⊂ WY (X) ⊂ C.
We now insert a lemma.
Lemma 2. Suppose (i)(a) is (...truncated)
