An extended version of Schur-Cohn-Fujiwara theorem in stability theory

Yongjian HU, Xuzhou ZHAN, Gongning CHEN
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
c
Higher Education Press and Springer-Verlag Berlin Heidelberg 2015
Abstract This paper is concerned with root localization of a complex
polynomial with respect to the unit circle in the more general case. The classical
Schur-Cohn-Fujiwara theorem converts the inertia problem of a polynomial
to that of an appropriate Hermitian matrix under the condition that
the associated Bezout matrix is nonsingular. To complete it, we discuss an
extended version of the Schur-Cohn-Fujiwara theorem to the singular case of
that Bezout matrix. Our method is mainly based on a perturbation technique
for a Bezout matrix. As an application of these results and methods, we further
obtain an explicit formula for the number of roots of a polynomial located on
the upper half part of the unit circle as well.
Keywords Inertia of polynomial, inertia of matrix, Bezout matrix, Schur-
Cohn-Fujiwara theorem, Schur-Cohn matrix
MSC 12D10, 15A57
1 Introduction
Let f(z) be a complex polynomial of degree n 1:
f(z) :=
n
i=0
fizi, fi ∈ C, fn = 0. (1.1)
A fundamental problem in the theory of stability is to count the number of
roots of such an f(z) in the unit disc of the complex plane C (see, e.g., [5–
7,9–11]). Denote by π◦(f), ν◦(f), and δ◦(f) the numbers of roots (counting
for multiplicities) of f(z) located in the open unit disc, the exterior of the
closed unit disc, and on the unit circle T := {z : |z| = 1} of the complex
plane C, respectively. The triple In◦(f) := (π◦(f), ν◦(f), δ◦(f)) is referred to
Received May 9, 2014; accepted January 28, 2015
Corresponding author: Yongjian HU, E-mail: yongjian@bnu.edu.cn
1114 Yongjian HU et al.
as the inertia of f(z) with respect to T. Moreover, the inertia of an n × n
complex matrix A with respect to the imaginary axis iR is defined by the triple
In(A) := (π(A), ν(A), δ(A)), where π(A), ν(A), and δ(A) stand for the numbers
of eigenvalues (counting for multiplicities) of A lying in the open right halfplane,
the open left half-plane of the complex plane C, and on iR, respectively.
For a pair of complex polynomials g(z) and h(z) with the maximal degree
n, by definition, the Bezout matrix B(g, h) is generated via the bilinear form
g(z)h(w) − g(w)h(z)
z − w
= (1, z, . . . , zn−1)B(g, h)(1,w, . . . ,wn−1)T, (1.2)
where vT means the transpose of a vector v. The Bezout matrix defined as in
(1.2) appears naturally when discussing the inertia problem of polynomials, has
many remarkable properties and applications in system theory and control, and
is closely related to many special matrices, such as Hankel, Toeplitz, Loewner,
and Vandermonde matrices (see, e.g., [1–4,10]).
Given a polynomial f(z) of form (1.1), the reciprocal polynomial f∗(z) of
f(z) (with respect to T and degree n) is defined by
f
∗(z) := znf(z−1) = f0zn + f1zn−1 + · · · + fn−1z + fn. (1.3)
If f∗(z) = cf (z) for some 0 = c ∈ C, then we say that the polynomial f(z)
is symmetric (with respect to T), whereby a symmetric polynomial f(z) has
only nonzero roots symmetric with respect to T. Moreover, it is obvious that
B(f, f∗) = 0 if f(z) is symmetric.
The relation between the inertia of a polynomial with respect to T and the
inertia of an appropriate Hermitian matrix with respect to iR is revealed by a
celebrated result due to Fujiwara (see, e.g., [5], [10, pp. 466, 467]). Here and in
the following, Pk is the rotation matrix of order k, i.e.,
Pk := (δk−i+1,j)1 i,j k, k 1.
Theorem 1 (Schur-Cohn-Fujiwara) Let f(z) be a complex polynomial of the
form (1.1), and let f∗(z) be the reciprocal polynomial of f(z). Then
˘B
:= PnB(f, f
∗)
is a Hermitian matrix and, when it is nonsingular, we have
In◦(f) = In(˘B), δ
◦(f) = δ(˘B) = 0. (1.4)
As a rule, the Hermitian matrix ˘B given in Theorem 1 is called the Schur-
Cohn matrix of the polynomial f(z).
In Section 2, we will study a version of the Schur-Cohn-Fujiwara theorem
in the singular case, i.e., when
detB(f, f
∗) = 0,
An extended version of Schur-Cohn-Fujiwara theorem in stability theory 1115
or equivalently,
det˘B = 0.
In particular, we present two explicit formulas of (1.4)-type in the singular
situation (see Theorems 2 and 3 below). Our strategy is based on a special
factorization of f(z), and a perturbation technique for a Bezout matrix, which
are much different from those given in [6–9,11]. Using our results and methods
here, in Section 3, we can further obtain a formula for the number of the roots
of a polynomial located on the upper half part of the unit circle (see Theorem 5).
2 A singular version of Schur-Cohn-Fujiwara theorem
Given a polynomial f(z) of form (1.1), let
p(z) := gcd(f(z), f
∗(z)).
Then, by definition, p(z) is a monic and symmetric polynomial. If p(z) = 1,
then the Bezout matrix B(f, f∗) is nonsingular and our discussion reduces to
the Schur-Cohn-Fujiwara theorem (Theorem 1). If p(z) = 1, then the Bezout
matrix B(f, f∗) is singular and the original Schur-Cohn-Fujiwara theorem will
not be applied directly. In order to establish an extension of the Schur-Cohn-
Fujiwara theorem to the singular case of B(f, f∗), we begin with a factorization
of p(z) into a power product of its factors with only simple roots in the complex
plane (Lemma 1 below). We then use it to formulate the inertia In◦(f) in terms
of the inertia of a sequence of Hermitian matrices in that singular case.
The following procedure, initiated from [12, pp. 65–68], will bring forth a
special factorization of p(z). With the Euclidean algorithm for polynomials, we
first compute recurrently
m1(z) := gcd(p(z), p (z)),
m2(z) := gcd(m1(z),m
1(z)),
. . . . . . ,
ms−1(z) := gcd(ms−2(z),m
s−2(z)) = 1,
ms(z) := gcd(ms−1(z),m
s−1(z)) = 1,
(2.1)
where s is a certain positive integer, determined by f(z). Define in turn
q1(z) := p(z)
m1(z), qk(z) := mk−1(z)
mk(z) , k= 2, . . . , s. (2.2)
These qk(z) are monic polynomials with only simple roots in the complex plane.
One may check that the roots of qk+1(z) are also the roots of qk(z). Thus,
qk+1(z) | qk(z), k= 1, . . . , s − 1.
1116 Yongjian HU et al.
Now, we write down
pk(z) := qk(z)
qk+1(z), k= 1, . . . , s − 1; ps(z) := qs(z). (2.3)
Let
Δ := {k | deg pk 1, k = 1, . . . , s}.
Then (2.3) results in immediately a special factorization of p(z) as follows.
Lemma 1 Let f(z) and f∗(z) be as before, let B(f, f∗) be singular, let p(z) =
gcd(f(z), f∗(z)), and let pk(z) be defined by (2.1)–(2.3). Then the set Δ is nonempty,
and
p(z) =

k∈Δ
pkk
(z). (2.4)
Moreover, these pk(z) are coprime symmetric polynomials with only simple roots
in the complex plane.
Proof As was mentioned above, p(z) = gcd(f(z), f∗(z)) is then a monic
symmetric polynomial, and thus, p(z) can be represented as
p(z) =

i
(z − αi)d(αi)

j
((z − βj)(z − β
−1
j ))d(βj ), (2.5)
where αi and βj are the pairwise different roots of p(z) on and inside T, with
multiplicities d(αi) and d(βj ), respectively. Inserting (2.5) into (2.1)–(2.3), we
immediately obtain the following factorizations of pk(z):
pk(z) =

i : d(αi)=k
(z − αi)

j : d(βj)=k
(z − βj)(z − β
−1
j ), k∈ Δ. (2.6)
The required assertion (2.4) follows directly from (2.5) and (2.6). Moreover,
from the factorizations (2.6), it easily follows that each factor pk(z) is a
symmetric polynomial and has only simple roots in the complex plane, which
are different from the roots of the remaining factors.
By use of Theorem 1, Lemma 1, and a perturbation technique for the inertia
of each pk(z) (k ∈ Δ) (or of the related Bezout matrix), we can obtain an
extension of the Schur-Cohn-Fujiwara theorem to the case when the Bezout
matrix B(f, f∗) is singular.
Theorem 2 Let B(f, f∗) be singular, let p(z) = gcd(f(z), f∗(z)) be of degree
m with factorization as in (2.4), and let p0(z) be the complex polynomial such
that f(z) = p0(z)p(z). Then B0 := Pρ0B(p0, p∗
0) and ˘Bk := PρkB(zp
k, p∗
k) (k ∈
Δ) are nonsingular Hermitian matrices in which ρk := deg pk(z) (k ∈ {0}∪Δ),
and
In◦(f) = In(B0) +

k∈Δ
kν(˘Bk),

k∈Δ
kν(˘Bk),m − 2

k∈Δ
kν(˘Bk)

. (2.7)
An extended version of Schur-Cohn-Fujiwara theorem in stability theory 1117
Proof Since f(z) = p0(z)p(z) in which, by Lemma 1, p(z) has the factorization
as in (2.4), f(z) can be represented as
f(z) = p0(z)

k∈Δ
pkk
(z),
moreover, p0(z) is a polynomial satisfying
gcd(p0(z), p
∗
0(z)) = 1,
so that detB(p0, p∗
0) = 0. This factorization of f(z) gives at once
In◦(f) = In◦(p0) +

k∈Δ
kIn◦(pk). (2.8)
By Theorem 1 and the symmetry of pk(z) (k ∈ Δ), B0 = Pρ0B(p0, p∗
0) (ρ0 =
deg p0(z)) is a nonsingular Hermitian matrix and
In◦(pk) =

In(B0), k= 0,
(π◦(pk), π◦(pk), deg pk(z) − 2π◦(pk)), k∈ Δ.
(2.9)
Combining (2.9) and (2.8), from
m = degp(z) =

k∈Δ
k deg pk(z),
we deduce that
In◦(f) = In(B0) +

k∈Δ
kπ
◦(pk),

k∈Δ
kπ
◦(pk), m− 2

k∈Δ
kπ
◦(pk)

.
This indicates that to prove assertion (2.7), it suffices to verify that for each
k ∈ Δ, ˘Bk is a nonsingular Hermitian matrix such that π◦(pk) = ν(˘Bk).
To do it, for k ∈ Δ, we define
pk,r(z) := pk(rz),
where r is real. Observe that when r is close enough to 1 from the left,
π
◦(pk) = π
◦(pk,r), gcd(pk,r(z), p
∗
k,r(z)) = 1,
and therefore, the Bezout matrix Bk(r) := B(pk,r, p∗
k,r) (k ∈ Δ) is nonsingular.
By Theorem 1, for such real r and for each k ∈ Δ, we have
In◦(pk,r) = In(PρkBk(r)), (2.10)
in which PρkBk(r) is the Schur-Cohn matrix of pk,r(z). Since each entry of Bk(r)
is a polynomial of r and
Bk(1) = B(pk, p
∗
k) = 0,
1118 Yongjian HU et al.
we have in turn
PρkBk(r) = (r − 1)PρkB

k(1) + o(r − 1) = (r − 1)(PρkB

k(1) + o(1)),
where k ∈ Δ, B
k(1) is the first derivative of Bk(r) at r = 1, o(1) → 0 as r → 1−,
and hence, PρkB
k(1) is a Hermitian matrix. This implies that
In(PρkBk(r)) = In(−PρkB

k(1) + o(1)), k∈ Δ. (2.11)
Now, we focus on B
k(1). Observe that, for k ∈ Δ, by definition (see (1.2)),
(1, z, . . . , zρk−1)B

k(1)(1,w, . . . ,wρk−1)T
=
pk,r(z)p∗
k,r(w) − pk,r(w)p∗
k,r(z)
z − w

r

r=1
= pk(0)
pk(rz)rρkpk(wr−1) − pk(rw)rρkpk(zr−1)
z − w

r

r=1
= 2pk(0)
zp
k(z)pk(w) − wp
k(w)pk(z)
z − w

= 2(1, z, . . . , zρk−1)B(zp

k, pk(0) pk)(1,w, . . . ,wρk−1)T
= 2(1, z, . . . , zρk−1)B(zp

k, p
∗
k)(1,w, . . . ,wρk−1)T.
This yields
B

k(1) = 2B(zp

k, p
∗
k), k∈ Δ.
Then
˘B
k = PρkB(zp

k, p
∗
k) =
1
2 PρkB

k(1)
is Hermitian (since PρkB
k(1) is Hermitian). By (2.10) and (2.11), for k ∈ Δ,
we have
In◦(pk,r) = In(−2PρkB(zp

k, p
∗
k) + o(1)) = In(−˘Bk + o(1)). (2.12)
For k ∈ Δ, pk(z) has only simple roots, p∗
k(z) = pk(0) pk(z), and pk(0) = 0, so
that zp
k(z) and p∗
k(z) are coprime. Then ˘Bk = PkB(zp
k, p∗
k) is a nonsingular
Hermitian matrix for k ∈ Δ. Note that the eigenvalues of a square matrix are
continuous functions of its entries. Then we derive that
In(−˘Bk + o(1)) = In(−˘Bk), k∈ Δ.
From (2.12), it follows that
π
◦(pk) (=π
◦(pk,r) for those r mentioned above) = π(−˘Bk) = ν(˘Bk)
for each k ∈ Δ, as desired.
According to the bilinearity of the Bezout matrix, we can obtain another
formula of In◦(f), which is similar to and simpler than (2.7) in Theorem 2.
An extended version of Schur-Cohn-Fujiwara theorem in stability theory 1119
Theorem 3 Let f(z), p(z), p0(z), m, B0, and pk(z) (k ∈ Δ) be the same as
those given in Theorem 2. Then
ˆB
k := Pρk−1B(p

k, (p

k)∗), k∈ Δ,
are nonsingular Hermitian matrices in which ρk = degpk(z), and
In◦(f) = In(B0) +

k∈Δ
kν(ˆBk),

k∈Δ
kν(ˆBk),m − 2

k∈Δ
kν(ˆBk)

. (2.13)
Proof Observe that
p
∗
k(z) = pk(0) pk(z), k∈ Δ,
since pk(z) is symmetric. By taking derivative on both sides of this equality,
we can obtain the following equality:
p
∗
k(z) = ρ
−1
k [(p

k)∗(z) + pk(0) zp

k(z)], k∈ Δ. (2.14)
Let
˘B
k = PρkB(zp

k, p
∗
k), k∈ Δ,
where ρk = degpk(z). Then Theorem 2 says that ˘Bk is nonsingular Hermitian
for k ∈ Δ. From (2.14), together with the bilinearity of the Bezout matrix, it
follows that for k ∈ Δ,
˘B
k = PρkB(zp

k, p
∗
k)
= ρ
−1
k PρkB(zp

k, (p

k)∗ + pk(0) zp

k)
= ρ
−1
k PρkB(zp

k, (p

k)∗)
= ρ
−1
k PρkB(zp

k, (zp

k)∗), (2.15)
so that PρkB(zp
k, (zp
k)∗), and therefore, Pρk−1B(p
k, (p
k)∗) are nonsingular
Hermitian. Thus, by use of Theorem 1 twice, we have
ν(˘Bk) = ν(PρkB(zp

k, (zp

k)∗))
= ν
◦(zp

k)
= ν
◦(p

k)
= ν(Pρk−1B(p

k, (p

k)∗))
= ν(ˆBk).
Inserting (2.15) into (2.7), we obtain (2.13) immediately.
It should be mentioned that from the proof of Theorem 3, we see that
˘B
k (k ∈ Δ) given in Theorem 2 is actually the Schur-Cohn matrix of the
polynomial ρ
−1/2
k zp
k(z). As much, ˆBk (k ∈ Δ) is, by definition, the Schur-Cohn
matrix of the polynomial p
k(z).
1120 Yongjian HU et al.
3 Numberofrootsof f(z) on upper half part of unit circle
We use the symbol δ◦
+(g) to designate the number of roots of a complex
polynomial g(z) located on the upper half part of the unit circle (including ±1).
Let f(z) and f∗(z) be of forms (1.1) and (1.3), respectively. By Theorem 1,
if the Bezout matrix B(f, f∗) is nonsingular, then δ◦(f) = 0, and thus, δ◦
+(f) =
0 as well. In this section, we deduce an explicit formula of δ◦
+(f) in the case
of the Bezout matrix B(f, f∗) to be singular, based on the factorization of
p(z) = gcd(f(z), f∗(z)) (see Lemma 1) and the extended version of Schur-Cohn-
Fujiwara theorem (see Theorem 2), together with a perturbation technique for
f(z), or for its Schur-Cohn matrix.
The following theorem provides us a formula of δ◦
+(f) for a symmetric
polynomial f(z) which has only simple roots unlike ±1 in the complex plane.
Theorem 4 Let f(z) be a symmetric polynomial of degree n, which has only
simple roots in the complex plane. Let
g(z) := i((z2 − 1)f
(z) − nzf(z)).
If f(1)f(−1) = 0, then both PnB(g, f∗) and Pn(zf , f∗) are nonsingular
Hermitian matrices, and
δ
◦
+(f) = ν(PnB(g, f
∗)) − ν(PnB(zf

, f
∗)). (3.1)
Proof Let
fε(z) := f(z − iε),
where ε is a sufficiently small positive number. We check easily that
gcd(fε, f
∗
ε) = 1, δ
◦
+(f) = ν
◦(fε) − ν
◦(f).
By Theorems 1 and 2, we have in turn PnB(fε, f∗
ε) and PnB(zf , f∗) are both
nonsingular Hermitian, and
δ
◦
+(f) = ν(PnB(fε, f
∗
ε )) − ν(PnB(zf

, f
∗)). (3.2)
Note that
B(fε, f
∗
ε )|ε=0 = 0
and that each entry of B(fε, f∗
ε ) is a continuous and differential function of ε.
Thus,
B(fε, f
∗
ε) = ε(B
(fε, f
∗
ε )|ε=0 + o(1)), (3.3)
which implies that PnB (fε, f∗
ε ) is Hermitian. Furthermore, by the symmetry
of f(z), we have
(1, z, . . . , zn−1)B
(fε, f
∗
ε )|ε=0(1,w, . . . ,wn−1)T
=
f(z − iε)wnf(w−1 − iε) − znf(z−1 − iε) f(w − iε)
z − w

ε

ε=0
An extended version of Schur-Cohn-Fujiwara theorem in stability theory 1121
= i
f (w)f∗(z) − f (z)f∗(w)
z − w
− nzf(z)f∗(w) − nwf(w)f∗(z)
z − w
+ z2f (z)f∗(w) − w2f (w)f∗(z)
z − w

= i
((z2 − 1)f (z) − nzf(z))f∗(w) − f∗(z)((w2 − 1)f (w) − nwf(w))
z − w
= (1, z, . . . , zn−1)B(g, f
∗)(1,w, . . . ,wn−1)T.
This means that
B
(fε, f
∗
ε )|ε=0 = B(g, f
∗),
and thus,
PnB
(fε, f
∗
ε )|ε=0 = PnB(g, f
∗).
Since f(z) has only simple roots on the complex plane and f(0)f(1)f(−1) = 0,
we have
gcd(g, f
∗) = gcd(g, f(0)f) = 1,
and thus, PnB(g, f∗) is a nonsingular Hermitian matrix. Then by (3.3), for
sufficiently small ε > 0, we have
ν(PnB(fε, f
∗
ε )) = ν(PnB(g, f
∗) + o(1)) = ν(PnB(g, f
∗)). (3.4)
Inserting (3.4) into (3.2), we obtain (3.1) immediately.
Now, let f(z) be a polynomial of form (1.1) such that B(f, f∗) is singular,
and let p(z) = gcd(f(z), f∗(z)) have the factorization as in (2.4). Then
δ
◦
+(f) = δ
◦
+(p) =

k∈Δ
kδ
◦
+(pk).
Applying Theorem 4 to each symmetrical factor pk(z) of p(z) with pk(±1) =
0 (k ∈ Δ), we can formulate δ◦
+(f) in terms of the inertia of a sequence of
Hermitian matrices.
Theorem 5 Let B(f, f∗) be singular, and let p(z) = gcd(f(z), f∗(z)) have
the factorization as in (2.4). Let
gk(z) := i(z2 − 1)p

k(z) − ρkzpk(z), ρk = degpk(z), k∈ Δ.
If f(1)f(−1) = 0, then for k ∈ Δ, both PρkB(gk, p∗
k) and Pρk (zp
k, p∗
k) are
nonsingular Hermitian matrices, and
δ
◦
+(f) =

k∈Δ
k(ν(PρkB(gk, p
∗
k)) − ν(PρkB(zp

k, p
∗
k))).
Remark that if f(z) vanishes at z = 1 or at z = −1, it can be factorized as
f(z) = (z − 1)k(z + 1)lf1(z),
1122 Yongjian HU et al.
in which k, l are nonnegative integers such that k+l > 0, and f1(1)f1(−1) = 0.
Observe that, by definition,
δ
◦
+(f) = δ
◦
+(f1) + k + l.
Then Theorem 5 is applicable to the symmetric polynomial f1(z).
Acknowledgements This work was supported by the National Natural Science
Foundation of China (Nos. 11071017, 11271045) and the Program for New Century
Excellent Talents in University.
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