VNU Journal of Mathematics – Physics, Vol. 29, No. 1 (2013) 18-32
Some Remarks on the Complex Stability Radius of
Differential-Algebraic Equations
Le Huy Hoang*
Faculty of Informatics, National University of Civil Engineering, Dong Tam, Hanoi, Vietnam
Received 05 December 2012
Revised 28 February 2013; accepted 15 March 2013
Abstract. This paper is concerned with the robust stability of linear differential-algebraic
equations (DAEs). A system of linear DAEs subjected to structured perturbation is considered.
Computable formulas of the complex stability radius are given and analysed. A comparison of our
formula to previous results is given.
Keywords: linear differential-algebraic equations, index of matrix pencil, asymptotic stability,
structured perturbation, complex stability radius.
1. Introduction*
Differential-algebraic equations (DAEs) play an important roles in mathematical modeling of reallife problems arising in a wide range of applications, for example, multibody mechanics, prescribed
path control, eletrical design, biology, biomedicine, see [1, 2] and references therein. On the other
hand, the robustness issue is a crucial problem for the application of control theory, for example, one
of the basic goal of feedback control is to enhance system robustness. Robust stability is also an
important topic in linear algebra as well as in numerical analysis.
Consider a linear DAE
Ey(x) = Ay(x),
where A, E  K
nn
, K
or
(1.1)
. The leading coefficient matrix E is singular.
Definition 1.1. (see [1]) The matrix pencil {E , A} is said to be regular if there exists t  such
that the determinant of ( A  tE ) , denoted by det( A  tE ) , is different from zero. We also say that
(1.1) is regular. Otherwise, if det( A  tE )  0 ,
t 
_______
*Tel.: +84-973633002
Email: lehuyhoangxd@gmail.com
18
, we say that {E , A} is irregular.
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L.H. Hoang / VNU Journal of Mathematics-Physics, Vol. 29, No. 1 (2013) 18-32
If {E , A} is regular, then a complex number t is called a (generalized finite) eigenvalue of {E , A}
if det( A  tE )  0 . The set of all eigenvalues is called the spectrum of the pencil {E , A} and denoted
by  ( E , A) .
If E is singular and {E , A} is regular, then we say that {E , A} has the eigenvalue  .
Suppose that the matrix pencil {E , A} is regular. Then the pairs can be transformed to Kronecker
canonical form i.e there exist nonsingular matrices P , Q such that
I
PEQ   r
O
O
J O 
,
PAQ

O I  ,
N 
nr 

(1.2)
where N is a nilpotent matrix of index k (see [1, 2]). If N is a zero matrix, then k = 1. Furthermore,
we may assume without loss of generality, that N and J are upper triangular. If {E , A} is regular,
then the nilpotency index of N in (1.2) is called the index of matrix pencil {E , A} and we write
index{E , A}  k .
In particular, a regular index-one system can be given by the form
E
E   11
O
E12 
 A11
,
A

A
O 
 21
A12 
,
A22 
1
A21 are square and of full rank (or invertible matrices).
where A22 and E11  E12 A22
Now, we give the definition of asymptotic stability of the solution of (1.1), see [2].
Definition 1.2. Suppose that {E, A} is regular. Let Q be a projector onto the subspace of consistent
initial conditions. Let P = I - Q. We say that the null solution of (1.1) is stable if for any ò  0 , there
exists   0 such that for an arbitrary vector y0  n satisfying || y0 ||  , the solution of the initial
value problem
 Ey( x)  Ay( x), x  [0, ),

 P( y(0)  y0 )  0
exists uniquely and the estimate || y ( x) || ò holds for all x  0 .
The null solution is said to be asymptotically stable if it is stable and lim || y ( x) || 0 for solution y
x 
of (1.1). If the null solution, of (1.1) is asymptotically stable, we say that system (1.1) is
asymptotically stable.
Theorem 1.1. (see [3]) The null solution, of system (1.1) is asymptotically stable if and only if the
eigenvalues of the matrix pencil {E, A} all have negative real part.
If the eigenvalues of the matrix pencil {E , A } all have negative real part, then the matrix pencil
{ E , A } is said to be stable.
Now, let us suppose that system (1.1) is asymptotically stable and consider the perturbed system
(E + BEAEC)y’ = (A + BAAAC)y,
(1.3)
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L.H. Hoang / VNU Journal of Mathematics-Physics, Vol. 29, No. 1 (2013) 18-32
where BA  K n p1 , BE  K n p2 , C  K qn are given matrices,  A  K p1 q ,  E  K p2 q ( K 
or K 
) are uncertain perturbations. BA  AC , BE  E C are called structured perturbations. Denote
 
   A  , we define the set of "bad" (destabilizing) perturbations
 E 

the matrix pencil {( E  BE  E C ), ( A  BA AC )}
VK    K ( p1  p2 )q
.
is either irregular or unstable


Definition 1.3. Let, the system (1.1) be asymptotically stable. The structured stability radius for
(1.1) is defined by rK  inf   |   VK  , where · is a matrix norm induced by vector norms in
K( p1  p2 )q .
Depending on K 
or K  , we talk about the complex or the real stability radius,
respectively. Obviously, we have the estimate r  r The problem of computing the stability radius
for ODEs was introduced in [4-7]. Later, the result was extended to DAEs in [8-12], The aim of this
paper is to compute a general formula of the complex stability radius. Moreover, a comparison of our
formula to previous results is given.
The outline of this paper is as follows. Firstly, the complex structured stability radius for (1.1) is
given. Furthermore, we show the details on stability and structure robustness of systems of index one.
Finaly, we show that the formula of the complex stability radius by Byers and Nichols in [8] can be
obtained as a special case of our result.
2. Main result
Theorem

2.1.
The
r  sup G1 (t ) G2 (t )
ti

complex
structured
stability
radius
for
(1.1)
is
given
1
,
(2.4)
where G1 (t )  C ( A  tE ) 1 BA , G2 (t )  tC ( A  tE ) 1 BE .
Proof. To prove this theorem, we use the technique that is same as in [9]. First, we prove that


.


.
r  sup G1 (t ) G2 (t )
Re t 0
To this end, we prove that
r  sup G1 (t ) G2 (t )
Re t 0
1
1
There are two cases in which {( E  BE  E C ),( A  BA AC )} is either unstable or irregular.
by
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L.H. Hoang / VNU Journal of Mathematics-Physics, Vol. 29, No. 1 (2013) 18-32
{( E  BE  E C ),( A  BA AC )} be unstable, then there exists
t  (( E  BE  E C ),( A  BA AC )) and Re(t )  0 . There exists x  0 satisfying,
The
first
case.
Let
( A  BA AC ) x  t ( E  BE  E C ) x

( A  tE ) x  ( BA AC  tBE  E C ) x
 
x  ( A  tE )1   BA tBE   A  Cx
 E 
 Cx  C ( A  tE )1  BA tBE  Cx.

Given u  Cx , we have, u  G1 (t ) G2 (t ) u.

Hence,   sup G1 (t ) G2 (t ) 
Re t 0
or

1
,

r  sup G1 (t ) G2 (t )
Re t 0

1
.
The second case. Let {( E  BE  E C ),( A  BA AC )} be irregular which means that for any t 
we have det(( A  BA AC )  t ( E  BE  E C ))  0 . Given t such that Re(t )  0 , then there exists
x  0 satisfying
t ( E  BA AC ) x  ( A  BE  E C ) x.
Similarly to the first case, we obtain

  sup G1 (t ) G2 (t )
Re t 0
It is clear that, in any case,

r  sup G1 (t ) G2 (t )
Ret 0

r  sup G1 (t ) G2 (t )
Re t 0
.
1
Now, we prove the inverse inequality


1
.
(2.5)

1
.
Indeed, for any ò  0 , there exists t 0 having Re(t0 )  0 such that
G1 (t0 )
G2 (t0 )
1

 sup G1 (t ) G2 (t ) 
Re t 0

1
 ò.
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L.H. Hoang / VNU Journal of Mathematics-Physics, Vol. 29, No. 1 (2013) 18-32
1
 A 
such that   G1 (t0 ) G2 (t0 )  . There

 E 
We construct a destabilizing perturbation   
exists a vector x 
p1  p2
, x  1 such that
G1 (t0 )
G2 (t0 )  x  G1 (t0 ) G2 (t0 )  . Invoking a
corollary of the Hahn-Banach theorem, there exists a functional y* 
q
, y *  1 such that
y* G1 (t0 ) G2 (t0 )  x  G1 (t0 ) G2 (t0 )  x  G1 (t0 ) G2 (t0 )  .
Let us define   G1 (t0 ) G2 (t0 ) 
1
xy* , then
 G1 (t0 ) G2 (t0 ) x  G1 (t0 ) G2 (t0 ) 
1
xy* G1 (t0 ) G2 (t0 )  x
 G1 (t0 ) G2 (t0 )
1
x G1 (t0 ) G2 (t0 )  .
We deduce   G1 (t0 ) G2 (t0 ) 
1
.
On the other hand, from the definition of  , we have   G1 (t0 ) G2 (t0 ) 
Thus,   G1 (t0 ) G2 (t0 ) 
1
1
.
.
We show that the perturbed system will be either unstable or irregular.
 G1 (t0 ) G2 (t0 )  x
 x
   C ( A  t0 E ) 1 BA t0C ( A  t0 E ) 1 BE  x 
 C ( A  t0 E ) 1   BA t0 BE  x

Multiplying
u  ( A  t0 E)
1
both
BA
sides
with
x
x.
( A  t0 E )1  BA t0 BE 
from
 ( A  t0 E ) 1   BA
t0 BE  Cu
the
left,
denoting
t0 BE  x , we obtain
u

( A  t0 E )u


( A  t0 E )u

 ( A  BA  AC )u 
 
t0 BE   A  Cu
 E 
 BA  ACu  t0 BE  E Cu
  BA
t0 ( E  BE  E C )u.
We have u  0 because x  0 . It is obtained that either t0   (( E  BE  E C ), ( A  BA AC )) or
(E  BE EC),( A  BA AC) is irregular.
Because Re(t0 )  0 then the perturbed system is either unstable or irregular. It is clear that
r    G1 (t0 ) G2 (t0 )
deduce
1

 sup G1 (t ) G2 (t ) 
Re t 0

1
 ò. Because ò is arbitrary, we
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L.H. Hoang / VNU Journal of Mathematics-Physics, Vol. 29, No. 1 (2013) 18-32

r  sup G1 (t ) G2 (t )
Re t 0
From (2.5) and (2.6) we have


1
r  sup G1 (t ) G2 (t )
Re t 0
.

(2.6)
1
.
To complete this proof, we note that G1 (t ) G2 (t ) is analytic in
principle, their least upper bound is attained in i

Re t 0
 


ti
, due to the maximum
(at a finite point or at infinity). Hence,
sup G1 (t ) G2 (t )
Finaly, we have r  sup G1 (t ) G2 (t )

\
1
 sup G1 (t ) G2 (t ) 
ti

1
.
1
,
where G1 (t )  C ( A  tE ) 1 BA , G2 (t )  tC ( A  tE ) 1 BE .
□
Remark 2.1.
If E is nonsingular matrix, then
lim G1 (t ) G2 (t )  lim  C ( A  tE ) 1 BA tC ( A  tE ) 1 BE 
t 
t 

 lim  C ( A  tE ) 1 BA
t 

1

C ( A  E ) 1 BE   .
t

If E is singular matrix, then
( J  tI r ) 1

O

 PB ,
1
k 1
G1 (t )  C ( A  tE ) BA  CQ 
A
i
O
I n  r   (tN )


i 1

1

1
O
( t J  I r )

 PBE , t  0.
G2 (t )  tC ( A  tE ) 1 BE  CQ 
k 1
i 

O
t ( I n  r   (tN ) )


i 1
Thus, the complex structured stability radius for (1.1) is strictly positive only if the
index{E , A}  0 . Moreover, for index{E , A}  1 , the radius is positive when the structured
perturbation of the leading term has influence only on the differential part. And from now on, we
consider the system having index one with suitable structured perturbations of the leading term.
Lemma 2.1. Let, {E, A} be regular.
If index{E , A}  1 then deg det( A  tE)  rankE .
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L.H. Hoang / VNU Journal of Mathematics-Physics, Vol. 29, No. 1 (2013) 18-32
If index{E , A}  1 then deg det( A  tE )  rankE .
Proof. Let pencil {E, A} have canonical form,
 J O  1
 I O  1
E  P 1  r
Q , A  P 1 
Q .

O N 
O I n  r 
• If k  0 then E  P 1 I nQ 1 , A  P 1 JQ 1 . It means that rankE  n
det( A  tE )  det( P 1Q 1 ) det( J  tI n ).
Because det( J  tI n ) is polynomial of degree n , so deg det( A  tE )  rankE .
 J  tI r
 O
• If k  1 , then N is null matrix, and A  tE  P 1 
O  1
Q .
I nr 
Hence, det( A  tE )  det( P 1Q 1 ) det( J  tI r ).
Because pencil {E , A} is regular, so det( A  tE )  0 , or det( J  tI r ) is polynomial of degree r .
On ther other hand, rankE  r , so deg det( A  tE)  rankE .
•
If
k  1,
 J  tI r
A  tE  P 1 
 O
N
then
is
in
Jordan
form,
N  diag  J1 ,
, Jl  ,
and
O  1
Q ,
MT 
where MT is upper triangular matrix which have diagonal elements equal to one. We deduce
det( A  tE )  det( P 1Q 1 ) det( J  tI r ).
Because pencil {E , A} is regular, so det( A  tE )  0 , or det( J  tI r ) is polynomial of degree
r . On ther other hand, rankE  r , so deg det( A  tE)  rankE .
□
To prove the following lemma, the technique is used in the proof of Theorem 2.1 can be applied.
Thus, we obtain the result.
Lemma 2.2. Let M 
nn
n p
, T
where H 
, 
p q
is nonsingular matrix. If
qn
d  inf   , M  H T is singular ,
are given matrices, then d  TM 1 H
1
.
Without loss of generality, the system having index one can be simplified as follows
 · 
 I O   y1   A11
O O   ·    A

 y
 21
 2
A12   y1 
,
A22   y2 
where A22 is invertible. From the Remark 2.1, for index{E , A}  1 , the radius is positive when the
structured perturbation of the leading term has influence only on the differential part. So that, we
consider the structured perturbations with
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L.H. Hoang / VNU Journal of Mathematics-Physics, Vol. 29, No. 1 (2013) 18-32
B 
B
BA   1  , BE    , C  C1 C2  .
O 
 B2 
 A 

 E 
Theorem 2.2. Let pencil {E , A} be regular and index one. For any   
( p1  p2 )q
satisfies   r , we have
index{( E  BE  E C ), ( A  BA  AC )}

index{E , A}.
deg det(( A  BA  AC )  t ( E  BE  E C ))  deg det( A  tE ) .
Proof. Consider the perturbed system
B E C2   y1   A11  B1 AC1

O   y2   A21  B2  AC1
 I  B E C1

O

The
perturbed
system
has
index
one
if
A12  B1 AC2   y1 
.
A22  B2  AC2   y2 
and
only
if
A22  B2  AC2 ,
and
1
I  B E C1  B E C2 ( A22  B2  AC2 ) ( A21  B2  AC1 ) are invertible.
Denote
R1  inf   A , A22  B2  AC2 is singular .
1
R1  C2 A22
B2
1
Using
Lemma
2.2,
we
.
Next, we prove inequality


1
sup G1 (t )
ti
 C2 A221B2
1
,
i.e
lim C ( A  tE )1 BA  C2 A221B2 .
t 
We have (see [13,14])
 A  tI
A  tE   11
 A21
A12 
I


A22 
O
A12 A221   A11  tI  A12 A221 A21

O
I 
O
O  I
  1
,
A22   A22 A21 I 
So
O  ( A11  tI  A12 A221 A21 )1
 I
( A  tE )1   1

O
  A22 A21 I  
O   I  A12 A221 

.
A221  O
I 
We deduce
1
1
1
1
C (tE  A) 1 BA  (C1  C2 A22
A21 )(tI  A11  A12 A22
A21 ) 1 ( B1  A12 A22
B2 )  C2 A22
B2 .
1
B2 .
We have lim C ( A  tE )1 BA  C2 A22
t 
Hence,
obtain
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L.H. Hoang / VNU Journal of Mathematics-Physics, Vol. 29, No. 1 (2013) 18-32


1
r  sup G1 (t )
ti
 R1.
This means that A22  B2  AC2 is nonsingular if   r .
On the other hand, we have
 I  B E C1
A  B  C
2 A 1
 21
B E C2   I B E C2 ( A22  B2  AC2 )1 
= 

A22  B2  AC2  O
I

 I  B E C1  B E C2 ( A22  B2  AC2 )1 ( A21  B2  AC1 )

O


O
A22  B2  AC2 

I
O

( A  B  C )1 ( A  B  C ) I  .
2 A 2
21
2 A 1
 22

It is easy to see that I  B E C1  B E C2 ( A22  B2  AC2 ) 1 ( A21  B2  AC1 ) is singular if and only if
 I  B E C1
A  B  C
2 A 1
 21
B E C2    I
 or
A22  B2  AC2    A21
O  O

A22   B2

B   A 
C1 C2   is singular.




O   E 

Denote
   I
R2  inf {  A  . 
  E   A21
O  O

A22   B2
B   A 
C C2  singular. }
O    E  1
Using Lemma 2.2, we have
1
R2  C2 A22
B2
1
(C1  C2 A22
A21 ) B 
1
.
Now, we need to prove
1
1
lim C ( A  tE )1 BA tC ( A  tE )1 BE  C2 A22
B2 (C1  C2 A22
A21 ) B .
t 

From ( A  tE )1  
I
1
  A22 A21
O  ( A11  tI  A12 A221 A21 )1

I  
O
O   I  A12 A221 

.
A221  O
I 
We deduce
1
1
1
C ( A  tE ) 1 BA  (C1  C2 A22
A21 )(tI  A11  A12 A22
A21 ) 1 ( B1  A12 A22
B2 )  C2 ( A22 ) 1 B2 ,
1
1
tC ( A  tE ) 1 BE  t (C1  C2 A22
A21 )( A11  tI  A12 A22
A21 ) 1 B.
We obtain
1
1
lim C ( A  tE )1 BA tC ( A  tE )1 BE  C2 A22
B2 (C1  C2 A22
A21 ) B .
t 

Thus, r  sup G1 (t ) G2 (t ) 
ti

1
 R2 .
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L.H. Hoang / VNU Journal of Mathematics-Physics, Vol. 29, No. 1 (2013) 18-32
In conclusion, if
  r , then index{( E  BE  E C ),( A  BA AC )}  1 . Using Lemma 1.1, we
have
deg det(( A  BA AC)  t ( E  BE  EC))  rank( E  BE  EC)  rankE.
□
The algebraic structure of an index one DAEs is characterized by the index and the number of the
finite eigenvalues of the pencil. We denote


V   


( p1  p2 )q
the pencil {( E  BE  E C ), ( A  BA  AC )}

is either irregular or unstable,
.

or its algebraic structure changes

The algebraic structure of the pencil {( E  BE  E C ),( A  BA AC )} changes if its index, or the
number of finite singular values, or both change . Now, for a DAEs of index one, the complex
structured stability radius can be redefined as following r  inf
  |   V .
By the Theorems 2.1 and 2.2, the following result immediately follows.
Theorem 2.3. Using the same assumption as in the Theorem. (2.1) and (2.2), we have

r  r  sup G1 (t ) G2 (t )
ti

1
,
(2.7)
where G1 (t )  C ( A  tE ) 1 BA , G2 (t )  tC ( A  tE ) 1 BE .
Example 2.1. For sake of simplicity, we use the maximum norm as the matrix norm.
Consider
 1 0 0 
 1 0 0


A   0 1 0  , E  0 0 1 .
 0 0 1
0 0 0 
 1 0 0
It is easy to verify that index{E , A}  2 ,  ( E, A)  1 . With C  I , BA   2 0 0  ,


 0 0 0 
1 0 0 
BE  0 0 0  , we have
0 0 0 
 1

 t

 t  1 0 0
 t  1 0 0


G1 (t )  C ( A  tE ) 1 BA   2
0 0  , G2 (t )  tC ( A  tE ) 1 BE   0 0 0  .
 0
 0
0 0
0 0








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L.H. Hoang / VNU Journal of Mathematics-Physics, Vol. 29, No. 1 (2013) 18-32
 1
t 
 sup max 
 2,
  3.
t 1 
ti
ti
 t 1
1
The least upper bound is attained at t  0 . So, r  , we construct the destabilizing
3
1
1 1


 3 3 0
0 0 3 




perturbations
We
deduce
 A   0 0 0 ,
 E  0 0 0  .
 0 0 0
0 0 0 








 (( E  BE EC),( A  BA AC))  0 which means that the system is unstable.
Hence, sup G1 (t ) G2 (t ) 

In the case which the least upper bound attained at  , we consider arbitrary sequence t n
proceeding to
  nA 
. The
n 
 E 
 . Then, for each t n , we construct the destabilizing perturbations 
  nA 
sequence  n  which have limits being the stability radius. But the perturbations that have norm
 E 
equal to the stability radius only change the algebraic structured. To verify that, we consider following
Example 2.2. Consider
 1

1
 1 2 

A 2
, E
.


2 4 

 1 0 
It is easy to verify that
 1 0
BE  
,
 2 0
we
 1
index{E , A}  1 ,  ( E , A)    . With C  I  BA
 2
 4t
 2t  1
G1 (t )  C ( A  tE ) 1 BA  
have
 1

 2t

0

G2 (t )  tC ( A  tE ) BE  2t  1
.


0
 0
1
Thus, sup G1 (t ) G2 (t ) 
ti

 2t
4t
3
 sup max 
,
 1,   3,
2
ti
 2t+1 2t  1
and

1
,
1
2 
29
L.H. Hoang / VNU Journal of Mathematics-Physics, Vol. 29, No. 1 (2013) 18-32
the least upper bound attained at t   . Considering the sequence tn  in , for each t n , we construct
1
1



1
1 
n
2
2

the destabilizing perturbations  A 
3  2
3  2 ,  nE  O . It is easy to see that
4n
4n 



0
0


n
 A 
n
n
 n  is a destabilizing perturbations i.e the pencil {( E  BE  E C ), ( A  BA  AC )} is unstable. On
 E 
1 1
 A 
n
the other hand, we have  A   A   3 3  ,  nE   E  O when n   . But   only


 E 
0 0
changes the algebraic structured because  (( E  BE  nE C ), ( A  BA  nAC ))   . This can be explained
that the least upper bound is not attained at a finite point so the greatest lower bound of the
destabilizing perturbations is not attained. Althought, the limits of the sequence of perturbations is
existence. Now, we will show that our result can be used to obtain the result of Ralph Byers, N.K.
Nichols in [8]. Firstly, we can repeat some results from [8]
Definition 2.1. ( see 8] ) The radius of stability of the stable regular pencil { A, E} is given by
{( E   E ), ( A   A)} is either unstable or irregular 

or algebraic structure changes


where ||·||F denotes the Frobenius norm.

 ( A, E )  inf || [ A |  E ] ||F
Lemma 2.3. ( see [8] ) If {E , A} is regular and of index less than or equal to one, then there exists
an orthogonal matrix P and a permutation matrix Q such that
E
PEQ   11
O
E12 
A
, PAQ   11

O
 A21
A12 
,
A22 
  A11
where rank ( E11 , E12 )  rankE  k , rankA22  n  k and  ( A, E )    
  A21
A12   E11
,
A22   O
E12  
.
O  
 A  A12 
 E11  E12 
, P AQ   11
.

O 
 O
 A21  A22 
Furthermore, P EQ  
 2  2  1,
define
for
the
matrix
function
 ,
,
 A  A12  i E11 i E12 
H ( ,  )   11

.
A22   O
O 
 A21
Theorem 2.4. ( see [8] ) If {E , A} is stable, regular, and of index less than or equal to one, then
We
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L.H. Hoang / VNU Journal of Mathematics-Physics, Vol. 29, No. 1 (2013) 18-32
 ( A, E )  inf
 min H ( ,  ), (2.8)
,
 2  2 1
To show (2.7) implies (2.8), we consider the pencil with index one in the form
E12 
A
, A   11

O
 A21
E
E   11
O
A12 
,
A22 
and
 I O
 I O
 I O
BA  
, BE  
, C


.
O I 
O O 
O I 
Theorem 2.5. If the matrix pencil {E , A} is regular, stable, and has index one, and the matrix
norm is Euclidean norm, then the complex structured stability radius is
r  inf  min H ( ,  ) ,
 ,
 2  2 1
Proof. Using the formula of the complex structured stability radius from Theorem 2.1, we have

 A  tE11

r  sup  11
ti
 A21


1
A12  tE12    I
A22   O
1
O
I
tI O  

 .

O O 

We shall consider two cases.
• The first case. If the least upper bound is attained at t 0 , then we set t0  i
  0,
and
    1 . Thus,
2
2


r 




 A11  i  E11

A21

A22
On the other hand,


 A11  i  E11

A21

2
1
  
A12  i E12    I



 A i E
 max   11  11
 
A21

A22
O
 
  O
I


i I O



O
O
1
A12  i

 
E
I
 12  
A22
 
  O
O
I
1
1

 
A12  i E12    I

i
2


I O

O

O
 
  O

, where  ,   ,

O
I

 
i I O 
.

 
O
O 
2
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L.H. Hoang / VNU Journal of Mathematics-Physics, Vol. 29, No. 1 (2013) 18-32
 I
 O

 
 i I
 
 O
O

 I   
  A11  i E11


O   
A21


O 

1


 
A12  i E12   

 
*

  

A22
where max 
· is the largest eigenvalue. Hence,


 A11  i  E11

A21

A22


  A11  i E11
 max 

 
A21



 A i E
 max   11  11
 
A21


  I
 max  

  O

2
1
  
A12  i E12    I

O
 
  O


i I O


I
O

O
2
1
1 * 

 1
 




A12  i E12   2 I O   A  i E A  i E

11
11
12
12

  






 
  O
A22
 
I   

A21

  

A22
1
A12  i



  

1



 1

E12   I O   1 I O    A11  i  E11 A12  i  E12  


 




 

 

*
  O
A22
I   

IO
A21
A22
1
   


A12  i E12      I O   A11  i E11


     O I  
A22
A
   
21




O   A11  i E11

I  
A21

1

 A11  i E11  A12  i E12 
 max 

A21
A22



1

   

A12  i E12  


 
*

   

A22
*
  A11  i E11  A12  i E12  1 

 

A21
A22
 



.

We deduce


r   max ( H ( ,  ))1
1
where  max 
· is largest singular value.
• The second case. If the least upper bound is attained at  , then we set t  i
| t |  where |  | 0 . Similarary with the previous case, we also have
 A  tE11
lim  11
t 
 A21
1
A12  tE12    I
A22   O
 A  iE11  A12  iE12 
 lim  11

 0
A22
 A21

Hence,
O tI O 
 I O O 
1
 iE iE12 
  11
A22 
 A21
1


  max ( H (0,1))1 .
1

, 
, and
32
L.H. Hoang / VNU Journal of Mathematics-Physics, Vol. 29, No. 1 (2013) 18-32


r   max ( H (0,1))1
1
where  max 
· is largest singular value.


From the both cases, we deduce r   sup  max ( H ( ,  )) 1
 2 ,2
  1

i.e






1
r  inf  min H ( ,  )
 ,
 2  2 1
where  min 
· is smallest singular value.
Remark 2.2. Using the fact that a rank-one destabilizing perturbation can be constructed, an
alternative proof for Theorem. 2.5 can be given. The Frobenius norm gives a upper bound for the
Euclidean, norm. Note that, for a rank one matrix, the Euclidean, and the Frobenius norms are equal.
Then, we can. show that, the formula of the complex stability radius given, in [8] and in Theorem 2.3
are the same.
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