Distance problems for dissipativ e Hamiltonian systems and
related matrix p olynomials
C. Mehl ‡∗ V. Mehrmann ‡ ∗ M. W o jt ylak † §
Jan uary 24, 2020
De dic ate d to Paul V an Do or en on the o c c asion of his 70th birthday
Abstract
W e study the c haracterization of sev eral distance problems for linear differen tial-
algebraic systems with dissipativ e Hamiltonian structure. Since all mo dels are only ap-
pro ximations of realit y and data are alw ays inaccurate, it is an imp ortan t question whether
a giv en mo del is close to a ’bad’ mo del that could b e considered as ill-p osed or singular.
This is usually done b y computing a distance to the nearest mo del with suc h prop erties.
W e will discuss the distance to singularit y and the distance to the nearest high index
problem for dissipativ e Hamiltonian systems. While for general unstructured differen tial-
algebraic systems the c haracterization of these distances are partially op en problems, w e
will sho w that for dissipativ e Hamiltonian systems and related matrix p olynomials there
exist explicit c haracterizations that can b e implemen ted n umerically .
Keyw ords. distance to singularit y , distance to high index problem, distance to instabil-
it y , dissipativ e Hamiltonian system, differen tial-algebraic system, matrix p encil, Kroneck er
canonical form,
AMS sub ject classification 2014. 15A18, 15A21, 15A22
1 In tro duction
W e study sev eral distance problems for linear systems of differ ential-algebr aic e quations
(D AEs) of the form
E ˙ x = ( J − R ) Qx, (1)
with constan t co efficien t matrices E , Q, J, R ∈ R n,n and a differen tiable state function x :
R → R n , see also [3, 16, 22, 30, 31, 32, 33, 34] for definitions and a detailed analysis of
suc h systems in different generalit y and their relation to the more general p ort-Hamiltonian
‡ Institut f ¨ ur Mathematik, MA 4-5, TU Berlin, Str. des 17. Juni 136, D-10623 Berlin, FR G.
{ mehl,mehrmann } @math.tu-berlin.de .
† Inst ytut Matemat yki, Wydzia l Matemat yki i Informat yki, Uniw ersytet Jagiello ´ nski, Krak´ ow, ul.
Lo jasiewicza 6, 30-348 Krak´ ow, P oland [email protected] .
§ Supp orted b y the Alexander v on Hum b oldt F oundation.
∗ P artially supp orted b y Deutsche F orschungsgemeinschaft through the Excellence Cluster Ma th + in Berlin
and Priorit y Program 1984 ’Hybride und m ultimo dale Energiesysteme: System theoretische Methoden f ¨ ur die
T ransformation und den Betrieb k omplexer Netze’.
1

systems . A system of the ab o v e form is called linear time-in v arian t dissip ative Hamiltonian
(dH) differ ential-algebr aic e quation (dHD AE) system if
E > Q ≥ 0 , J = − J > , R = R > ≥ 0 , (2)
where A > denotes the transp ose of a matrix A and for a symmetric matrix A = A > b y
A > 0 ( A ≥ 0) w e denote that A is p ositiv e definite (p ositiv e semidefinite). Suc h dHDAE
systems generalize linear time-in v arian t ordinary dissip ative Hamiltonian systems (the case
where E = I ) is the iden tit y) and linear Hamiltonian systems , the case that E = I and
R = 0. The asso ciated quadratic Hamiltonian is giv en b y H ( x ) = 1
2 x > E > Qx and satisfies the
dissip ation ine quality H  x ( t 1 )  − H  x ( t 0 )  ≤ 0 for t 1 ≥ t 0 . In many applications the matrix
Q can b e c hosen to b e the iden tit y matrix, i.e., Q = I , s ee [3, 16, 31, 32, 35], and this is the
case that w e study in this pap er. In Section 6.3 w e pro vide an analysis ho w the general case
(1) can b e transformed to this situation.
The system prop erties of (1) (with Q = I ) can b e analyzed b y in v estigating the corre-
sp onding dH matrix p encil
L ( λ ) := λE − ( J − R ) . (3)
In our analysis w e focus on systems with real co efficien ts. Some of our results can also easily
b e extended to the case of complex co efficien ts, but in some o ccasions w e mak e explicit or
implicit use of the fact that sk ew-symmetric matrices ha v e a zero diagonal whic h is not true
for sk ew-Hermitian matrices.
In man y practical cases, see, e.g., [3, 20], the underlying system is of second order form
M ¨ x − ( G − D ) ˙ x + K x = 0 with the underlying quadratic matrix p olynomial
P ( λ ) := λ 2 M − λ ( G − D ) + K (4)
where M , G, D , K ∈ R n,n satisfy M = M > , D = D > , K = K > ≥ 0 and G = − G > . As w e will
see b elo w, this can b e view ed as a generalization of the dH structure to second order systems.
The second order case can b e easily rewritten in first order dH form but w e will treat the
problem directly in second order, and w e will also discuss appropriate higher degree matrix
p olynomials P ( λ ) with an analogous structure.
Linear time in v arian t systems with the describ ed structure are v ery common in all areas
of science and engineering [3, 31, 35] and t ypically arise via linearization arround a stationary
solution. Ho w ev er, since all mathematical mo dels of ph ysical systems are usually only ap-
pro ximations of realit y and data are t ypically inaccurate, it is an imp ortan t question whether
a giv en mo del is close to a mo del with ’bad prop erties’ suc h as an ill-p osed mo del without
or with non-unique solution. T o answ er suc h questions for dH and p ort-Hamiltonian systems
has b een an imp ortan t researc h topic in recen t y ears, see, e.g., [1, 2, 16, 17, 18, 27, 28].
T o classify whether a mo del is close to a ’bad mo del’, one usually computes the distance
to the nearest mo del with the ’bad’ prop ert y . In this pap er w e will discuss the distance to the
set of singular matrix p olynomials, i.e., those with a determinant that is iden tically zero, and
the distance to the nearest high-index problem, i.e., a problem with Jordan blo cks associated
to the eigen v alue ∞ of size bigger than one.
While for general unstructured D AE systems the characterization of these t w o distances is
v ery difficult and partially op en [4, 5, 7, 21, 29], the picture c hanges if one considers structur e d
distanc es , i.e., distances within the set of linear constant coefficient dHD AE systems. In this
pap er, w e will mak e use of previous results from [16, 30] to deriv e explicit characterizations
for computing these distances in terms of n ull-spaces of sev eral matrices.
2

W e use the follo wing notation. By k X k F w e denote the F rob enius norm of a (p ossibly
rectangular) matrix X , w e extend this norm to matrix p olynomials P ( λ ) = P k
j =0 λ j X j b y
setting k P ( λ ) k F = k [ X 0 , . . . , X k ] k F . By λ min ( X ) w e denote the smallest eigen v alue of a
p ositiv e semidefinite matrix X .
The pap er is organized as follo ws. In Section 2 w e recall a few basic results ab out linear
time-in v arian t dH systems. In Section 3 w e presen t the different distances and state the main
results for first order systems. Instead of immediately presen ting the corresp onding pro ofs, w e
first consider related distance problems for a more general p olynomial structure in Section 4.
These distance c haracterizations are then sp ecialized in Section 5 to pro ve the main results for
the first order case. In Section 6 w e consider corresp onding distances for analogous quadratic
matrix p olynomials and also sho w ho w different r epresen tations of the first order case can b e
related.
2 Preliminaries
W e will mak e use of the Kronec k er canonical form of a matrix p encil [15]. Let us denote b y
J n ( λ 0 ) the standard upp er triangular Jordan blo c k of size n × n asso ciated with the eigen v alue
λ 0 and let L n denote the standard righ t Kronec ker block of size n × ( n + 1), i.e.,
L n = λ 


1 0
. . . . . .
1 0


 − 


0 1
. . . . . .
0 1


 and J n ( λ 0 ) = 





λ 0 1
. . . . . .
. . . 1
λ 0






.
Theorem 1 (Kronec k er canonical form) L et E , A ∈ C n,m . Then ther e exist nonsingular
matric es S ∈ C n,n and T ∈ C m,m such that
S ( λE − A ) T = diag( L  1 ,..., L  p , L >
η 1 ,..., L >
η q , J λ 1
ρ 1 ,..., J λ r
ρ r , N σ 1 ,..., N σ s ) , (5)
wher e p, q , r , s,  1 , . . . ,  p , η 1 , . . . , η q , ρ 1 , . . . , ρ r , σ 1 , . . . , σ s ∈ N and λ 1 , . . . , λ r ∈ C , as wel l as
J λ i
ρ i = I ρ i − J ρ i ( λ i ) for i = 1 , . . . , r and N σ j = J σ j (0) − I σ j for j = 1 , . . . , s . This form is
unique up to p ermutation of the blo cks.
F or real matrices (the case w e discuss), a real v ersion of the Kronec k er canonical form is
obtained under real transformation matrices S, T . In this case the blo c ks J λ j
ρ j with λ j ∈ C \ R
ha ve to be replaced with corresp onding blo c ks in real Jordan canonical form asso ciated to
the corresp onding pair of conjugate complex eigen v alues, but the other blo c ks ha v e the same
structure as in the complex case. An eigen v alue is called semisimple if the largest asso ciated
Jordan blo c k has size one.
The sizes η j and  i of the rectangular blo c ks are called the left and right minimal indic es
of λE − A , resp ectiv ely . The matrix p encil λE − A , E , A ∈ C n,m is called r e gular if n = m
and det( λ 0 E − A ) 6 = 0 for some λ 0 ∈ C , otherwise it is called singular . A p encil is singular
if and only if it has blo c ks of at least one of the t yp es L ε j or L >
η j in the Kronec ker canonical
form.
The v alues λ 1 , . . . , λ r ∈ C are called the finite eigen v alues of λE − A . If s > 0, then
λ 0 = ∞ is said to b e an eigen v alue of λE − A . (Equiv alen tly , zero is then an eigen v alue of
the rev ersal λA − E of the p encil λE − A .) The sum of all sizes of blo c ks that are asso ciated
3

with a fixed eigen v alue λ 0 ∈ C ∪ {∞} is called the algebr aic multiplicity of λ 0 . The size of
the largest blo c k N σ j is called the index ν of the p encil λE − A , where, b y con v en tion, ν = 0
if E is in v ertible. The p encil is called stable if it is regular and if all eigen v alues are in the
closed left half plane, and the ones lying on the imaginary axis (including infinit y) ha v e the
largest asso ciated blo c k of size at most one. Otherwise the p encil is called unstable .
The follo wing result was sho wn in [30]. W e state the result in full generalit y , but clearly
all statemen ts also hold for the sp ecial case that E , Q, J, R are real and that Q = I whic h is
the case considered in this pap er.
Theorem 2 L et E , Q ∈ C n,m satisfy E H Q = Q H E ≥ 0 and let al l left minimal indic es of
λE − Q b e e qual to zer o (if ther e ar e any). F urthermor e, let J, R ∈ R m,m b e such that we have
J = − J H , R ≥ 0 . Then the fol lowing statements hold for the p encil L ( λ ) = λE − ( J − R ) Q .
(i) If λ 0 ∈ C is an eigenvalue of L ( λ ) then Re( λ 0 ) ≤ 0 .
(ii) If ω ∈ R \ { 0 } and λ 0 = iω is an eigenvalue of L ( λ ) , then λ 0 is semisimple. Mor e over, if
the c olumns of V ∈ C m,k form a b asis of a r e gular deflating subsp ac e of L ( λ ) asso ciate d
with λ 0 , then RQV = 0 .
If, additional ly, Q is nonsingular then the pr evious statement holds for λ 0 = 0 as wel l.
If Q is singular then λ 0 = 0 ne e d not b e semisimple, but if L ( λ ) is r e gular, then Jor dan
blo cks asso ciate d with λ 0 = 0 have size at most two.
(iii) The index of L ( λ ) is at most two.
(iv) A l l right minimal indic es of L ( λ ) ar e at most one (if ther e ar e any).
(v) If in addition λE − Q is r e gular, then al l left minimal indic es of L ( λ ) ar e zer o (if ther e
ar e any).
Pr o of . F or the pro of see [30]. The additional statemen t in (ii) on the eigen v alue λ 0 = 0
w as not presen ted in [30], but it follo ws in a straigh tforw ard manner from [30, Theorem 6.1]
and the pro of of [30, Corollary 6.2].
Theorem 2 illustrates that the sp ecial structure of dH systems imp oses man y restrictions
in the sp ectral data and this has also an adv an tage when determining the distances to the
nearest ’bad’ problem. In particular, Theorem 2 implies that the distance to instabilit y and
the distance to higher index coincide for a p encil L ( λ ) with Q nonsingular.
The follo wing well-kno wn lemma, see [6, 23] (also stated for the general complex case),
will b e needed in order to mak e statemen ts ab out the index of a matrix p encil in sp ecial
situations.
Lemma 3 L et E , A ∈ C n,n b e matric es of the form
E =  E 11 0
0 0  and A =  A 11 A 12
A 21 A 22  ,
wher e E 11 is invertible.
(i) If A 22 is invertible, then the p encil λE − A is r e gular and has index one;
(ii) if A 22 is singular, then the p encil λE − A is singular or has an index gr e ater than or
e qual to two.
4

3 Problem statemen t and main results for dHD AE systems
W e are in terested in the follo wing distance problems for matrix p encils L ( λ ) of the form (3)
under p erturbations that preserv e the sp ecial structure of the p encil.
Definition 4 L et L denote the class of squar e n × n r e al matrix p encils of the form (3) . Then
1) the structured distance to singularity is define d as
d L
sing  L ( λ )) := inf  
 ∆ L ( λ ) 
 F   L ( λ )+∆ L ( λ ) ∈ L and is singular  ; (6)
2) the structured distance to the nearest high-index problem is define d as
d L
hi  L ( λ )) := inf  
 ∆ L ( λ ) 
 F   L ( λ )+∆ L ( λ ) ∈ L and is of index ≥ 2  ; (7)
3) the structured distance to instability is define d as
d L
inst  L ( λ )  := inf  
 ∆ L ( λ ) 
 F   L ( λ )+∆ L ( λ ) ∈ L and is unstable  . (8)
Note that all defined distances are meaningful, as for each matrix X ∈ R n,n the decomp osition
in to a sum X = X 1 + X 2 of a skew-symmetric matrix X 1 = 1
2 ( X − X > ) and symmetric matrix
X 2 = 1
2 ( X + X > ) is unique. F urthermore, w e ha v e k X k 2
F = k X 1 k 2
F + k X 2 k 2
F = 
 [ X 1 , X 2 ] 
 2
F
due to the trace of X >
1 X 2 b eing zero. Th us, the constrain t L ( λ ) + ∆ L ( λ ) ∈ L in (6)–(8) is
the same as writing
∆ L ( λ ) = λ ∆ E − (∆ J − ∆ R ) ,
with ∆ J = − ∆ >
J and E + ∆ E , R + ∆ R ≥ 0, and w e ha v e 
 [∆ J , ∆ R , ∆ E ] 
 F = 
 [∆ L ( λ )] 
 F .
The p ositivit y conditions for E + ∆ E , R + ∆ R are c rucial. Examples presen ted in Section
5.2 sho w that they can neither b e omitted nor simplified to E + ∆ E , R + ∆ R b eing merely
symmetric.
Theorem 5 L et L ( λ ) = λE − ( J − R ) ∈ L . Then the fol lowing statements hold.
(i) The p encil L ( λ ) is singular if and only if k er J ∩ k er E ∩ k er R 6 = { 0 } . In that c ase ther e
exists an ortho gonal tr ansformation matrix U ∈ R n,n such that
U > E U =  E 11 0
0 0  , U > J U =  J 11 0
0 0  , U > RU =  R 11 0
0 0  ,
wher e the p encil λE 11 − ( J 11 − R 11 ) is r e gular and has the size ( n − r ) × ( n − r ) with
r = dim k er( E − J + R ) > 0 . In p articular, al l right and left minimal indic es of L ( λ ) in
its Kr one cker c anonic al form ar e zer o.
(ii) The index of L ( λ ) is at most two. F urthermor e, the fol lowing statements ar e e quivalent.
(a) F or any ε > 0 ther e exists a p encil λ e
E − ( e
J − e
R ) with e
E , e
R ≥ 0 , and e
J = − e
J > which
is r e gular and of index two such that

  E − e
E , J − e
J , R − e
R  
 F = 
  E − e
E , ( J − R ) − ( e
J − e
R )  
 F ≤ ε, (9)
i.e., L ( λ ) is in the closur e of the set of r e gular dH p encils of index two.
5

(b) ker E ∩ k er R 6 = { 0 } .
T o construct the p erturbations where the distance to singularit y ia ac hiev ed, we use the
follo wing ansatz. F or a matrix Y ∈ R n,n and a v ector u ∈ R n with k u k 2 = 1 w e define the
matrix
∆ u
Y = − uu > Y − Y uu > + uu > Y uu > , (10)
that will b e used at sev eral o ccasions during the pap er. Then w e obtain the follo wing char-
acterization of the distance to singularit y .
Theorem 6 L et λE − ( J − R ) ∈ L . Then the fol lowing statements hold.
(i) The distanc e to singularity (6) is attaine d with a p erturb ation ∆ E = ∆ u
E , ∆ J = ∆ u
J ,
and ∆ R = ∆ u
R as in (10) for some u ∈ R n with k u k 2 = 1 . The distanc e is given by
d L
sing  λE − ( J − R ) 
= min
u ∈ R n
k u k =1 q 2 k J u k 2 + 2 
 ( I − uu > ) E u 
 2 + ( u > E u ) 2 + 2 
 ( I − uu > ) Ru 
 2 + ( u > Ru ) 2
and is b ounde d as
p λ min ( − J 2 + R 2 + E 2 ) ≤ d L
sing  λE − ( J − R )  ≤ p 2 · λ min ( − J 2 + R 2 + E 2 ) . (11)
(ii) The distanc e to higher index (7) and the distanc e to instability (8) c oincide and satisfy
d L
hi  λE − ( J − R )  = d L
inst  λE − ( J − R ) 
= min
u ∈ R n
k u k =1 q 2 
 ( I − uu > ) E u 
 2 + ( u > E u ) 2 + 2 
 ( I − uu > ) Ru 
 2 + ( u > Ru ) 2
and ar e b ounde d as
p λ min ( E 2 + R 2 ) ≤ d L
hi  λE − ( J − R )  = d L
inst  λE − ( J − R )  ≤ p 2 · λ min ( E 2 + R 2 ) .
The pro ofs of Theorems 5 and 6 are giv en in Section 5.1, where they are obtained as simple
consequences of a general theory dev elop ed in Section 4.2 for matrix p olynomials with a
sp ecial symmetry structure. Before w e giv e the pro ofs, w e will first consider a more general
minimization problem in the next section.
4 General distance problems
In this section, w e presen t a solution to a quite general minimization problem. This will allo w
us to solv e the distance problems for dH p encils in tro duced in Section 3 as w ell as analogous
problems for structured matrix p olynomials with a dH lik e structure in a unified manner.
Theorem 5 states that b oth the distance to singularit y as w ell as to higher index for a
dH p encil as in (3) can b e expressed via the existence of a common k ernel of t w o or three
structured matrices, so that b oth problems can b e rein terpreted as a distance problem to the
common k ernel of matrices with symmetry and p ositivit y structures. This concept will no w
b e extended to more than three matrices.
6

4.1 Distance to the common k ernel of a tuple of structured matrices
Definition 7 Let S n
` denote the follo wing set of ( ` + 2)-tuples of n × n real matrices
S n
` :=  ( J, X 0 , . . . , X ` ) ∈ ( R n,n ) ` +2   J > = − J, X i = X T
i ≥ 0 , i = 1 , . . . , `  ,
where ` ≥ 0 and n ≥ 1 are fixed. F or a giv en tuple ( J, X 0 , . . . , X ` ) ∈ S n
` w e define the
structur e d distanc e to the c ommon kernel d S n
`
k er ( J, X 0 , . . . , X ` ) as
inf  
 [∆ J , ∆ X 0 ,..., ∆ X ` ] 
 F    ( J + ∆ J , X 0 + ∆ X 0 , . . . , X ` + ∆ X ` ) ∈ S n
` ,
k er( J + ∆ J ) ∩ T `
i =0 k er( X i + ∆ X i ) 6 = { 0 }  . (12)
In the follo wing, we often drop the dependence on ` and n in the notation for s implicit y , th us
writing d S
k er ( J, X 0 , . . . , X ` ).
Observ e that in determining d S
k er ( J, X 0 , . . . , X ` ) w e measure the distance to a closed set.
Lemma 8 The set of al l ( J, X 0 , . . . , X ` ) ∈ S n
` satisfying k er J ∩ ker X 0 ∩ · · · ∩ k er X ` 6 = { 0 }
is a close d subset in S n
` .
Pr o of . The pro of follo ws b y considering sequences of tuples ( J ( m ) , X ( m )
0 , . . . , X ( m )
` ) and a
con vergen t subsequence of a sequence of unit v ectors u m satisfying
J ( m ) u m = X ( m )
i u m = 0 , i = 1 , . . . `.
Before w e present the solution of the minimization problem, w e first dev elop equiv alen t con-
ditions for J, X 0 , . . . , X ` to ha v e a non trivial common k ernel.
Prop osition 9 L et ( J, X 0 , . . . , X ` ) ∈ S n
` . Then
k er J ∩ ker X 0 ∩ . . . k er X ` = k er( J > J + X 2
0 + · · · + X 2
` ) = k er( − J + X 0 + · · · + X ` ) . (13)
F urthermor e, ther e exists an ortho gonal matrix U ∈ R n,n such that
U > J U =  e
J 0
0 0  , U > X i U =  e
X i 0
0 0  , i = 0 , . . . , ` (14)
with some e
J , e
X 0 ,..., e
X ` ∈ R n − r ,n − r , wher e r = dim k er( − J + X 0 + · · · + X ` ) ≥ 0 and wher e
the matrix − e
J + e
X 0 + · · · + e
X ` is invertible.
Pr o of . The inclusion ker J ∩ k er X 0 ∩ . . . k er X ` ⊆ k er( J > J + X 2
0 + · · · + X 2
` ) is trivial. T o
pro ve the con v erse, let x ∈ k er( J T J + X 2
0 + · · · + X 2
` ) b e nonzero. Since eac h summand is
p ositiv e semidefinite, w e obtain X 2
0 x = · · · = X 2
` x = 0 and J 2 x = − J T J x = 0. Noting that
k er Y = k er Y 2 holds for an y symmetric or sk ew-symmetric matrix finishes the pro of.
The inclusion k er J ∩ k er X 0 ∩ . . . k er X ` ⊆ k er( − J + X 0 + · · · + X ` ) is again trivial. T o prov e
the con verse let x ∈ k er( − J + X 0 + · · · + X ` ) b e nonzero. Since x > J x = 0, w e obtain that
x > X 0 x + · · · + x > X ` x = 0 and since eac h of the matrices X 0 , . . . , X ` is p ositiv e semidefinite,
w e obtain X 0 x = · · · = X ` x = 0, whic h then implies J x = 0 as w ell.
T o pro v e the last assertion, let U b e an orthogonal matrix with last r columns spanning
the k ernel of − J + X 0 + · · · + X ` . Note that if u is one of those r last columns of U then (13)
implies that u > J = 0 and u > X i = 0 for i = 0 , . . . , ` , whic h sho ws the form ula (14).
7

Remark 10 W e highligh t that the nonegativit y assumption for the matrices X i is crucial for
the t w o non trivial inclusions in Prop osition 9. F or example, consider
J =  0 1
− 1 0  and X 0 =  1 0
0 − 1  .
Then J − X 0 is singular while the in tersection of the k ernels of J and X 0 is trivial.
Also note that while an arbitrarily large n um b er of symmetric p ositiv e semidefinite ma-
trices X 0 , . . . , X ` can b e considered, the results from Prop osition 9 are no longer true if a
second sk ew-symmetric matrix is in volv ed. F or example, consider the matrices
J 1 = 

0 1 0
− 100
0 0 0 
 and J 2 = 

0 0 1
0 0 0
− 100 

Then J 1 + J 2 is singular (in fact, ev en the p encil λJ 1 + J 2 is singular), but J 1 and J 2 do not
ha ve a common k ernel.
Giv en ( J, X 0 , . . . , X ` ) ∈ S n
` as in Prop osition 9, w e aim to c haracterize all p erturbations
that pro duce a non trivial common k ernel of the matrices J, X 0 , . . . , X ` while preserving their
individual structures. F or this, w e will use particular p erturbations whose sp ecial prop erties
will b e presen ted in the follo wing lemma.
Lemma 11 L et Y ∈ R n,n , let u ∈ R n,n b e a ve ctor with k u k 2 = 1 and let
∆ u
Y := − uu > Y − Y uu > + uu > Y uu > . (15)
Then the fol lowing statements hold.
(i) u ∈ k er( Y + ∆ u
Y ) , in p articular, Y + ∆ u
Y is singular.
(ii) rank ∆ u
Y ≤ 2 , and rank ∆ u
Y ≤ 1 if and only if u is a right or left eigenve ctor of Y .
(iii) k ∆ u
Y k 2
F = 
 ( I − uu > ) Y u 
 2
F + 
 u > Y ( I − uu > ) 
 2
F + ( u > Y u ) 2 .
(iv) If Y ≥ 0 then Y + ∆ u
Y ≥ 0 and k ∆ u
Y k 2
F = 2 
 ( I − uu > ) Y u 
 2
2 + ( u > Y u ) 2 .
(v) If Y > = − Y , then ∆ u
Y = − uu > Y − Y uu > and k ∆ u
Y k 2
F = 2 k Y u k 2 .
Pr o of . (i) immediately follo ws from ∆ u
Y u = − Y u . F or the pro of of (ii) let U ∈ R n,n b e an
orthogonal matrix with last column u . Then w e obtain
U T Y U =  Y 11 Y 12
Y 21 Y 22  and U T ∆ u
Y U =  0 − Y 12
− Y 21 − Y 22  (16)
for some Y 11 , Y 12 , Y 21 , Y 22 with Y 11 ∈ R n − 1 ,n − 1 which immediately sho ws that rank ∆ u
Y ≤ 2.
In particular, w e hav e rank∆ u
Y ≤ 1 if and only if Y 12 = 0 or Y 21 = 0 whic h is equiv alen t to u
b eing a righ t or left eigen vector of Y , resp ectiv ely . Moreov er, (iii) immediately follo ws from
the represen tation (16) using that
( I − uu > ) Y u =  Y 12
0  , uY ( I − uu > ) =  Y 21 0  , and Y 22 = u > Y u.
8

Finally , using the additional (sk ew-)symmetry structure, w e obtain (iv) and (v), where the
part Y + ∆ u
Y ≥ 0 in (iv) again follo ws from the represen tation (16).
W e highligh t that the first prop ert y of statemen t (iv) in Lemma 11 will b ecome essen tial in
what follo ws, b ecause it allo ws us to p erform a p erturbation that mak es a symmetric matrix
singular while sim ultaneously preserving the p ositive semidefiniteness of the matrix. With
these preparations, we obtain the follo wing theorem that c haracterizes structure-preserving
p erturbations to matrices with a non trivial common k ernel.
Theorem 12 L et ( J, X 0 , . . . , X ` ) ∈ S n
` , i.e., J > = − J and X >
i = X i ≥ 0 for i = 0 , . . . , ` .
F urthermor e, for any u ∈ R n , k u k 2 = 1 , c onsider the p erturb ation matric es
∆ u
J := − uu > J − J uu > and ∆ u
X i := − uu > X i − X i uu > + uu > X i uu > , i = 0 , . . . , `. (17)
Then the fol lowing statements hold.
(i) F or any ve ctor u ∈ R n , k u k 2 = 1 , we have
(∆ u
J ) > = − ∆ u
J , as wel l as (∆ u
X i ) > = ∆ u
X i and X i + ∆ u
X i ≥ 0 , i = 0 , . . . , `. (18)
F urthermor e, the kernels of the matric es J + ∆ u
J , X 0 + ∆ u
X 0 , . . . , X ` + ∆ u
X ` have a
nontrivial interse ction.
(ii) F or any ve ctor u ∈ R n , k u k 2 = 1 , we have
k ∆ u
J k 2
F = 2 k J u k 2 , and 
 ∆ u
X i 
 2
F = 2 
 ( I − uu > ) X i u 
 2 + ( u > X i u ) 2 , i = 0 , . . . , `.
(iii) L et ∆ J , ∆ X 0 ,..., ∆ X ` ∈ R n,n b e any p erturb ation matric es satisfying
∆ >
J = − ∆ J as wel l as ∆ >
X i = ∆ X i , and X i + ∆ X i ≥ 0 , i = 0 , . . . , `, (19)
and such that the kernels of the matric es J + ∆ J , X 0 + ∆ X 0 , . . . , X ` + ∆ X ` have a
nontrivial interse ction. Then
k ∆ u
J k F ≤ k ∆ J k F , and 
 ∆ u
X i 
 F ≤ k ∆ X i k F , i = 0 , . . . , `.
for some r e al ve ctor u with k u k 2 = 1
Pr o of . (i) and (ii) follo w immediately from Lemma 11. T o pro v e (iii), consider an y
p erturbation matrices ∆ J , ∆ X 0 ,..., ∆ X ` satisfying (19) suc h that the k ernels of the matrices
J + ∆ J , X 0 + ∆ X 0 , . . . , X ` + ∆ X ` ha v e a non trivial intersection. Then b y Prop osition 9, there
exists an orthogonal matrix U suc h that
U > ( J + ∆ J ) U =  e
J 0
0 0  , U > ( X i + ∆ X i ) U =  e
X i 0
0 0  , i = 0 , . . . , ` (20)
with some e
J , e
X 0 ,..., e
X ` ∈ R n − 1 ,n − 1 , not necessarily in v ertible, i.e., in con trast to (14) w e split
only one v ector from the in tersection of kernels. T ransforming and decomp osing accordingly ,
w e hav e
U > J U =  e
K t
− t > 0  , and U > X i U =  e
S i s i
s >
i r i  , i = 0 , . . . , ` (21)
9

for some sk ew-symmetric matrix e
K ∈ R n − 1 ,n − 1 , some symmetric matrices e
S i ∈ R n − 1 ,n − 1 ,
some r i ∈ R , s i ∈ R n − 1 ( i = 0 , . . . , ` ), and some t ∈ R n − 1 . Subtracting (21) from (20), w e
obtain that
U > ∆ J U =  e
J − e
K − t
t > 0  , and U > ∆ X i U =  e
X i − e
S i − s i
− s >
i − r i  , i = 0 , . . . , `. (22)
Observ e that for the particular c hoice u = U e n the p erturbations from (15) ha v e, b y (21),
the forms
U > ∆ u
J U = −  0 − t
t > 0  and U > ∆ u
X i U = −  0 s i
s >
i r i  , i = 0 , . . . , `. (23)
Since the F rob enius norm is in v arian t under real orthogonal transformations, we immediately
obtain that k ∆ u
J k F ≤ k ∆ J k F and 
 ∆ u
X i 
 F ≤ k ∆ X i k F for i = 0 , . . . , ` .
W e no w ha v e all ingredien ts to state and pro v e the solution of our general minimization
problem.
Theorem 13 L et ( J, X 0 , . . . , X ` ) ∈ S n
` , i.e., J > = − J and X >
j = X j ≥ 0 for j = 1 , . . . , ` .
Then the structur e d distanc e d S
k er ( J, X 0 , . . . , X ` ) to the c ommon kernel (12) is attaine d at
∆ J = ∆ u
J , ∆ X 0 = ∆ u
X 0 ,..., ∆ X ` = ∆ u
X ` b eing as in (17) for some u ∈ R n with k u k 2 = 1 .
Conse quently,
d S
k er ( J, X 0 , . . . , X ` ) = min
u ∈ R n , k u k =1 2 k J u k 2 +
`
X
i =1  2 
 ( I − uu > ) X i u 
 2 + ( u > X i u ) 2  ! 1 / 2
,
and in addition, we have the b ounds
q λ min ( − J 2 + X 2
0 + · · · + X 2
` ) ≤ d S
k er ( J, X 0 , . . . , X ` ) ≤ q 2 · λ min ( − J 2 + X 2
0 + · · · + X 2
` ) .
(24)
Pr o of . The first t w o statemen ts follo w directly from Theorem 12. It remains to pro v e
(24). F or this aim note that for ev ery u ∈ R n with k u k 2 = 1, w e ha v e
u > ( − J 2 + X 2
0 + · · · + X 2
` ) u = k J u k 2 + k X 0 u k 2 + · · · + k X ` u k 2
= k J u k 2 +
`
X
i =1  

 ( I − uu > ) X i u 

 2 + ( u > X i u ) 2  ! .
T aking the infim um o v er all u ∈ R n with k u k 2 = 1 sho ws (24).
Remark 14 In the sp ecial case λ min ( − J 2 + X 2
0 + · · · + X 2
` ) = 0 it immediately follo ws that
d S
k er ( J, X 0 , . . . , X ` ) = 0. This is in line with Prop osition 9, b ecause the singularit y of the
matrix − J 2 + X 2
0 + · · · + X 2
` is equiv alen t to the existence of a non trivial common k ernel of
the matrices J, X 0 , . . . , X ` .
10

4.2 Distance problems for structured matrix p olynomials
As a first application of the results from Subsection 4.1, w e will consider distance problems
for a particular class of structured matrix p olynomials. T o this end, recall from [15] that b y
definition a square matrix p olynomial P ( λ ) = P k
i =0 λ i Y i is singular if and only if det P ( λ ) ≡ 0.
Also recall that the c omp anion line arization
L ( λ ) = λ 




Y k
I
. . .
I




 + 




Y k − 1 . . . Y 1 Y 0
− I 0
. . . . . .
− I 0




 . (25)
of P ( λ ) is a str ong line arisation in the sense of [10]. In particular, L ( λ ) is singular if and
only if P ( λ ) is singular. F urthermore, as sho wn in [10], in the linearization the sp ectral
data for eigen v alues of P ( λ ) is preserv ed. Therefore, for the sak e of simplicit y , w e define the
notions of index and instability for the matrix p olynomial P ( λ ) via the resp ectiv e notions of
the Kronec k er canonical form of (25), cf. Section 2. W e then extend Definition 4 as follo ws.
Definition 15 Consider the class of matrix p olynomials
P n
k ,j := ( − λ j J +
k
X
i =0
λ i A i      J > = − J, A >
i = A i ≥ 0 ∈ R n,n , i = 0 , . . . , k ) ,
wher e n ≥ 1 , k , j ≥ 0 and, without loss of gener ality, j ≤ k . Then for P ( λ ) ∈ P n
k ,j
1) the structured distance to singularity is define d as
d P n
k,j
sing  P ( λ )) := inf  
 ∆ P ( λ ) 
 F   P ( λ )+∆ P ( λ ) ∈ P n
k ,j is singular  ; (26)
2) the structured distance to the nearest high index problem is define d as
d P n
k,j
hi  P ( λ )) := inf  
 ∆ P ( λ ) 
 F   P ( λ )+∆ P ( λ ) ∈ P n
k ,j is of index ≥ 2  ; (27)
3) the structured distance to instability is define d as
d P n
k,j
inst  P ( λ )  := inf  
 ∆ P ( λ ) 
 F   P ( λ )+∆ P ( λ ) ∈ P n
k ,j is unstable  . (28)
We often simply write d P
?  P ( λ )) inste ad of d P n
k,j
?  P ( λ )) for ? ∈ { sing , hi , inst } .
In other w ords, P n
k ,j consists of the set of matrix p olynomials of degree less than or equal to
k for whic h all co efficien ts are symmetric p ositiv e semidefinite except for the co efficien t at λ j
whic h is only assumed to hav e a p ositiv e semidefinite symmetric part. P articular examples
for this kind of matrix p olynomials are the dH p encils of the form (3), i.e., the set P n
1 , 0 ,
and quadratic matrix p olynomials of the form (4), i.e., the set P n
2 , 1 . Observ e that if b oth
P ( λ ) , P ( λ )+∆ P ( λ ) ∈ P n
k ,j then ∆ P ( λ ) m ust tak e the form
∆ P ( λ ) = − λ j ∆ J +
k
X
i =0
λ i ∆ A i ,
where ∆ >
J = − ∆ J and ∆ >
A i = ∆ A i . W e ha v e the follo wing theorem for c haracterizing the
distance to the nearest singular or high index matrix p olynomial.
11

Theorem 16 L et k ≥ 1 and j ∈ { 0 , . . . , k } and c onsider the set P n
k ,j of matrix p olynomials
P ( λ ) = − λ j J +
k
X
i =0
λ i A i .
with J, A 0 , . . . , A k ∈ R n,n , J > = − J and A >
i = A i ≥ 0 for i = 0 , . . . , k .
(i) If P ( λ ) ∈ P n
k ,j then the fol lowing statements ar e e quivalent:
(a) the p olynomial P ( λ ) is singular, i.e., det P ( λ ) ≡ 0 ;
(b) the matrix P (1) is singular;
(c) the kernels of the matric es J, A 0 , . . . , A k have a nontrivial interse ction.
(ii) If P ( λ ) ∈ P n
k ,j , then its distanc e to the set of singular matrix p olynomials in P n
k ,j e quals
the distanc e to the c ommon kernel of the matric es J, A 0 , . . . , A k , i.e.,
d P
sing ( P ( λ )) = d S
k er ( J, A 0 , . . . , A k ) . (29)
(iii) If max { n, k } > 1 , then the closur e of the set
I hi :=  P ( λ ) ∈ P n
k ,j   P ( λ ) is regular and has index greater than one  (30)
in P n
k ,j is e qual to
K := ( − λ j J +
k
X
i =0
λ i A i      k er A k ∩ ker A k − 1 6 = { 0 } ) if j < k (31)
or to
K := ( − λ k J +
k
X
i =0
λ i A i      k er J ∩ ker A k ∩ k er A k − 1 6 = { 0 } ) if j = k > 1 .
If max { n, k } = 1 or j = k = 1 , then I hi is empty.
(iv) L et P ( λ ) ∈ P n
k ,j . If max { n, k } > 1 , then the distanc e of P ( λ ) to the set of higher index
p olynomials in P n
k ,j e quals the distanc e to the r esp e ctive c ommon kernel
d P
hi ( P ( λ )) = ( d S
k er (0 , A k , A k − 1 ) if j < k ,
d S
k er ( J, A k , A k − 1 ) if j = k > 1 . (32)
If max { n, k } = 1 or j = k = 1 , then d P
hi ( P ( λ )) = ∞ .
Before w e giv e the pro of w e mak e a few remarks.
Remark 17 W e observ e the follo wing simple facts ab out the inclusions P n
k ,j ⊆ P n
k +1 ,j , k ≥ 0.
1) It is an immediate corollary from equation (29) that for P ( λ ) ∈ P n
k ,j one has
d P n
k,j
sing ( P ( λ )) = d P n
`,j
sing ( P ( λ )) , ` ≥ k .
12

2) Observe that the closest singular polynomial may be of low er degree than the original
one, e.g., let ε > 0 b e small and let
P ( λ ) = λ  ε 0
0 0  +  0 0
0 1  ∈ P 2
1 , 0 .
Then the closest singular p encil in P 2
1 , 0 is obtained b y removing the ε en try , and this is
a p encil of degree zero.
3) The (algebraic, geometric, partial) m ultiplicities of the eigen v alue infinit y and the index
of a matrix p olynomial P ( λ ) ∈ P n
k ,j are in v ariants with respect to the paramete r k and
not with resp ect to the degree of the p olynomial. F or example consider the matrix
p olynomial P ( λ ) = P k
i =0 λ i A i ∈ P n
k ,j with A 0 = I n and A i = 0 for i = 1 , . . . , k whic h
is a matrix p olynomial of degree zero. If P ( λ ) is considered to b e a matrix p encil (i.e.
k = 1) then it is of index one and the algebraic m ultiplicit y of ∞ is n . If, ho w ev er,
P ( λ ) is considered as a quadratic matrix p olynomial (i.e., k = 2), then it companion
linearization has the form
λ  0 0
0 I n  +  0 I n
− I n 0 
and it follo ws that the eigen v alue ∞ has algebraic m ultiplicit y 2 n and the index is
t wo. The fact that a consisten t sp ectral theory of matrix p olynomials is only p ossible
if the leading co efficien ts are allo wed to be zero is a w ell-kno wn fact in the theory of
matrix p olynomials (see [19]) and led to the in tro duction of the notion gr ade for the
parameter k in [26]. Consequen tly , there is in general no equalit y b et w een d P n
k,j
hi ( P ( λ ))
and d P n
`,j
hi ( P ( λ )) for ` > k whic h is also reflected by form ula (32).
Pr o of . (i) The implication (a) ⇒ (b) is trivial. Next, if P (1) = − J + A 0 + · · · + A k is
singular, then it follo ws from Prop osition 9 that the k ernels of J, A 0 , . . . , A k ha v e a non trivial
in tersection which, in turn, implies that P ( λ ) is singular as ob viously det P ( λ ) ≡ 0. Then (ii)
is an easy consequence of (i).
(iii) First, consider the case n = 1. If k = 1, then I hi is clearly empt y . If k > 1 and
P ( λ ) ∈ K , then J = A k = A k − 1 = 0 and the companion form of P ( λ ) is
L ( λ ) = λ 




0
1
. . .
1




 + 




0 A k − 2 . . . A 0
− 1 0
. . . . . .
− 1 0




 .
By Lemma 3 the index of L ( λ ) and hence that of P ( λ ) is at least t w o. If P ( λ ) is regular, w e
th us hav e P ( λ ) ∈ I hi . If P ( λ ) is singular, then it is identically zero and replacing A 0 with ε
and letting ε → 0, w e see that P ( λ ) is in the closure of I hi .
F or n > 1 w e distinguish the cases j < k and j = k .
Case j < k . First observ e that K is a closed set in P n
k ,j b y Lemma 8. Hence, to pro v e
the inclusion I hi ⊆ K it suffices to sho w that an y matrix p olynomial P ( λ ) ∈ I hi satisfies
k er A k ∩ ker A k − 1 6 = { 0 } . T o do this, supp ose on the con trary that for some P ( λ ) ∈ I hi w e
ha ve k er A k ∩ k er A k − 1 = { 0 } . If k er A k = { 0 } then the matrix p olynomial has no infinite
13

eigen v alues and hence is of index zero. Hence, we ma y assume that A k has a non trivial k ernel,
and then there exists an orthogonal congruence transformation so that
U > A k U =  e
A k 0
0 0  , U > J U =  J 11 J 12
− J >
12 J 22  , and U > A k − 1 U =  A 11 A 12
A >
12 A 22  , (33)
where e
A k ∈ R n − r ,n − r ( r > 0) is in v ertible and all three matrices are partitioned conformably .
In fact, replacing P ( λ ) b y U > P ( λ ) U if necessary , we ma y assume U = I n in what follo ws.
Since k er A k ∩ k er A k − 1 = { 0 } and since A k − 1 is p ositiv e semidefinite, it then follo ws that A 22
is in v ertible.
If j < k − 1 then the companion linearization (25) of P ( λ ) tak es the form
λ 
 e
A k
0
I ( k − 1) n

 + 

A 11 A 12 ∗
A >
12 A 22 ∗
∗ ∗ ∗ 
 , (34)
where in comparison to (25) the first blo c k row and column ha v e b een split in to t w o, the last
k − 1 blo c k ro ws and columns ha v e b een merged in to one, resp ectiv ely , and ∗ denotes a p ossibly
nonzero blo c k en try . Then it follo ws from Lemma 3 (applied to the p encil that is obtained
from (34) b y p erm uting the second and third blo ck ro ws and columns) that the companion
p encil and hence the matrix p olynomial P ( λ ) is of index one, since A 22 is in v ertible.
If j = k − 1, then the co efficien t of λ k − 1 in P ( λ ) is giv en b y A k − 1 − J and hence the
companion linearization (25) of P ( λ ) has the form
λ 
 e
A k
0
I ( k − 1) n

 + 

A 11 − J 11 A 12 − J 12 ∗
A >
12 + J >
12 A 22 − J 22 ∗
∗ ∗ ∗ 
 (35)
with the same con v en tions as for the p encil (34). But with A 22 inv ertible also A 22 − J 22 is
in vertible (see Proposition 9 applied with ` = 0 to J 22 and X 0 = A 22 ). Again, it follo ws from
Lemma 3 that the matrix p olynomial P ( λ ) is of index one.
F or the con v erse inclusion K ⊆ I hi consider a matrix p olynomial P ( λ ) ∈ K . F urthermore,
let u ∈ k er A k ∩ ker A k − 1 with k u k 2 = 1 and let U ∈ R n,n b e orthogonal with last column u .
Then
U > A k U =  e
A k 0
0 0  , U > A k − 1 U =  e
A k − 1 0
0 0  , U > J U =  J 11 v
− v > 0  , (36)
with e
A k , e
A k − 1 , J 11 ∈ R n − 1 ,n − 1 (not necessarily b eing in v ertible) and v ∈ R n − 1 . Note that the
en try 0 in (2 , 2) blo c k of U > J U is caused b y the sk ew-symmetry of J . Again, replacing P ( λ )
with U > P ( λ ) U if necessary , w e ma y assume without loss of generalit y that U = I n .
First assume that j < k − 1. F or small ε > 0 w e ha ve that e
A k + εI n − 1 ∈ R n − 1 ,n − 1
and A j − J + εI n ∈ R n,n are in vertible. Then the matrix p olynomial P ε ( λ ) that is obtained
from P ( λ ) b y replacing A k with A k + diag( εI n − 1 , 0) and A j − J with A j − J + εI n ∈ R n,n is
regular, b ecause at least one co efficien t (namely the co efficien t asso ciated with λ j ) is in v ertible.
F urthermore, the companion linearization of P ε ( λ ) tak es the form
λ 
 e
A k + εI n − 1
0
I ( k − 1) n

 + 
 e
A k − 1 0 ∗
0 0 ∗
∗ ∗ ∗ 
 . (37)
14

By Lemma 3 we see that this pencil, and hence P ε ( λ ) itself, has index greater than one.
No w let j = k − 1. F or sufficien tly small ε > 0 w e ha v e that e
A k + εI n − 1 ∈ R n − 1 ,n − 1 and
A k − 1 + εI n ∈ R n,n are in vertible and that v + εe 1 6 = 0. Let P ε ( λ ) b e the matrix p olynomial
obtained from P ( λ ) b y replacing A k and with A k + diag( εI n − 1 , 0) and A k − 1 with A k − 1 + εI n
as w ell as v in J with v + εe 1 6 = 0. Again, it follo ws that P ε ( λ ) is regular as the co efficien t at
λ k − 1 is regular (see Prop osition 9 applied with ` = 0 to J and X 0 = A k − 1 + εI n ). Then the
companion linearization of P ε ( λ ) tak e s the form
λ 
 e
A k + εI n − 1
0
I ( k − 1) n

 + 
 e
A k − 1 + εI n − 1 − J 11 − v − εe 1 ∗
v > + εe >
1 0 ∗
∗ ∗ ∗ 
 . (38)
By Lemma 3 we see that (38), and hence P ε ( λ ) itself, has index greater than one.
Letting ε → 0 w e see that in b oth cases j < k − 1 and j = k − 1 w e ha v e I hi 3 P ε ( λ ) →
P ( λ ) ∈ I hi .
Case j = k . If j = k = 1 and P ( λ ) = λ ( J + A 1 ) + A 0 ∈ P n
1 , 1 , then b y Theorem 2, zero
is a semisimple eigen v alue (if it is an eigen v alue) of the rev ersal λA 0 + J + A 1 of P ( λ ) and
hence the index of P ( λ ) is at most one. This sho ws that I hi is empty in that case.
Th us, assume that j = k > 1. T o sho w the inclusion I hi ⊆ K supp ose, as in the pro of of
(iii), that for some P ( λ ) ∈ I hi w e hav e k er J ∩ ker A k ∩ k er A k − 1 = { 0 } . If k er( A k − J ) = { 0 }
then the matrix p olynomial has no infinite eigen v alues and hence is of index zero. Hence, w e
ma y assume that A k − J has a nontrivial k ernel, whic h b y Prop osition 9 applied with ` = 0
to J and X 0 = A k implies that there exists an orthogonal congruence transformation U so
that
U > A k U =  e
A k 0
0 0  , U > J U =  J 11 0
0 0  , and U > A k − 1 U =  A 11 A 12
A >
12 A 22  , (39)
where e
A k − J 11 ∈ R n − r ,n − r ( r > 0) is in v ertible and all three matrices are partitioned con-
formably . In fact, replacing P ( λ ) by U > P ( λ ) U if necessary , w e ma y assume U = I n in what
follo ws. Since k er J ∩ k er A k ∩ ker A k − 1 = { 0 } and since A k − 1 is positive semidefinite, it follo ws
that A 22 is in v ertible. The companion matrix p encil (25) of P ( λ ) tak es the form
λ 
 e
A k − J 11
0
I ( k − 1) n

 + 

A 11 A 12 ∗
A >
12 A 22 ∗
∗ ∗ ∗ 
 (40)
and from Lemma 3 w e infer that the matrix p olynomial P ( λ ) is of index one.
F or the con v erse K ⊆ I hi let P ( λ ) ∈ K . Then w e ha v e the follo wing decomp osition
U > A k U =  e
A k 0
0 0  , U > A k − 1 U =  e
A k − 1 0
0 0  , U > J U =  J 11 0
0 0  ,
and U > A k − 2 U =  A 11 A 12
A >
12 A 22  , U > J U =  J 11 0
0 0  ,
with A k , A k − 1 , A 11 , J 11 ∈ R n − 1 ,n − 1 not necessarily in v ertible. Replacing e
A k − 2 with e
A k − 2 +
εI n − 1 , then for sufficien tly small ε w e w e get a family of regular p encils P ε ( λ ) of index at
least t w o and such that I hi 3 P ε ( λ ) → P ( λ ) ∈ I hi .
(iv) is an immediate consequence of (iii).
15

Remark 18 At first, it ma y come as a surprise that a matrix p olynomial P ( λ ) as in Theo-
rem 16 is already singular if P (1) is singular whic h means that 1 cannot b e an eigen v alue of a
regular P ( λ ) ∈ P n
k ,j . More generally , if α > 0 then replacing λ with λ
α and A i with α i A i sho ws
that if P ( α ) is singular, then P ( λ ) is already a singular matrix p olynomial. This generalizes
in a non trivial wa y the observ ation that an y scalar p olynomial with nonnegativ e co efficien ts
cannot ha ve real zeros that are p ositiv e unless it is the zero p olynomial.
Remark 19 The reason for not inv estigating the structured distance to instabilit y for P ( λ )
in Theorem 16 is the fact that in con trast to Theorem 6(ii) the distances to higher index and
instabilit y need not coincide for matrix p olynomials of degree larger than one. W e will return
to the distance to instabilit y for quadratic p olynomials in Section 6.2, b ecause that task is
still accessible b y the common kernel methods framework. This is due to a non trivial result,
Theorem 27 b elo w, whic h states that the only sp ectral p oin ts that may cause instabilit y are
zero and infinit y . How ev er, already for degree three the reason for instability ma y b e differen t,
since a p olynomial in P n
3 ,` migh t ha v e eigen v alues in the righ t half plane. F or example, the
scalar p olynomial p ( λ ) = λ 3 + 1 ∈ P 1
3 ,` (th us J = X 1 = X 2 = 0, X 0 = X 3 = 1) has t wo roots
in the righ t half plane.
5 Distance problems for first order dHD AE systems
In this section w e will revisit the distance problems for first order dHD AE systems form ulated
in Section 3. W e will first presen t the missing pro ofs whic h are no w easy consequences of the
extended results from the previous section. Then, w e will presen t t w o examples that sho w that
the structured distances for dHD AE systems may differ considerably from the corresponding
ones under arbitrary p erturbations.
5.1 Pro ofs of and commen ts on the main results in Section 3
Pro of of Theorem 5. (i) It follows from Theorem 16(i) ( k = 1, j = 0) that P ( λ ) is singular
if and only if the k ernels of E , J and R ha v e a nontrivial in tersection. Applying Theorem 9
( ` = 1) w e get the desired transformation U .
(ii) By Theorem 2 the index is at most t w o. Then it follo ws from Theorem 16(iii) ( k = 1,
j = 0) that P ( λ ) is in the closure of regular dH p encils of index 2 if and only if the k ernels of
E and R ha ve a non trivial in tersection.
Pro of of Theorem 6. (i) The pro of is obtained from Theorem 13 with k = 1, X 0 = E and
X 1 = R and Theorem 16, and using J > J = − J 2 .
(ii) First, it immediately follo ws from Theorem 2 that P ( λ ) is stable if and only if it is
regular and of index one. The pro of is then obtained from Theorem 13 with k = 1, X 0 = E
and X 1 = R and using J > J = − J 2 . By Theorem 2 an y p encil λE − ( J − R ) with E , R ≥ 0
and J > = − J , asso ciated with a dH system, is of index at most 2.
W e ha v e the follo wing immediate corollary of Theorems 5 and 6.
Corollary 20 F or J = − J > , E , R ≥ 0 one has the fol lowing estimate
2 · λ min ( − J 2 ) + d L
hi  λE − ( J − R )  2 ≤ d L
sing  λE − ( J − R )  2 ≤ 2 k J k 2 + d L
hi  λE − ( J − R )  2 .
In p articular, the set of singular dH p encils in P n
1 , 0 is c ontaine d in the closur e in P n
1 , 0 of the
set of index two r e gular dH p encils.
16

Remark 21 Consider the reversed pencil − λ ( J − R ) + E with J = − J > and E , R ≥ 0.
Statemen t (iii) of Theorem 16 sho ws that its structured distance to higher index d L
hi  − λ ( J −
R ) + E  equals its distance to singularit y d L
sing  − λ ( J − R ) + E  . This is in line with the
fact that ∞ can only b e a semisimple eigen v alue of − λ ( J − R ) + E as zero is a semisimple
eigen v alue of λE − ( J − R ), cf. Theorem 5. Then, in summary , we ha v e
d P
k er ( J, E , R ) = d P
hi  − λ ( J − R ) + E 
= d P
sing  − λ ( J − R ) + E 
= d P
sing  λE − ( J − R ) 
= d P
sing  λR − ( J − E )  .
Since our main fo cus is on distance problems in this pap er, it w as necessary to c haracterize
the closure of the set of dH p encils of index t w o in Theorem 5 (iii). Our next result giv es a
c haracterization of the set of regular dH p encils of index t w o.
Prop osition 22 A p encil L ( λ ) = λE − ( J − R ) with E , R ≥ 0 , J = − J > is r e gular of index
two if and only if ther e exists an ortho gonal matrix U such that
U > E U = 

E 11 0 0
0 0 0
0 0 0 
 , U > J U = 

J 11 J 12 J 13
− J 12 > J 22 0
− J >
13 0 0 
 , U > RU = 

R 11 R 12 0
R >
12 R 22 0
0 0 0 
 ,
(41)
wher e n = p + q + r , p, r > 0 , E 11 ∈ R p,p is invertible, J 22 − R 22 ∈ R q ,q is invertible, and
J 13 ∈ R p,r has ful l c olumn r ank.
Pr o of . Supp ose that the decomp osition (41) holds. As J 22 − R 22 is inv ertible and J 13
has full column rank, w e ha v e that − J + E + R is in v ertible. Hence, by Theorem 5(i) and
Theorem 16(i) the p encil is regular. Therefore, by Lemma 3 it is of index t w o. T o pro v e the
con verse implication first w e find an orthogonal transformation U 1 suc h that
U >
1 E U 1 =  E 11 0
0 0  , U >
1 J U 1 =  J 11 J 0
− J 0 > e
J  , U >
1 RU 1 =  R 11 R 0
R 0 > e
R  , (42)
with E 11 ∈ R p,p in vertible. By Lemma 3 the matrix e
J − e
R is singular. Applying Theorem 9
to e
J and e
R w e get an orthogonal transformation and a splitting of the last n − p ro ws and
columns, whic h com bined with (42) gives
U > E U = 

E 11 0 0
0 0 0
0 0 0 
 , U > J U = 

J 11 J 12 J 13
− J 12 > J 22 0
− J >
13 0 0 
 , U > RU = 

R 11 R 12 R 13
R >
12 R 22 0
R >
13 0 0 
 ,
(43)
with some orthogonal U . As R is p ositiv e semidefinite w e ha ve R 13 = 0. But then J 13 needs
to ha v e full column rank, otherwise the p encil w ould b e singular.
5.2 Structured vs. unstructured distances
In this subsection w e compare the structured and the unstructured distances. First note that
the statemen ts of Theorem 5 are not true without the structure assumptions on E and R .
While it is ob vious that (i) cannot hold for arbitrary p encils, we need a simple example to
dispro ve an unstructured analogue of (ii).
17

Example 23 Let n = 2, E = J = 0, R = I 2 . Then setting e
J = J , e
R = R and
e
E =  0 ε
0 0 
w e se e that λ e
E − ( e
J − e
R ) is regular and of index t w o (but not dissipativ e Hamiltonian).
Letting ε → 0, we find that λE − ( J − R ) is in the closure of regular p encils of index t w o
although k er E ∩ k er R = { 0 } .
T o analyze the distances in Theorem 6, recall that in [7] the (unstructured) distance to
singularit y was defined as
d sing ( λE − A ) := inf  k [∆ E , ∆ A ] k F   λ ( E + ∆ E ) + A + ∆ A is singular  .
Example 24 Let
E =  0 0
0 1  , J =  0 − 0 . 5
0 . 5 0  , R =  0 . 18 0 . 42
0 . 42 1 . 03  ≥ 0 .
Then (rounding the n umerical results to four digits) w e hav e
λ min ( − J 2 + E 2 + R 2 ) = 0 . 5819 , σ min  A
E  = 0 . 1908 , σ min  A E  = 0 . 6056 ,
where σ min stands for the smallest singular v alue. The first equalit y implies, b y Theorem
6(ii), that d L
sing ( λE − ( J − R ))) ≥ 0 . 5819, while the second and third equalit y imply , together
with Corollary 3 of [7], that d sing ( λE − A ) = 0 . 1908.
The next example sho ws mainly the same b eha viour, though w e refine the constrain ts.
Namely , w e sho w that if w e c hange the constrain t in the definition of (6) to E + ∆ E , R +
∆ R b eing symmetric (instead of b eing p ositiv e definite) then w e get an essen tially differen t
distance.
Example 25 Consider a dissipativ e Hamiltonian system (3) with co efficien ts
E = 





1
1
1
0
0






, J = 





0000 − 1
0001 1
000 − 1 − 1
0 − 1 1 0 ε
1 − 1 1 − ε 0






, R = 





α 0 0 0 1
0 α 0 1 1
0 0 α 1 1
0 1 1 ε 0
1 1 1 0 ε






,
where ε > 0 and
α = ε − 1 · 




 

0 1
1 1
1 1 
 





2
F
+ 1 .
By Theorem 1.1 of [12] suc h c hoice of α makes R > 0. Consider no w the p erturbation
∆ E = 0 , ∆ J = 

0 3 × 3
− ε
ε 
 , ∆ R = 

0 3 × 3
− ε
− ε 
 .
18

Clearly ∆ J = − ∆ >
J and ∆ R = ∆ >
R , and ∆ E = ∆ >
E . The p encil λE − b
A with
b
A := J + ∆ J − ( R + ∆ R ) = 





− α 0 0 0 − 2
0 − α 0 0 0
0 0 − α − 2 − 2
0 − 2 0 0 0
0 − 2 0 0 0






is no w singular, b ecause one easily c hec ks that det( λE − b
A ) ≡ 0. In this w a y , w e ha v e
constructed a p erturbation with
k [∆ J , ∆ R , ∆ E ] k F = 2 ε,
suc h that the p erturb ed p encil is singular, but the p erturbation is not structure-preserving,
b ecause the matrix R + ∆ R is no w indefinite. Observ e also that, unlik e in Theorem 5(i), E
and b
A do not ha v e a common right or left k ernel. This means that in the Kronec k er canonical
form there are left and righ t minimal indices of size at least one.
On the other hand, w e ha ve
J > J + E 2 + R 2 = 





3 + α 2 0 2 − ε α + ε
0 5 + α 2 0 α + 2 ε α
2 0 5 + α 2 α α + 2 ε
− ε α + 2 ε α 4+2 ε 2 4
α + ε α α + 2 ε 4 6 + 2 ε 2






and a simple Ma tlab calculation sho ws that w e ha v e λ min ( J > J + E 2 + R 2 ) 1 / 2 ≥ 0 . 81 for ε ∈
(10 − 1 , 10 − 6 ). Hence, the smallest p erturbation ∆ E , ∆ J , ∆ R that makes the p encil singular,
while k eeping R + ∆ R ≥ 0, E + ∆ E ≥ 0, and J + ∆ J sk ew-symmetric, satisfies
k [∆ J , ∆ R , ∆ E ] k F ≥ 0 . 81 ,
for ε ∈ (10 − 1 , 10 − 6 ) b y Theorem 6(i).
Note that the p encil λE − ( J − R ) from Example 23 also sho ws that Theorem 6(ii) do es not
hold for unstructured p erturbations. Indeed, that p encil is in the closure of regular p encils
of index 2, but d P
hi ( λE − ( J − R )) ≥ λ min ( E 2 + R 2 ) = 1.
As last example w e consider the analysis of distance problems in circuit sim ulation.
Example 26 A simple RLC netw ork, see, e.g., [3, 9, 13, 14], can b e mo deled b y a dHD AE
system of the form


G c C G >
c 0 0
0 L 0
0 0 0 

| {z }
=: E


˙
V
˙
I l
˙
I v 
 = 
 − G r R − 1
r G >
r − G l − G v
G >
l 0 0
G >
v 0 0 

| {z }
=: J − R


V
I l
I v 
 , (44)
where L > 0, C > 0, R r > 0 are real symmetric matrices describing inductances, capacitances,
and resistances, resp ectiv ely . The subscripts r , c, l , and v refer to the resistors, capacitors,
inductors, and v oltage sources, while V , I denote v oltage and curren t, resp ectiv ely . The
matrices G c , G l , G r , G v enco de the net w ork top ology , see [13] for details. Here, J and − R
19

are defined to b e the sk ew-symmetric and symmetric parts, resp ectiv ely , of the matrix on the
righ t hand side of (44). W e see that E , R ≥ 0 and J = − J T .
It w as sho wn in [13, Theorem 1] that the p encil λE − ( J − R ) is regular if and only if G v
has full column rank and
G 1 :=  G c G r G l G v 
has full ro w rank. Note that this equiv alence is no w a simple corollary of Prop osition 6(i).
Indeed, λE − ( J − R ) is singular if and only if the k ernels of the three matrices
E = 

G c C G >
c 0 0
0 L 0
0 0 0 
 , J = 

0 − G l − G v
G >
l 0 0
G >
v 0 0 
 , and R = 

G r R − 1
r G >
r 0 0
0 0 0
0 0 0 

ha ve a non trivial in tersection. Ha ving in mind that C , L , and R − 1
r are p ositiv e definite
matrices, we immediately obtain that x =  x >
1 x >
2 x >
3  > ∈ k er E ∩ ker J ∩ k er R if and
only if G >
c x 1 = 0, x 2 = 0 as w ell as G >
l x 1 = 0, G >
v x 1 = 0, G v x 3 = 0, and G >
r x 1 = 0 whic h in
turn is equiv alen t to
x >
1 G 1 = 0 , x 2 = 0 , and G v x 3 = 0 .
Th us, w e see that λE − ( J − R ) is singular if and only if either G 1 do es not ha v e full ro w
rank or G v do es not ha v e full column rank.
All this sho ws that regularit y of the p encil λE − ( J − R ) dep ends only on the net w ork
top ology , cf. Remark 1 in [13]. As one can exp ect, the distance to singularit y dep ends also
on the v alues of matrices L, C , R r . Observing that the matrix − J 2 + R 2 + E 2 has the form
− J 2 + R 2 + E 2 = 

( G c C G >
c ) 2 + ( G r R − 1
r G >
r ) 2 + G l G >
l + G v G >
v 0 0
0 L 2 + G >
l G l G >
l G v
0 G >
v G l G >
v G v 
 ,
w e obtain by (11) that the structured distance to singularit y is b ounded b y
λ min ( − J 2 + R 2 + E 2 ) 1 / 2 ≤ d L
sing  λE − ( J − R )  ≤ 2 · λ min ( − J 2 + R 2 + E 2 ) 1 / 2 ,
where λ min ( − J 2 + R 2 + E 2 ) is the minim um of the t w o v alues
λ min  ( G c C G >
c ) 2 + ( G r R − 1
r G >
r ) 2 + G l G >
l + G v G >
v  and λ min  L 2 + G >
l G l G >
l G v
G >
v G l G >
v G v  .
In this section w e hav e c haracterized the distances to singularit y , high index and instabilit y
for linear constan t co efficien t dH systems. In the next section w e extend these results to
quadratic matrix p olynomials.
6 Quadratic p olynomials, linearization, and remo ving Q
In this section, w e will sho w ho w the main results of the previous sections can b e applied to
more general situations. W e first study ho w the transformation (linearization) of structured
matrix p olynomials to dH p encils can b e p erformed and then apply the distance results to
quadratic matrix p olynomials. Finally w e discuss the more general dH p encils in (1) and
sho w ho w the multiplier Q can b e remo v ed.
20

6.1 Dissipativ e Hamiltonian linearisations
Consider a second order system of the form
M ¨ x − ( G − D ) ˙ x + K x = 0 (45)
with M , G, D , K ∈ R n,n satisfying M , D , K ≥ 0 and G = − G > . A companion linearization
(25) of (45) is given b y
L ( λ ) = λ  M 0
0 I  −  D − G K
− I 0  = λ  M 0
0 I  −  G I
− I 0  −  D 0
0 0   I 0
0 K  .
It satisfies the h yp othesis of Theorem 2 with
E =  M 0
0 I  , J =  G I
− I 0  , R =  D 0
0 0  , Q =  I 0
0 K  ,
and th us w e immediately ha ve the follo wing result.
Theorem 27 L et P ( λ ) = λ 2 M − λ ( G − D ) + K , wher e M , D , K , G ∈ C n,n with G H = − G
and M , D , K ≥ 0 . Then the fol lowing statements hold.
(i) A l l eigenvalues of P ( λ ) ar e in the close d left half c omplex plane and al l finite nonzer o
eigenvalues on the imaginary axis ar e semisimple.
(ii) The p ossible length of Jor dan chains of P ( λ ) , asso ciate d with either the eigenvalue ∞
or the eigenvalue zer o, is at most two.
(iii) A l l left and al l right minimal indic es of P ( λ ) ar e zer o (if ther e ar e any).
Pr o of . The proof of (i) and the statemen t in (ii) on the length of the Jordan c hains
asso ciated with the eigen v alue ∞ follo ws immediately from Theorem 2, while the remaining
statemen t of (ii) then follo ws b y applying the already pro v ed part of (ii) to the reversal
λ 2 K + λ ( G − D ) + M of P ( λ ), whic h has the same structure. T o see (iii), observ e that b y
Theorem 2 the left minimal indices of L ( λ ) are all zero and the righ t minimal indices of L ( λ )
are at most one. By [10, Theorem 5.10] the left minimal indices of P ( λ ) coincide with those
of L ( λ ), and if ε 1 , . . . , ε k are the righ t minimal indices of P ( λ ), then ε 1 + 1 , . . . , ε k + 1 are the
righ t minimal indices of L ( λ ). This implies that all minimal indices of P ( λ ) are zero (if there
are an y).
Remark 28 One may be tempted to use the results on dH p encils with Q = I to pro v e
Theorem 27, i.e., to apply Theorem 5 instead of Theorem 2 and also to extend the result
to p olynomials of higher degree. Ho w ev er, as the constan t term of the companion form (25)
con tains a principal submatrix  Y k Y k − 1
− I n 0  ,
whic h has a p ositiv e semidefinite symmetric part only when Y k − 1 = I , in general one needs
to consider differen t linearizations, e.g., L ( λ ) to b e from one of the classical linearizations
classes in [24, 25] or a so-called Fie d ler line arization , see, e.g., [11]. Ho w ev er, one still meets
the follo wing general obstacles.
21

1) As we ha v e seen in Remark 19, a p olynomial from P n
k ,j ma y ha v e sp ectrum in the righ t
half plane if k > 2. Hence, no linearization of such a polynomial satisfies the hypothesis
of Theorem 2.
2) The index of a p olynomial from P n
k ,j ma y b e larger than t w o if k > 2; consider e.g.
P ( λ ) = λ 3 X 3 + λ 2 X 2 + λX 1 + λX 0 − J with X 3 = X 2 = X 1 = J = 0 and X 0 = 1. The
companion form is
λ 

0
1
1 
 + 

0 0 1
− 1
− 1 
 ,
whic h corresp onds to a blo c k of size three at ∞ , i.e., P ( λ ) is of index 3. Hence, none of
its index preserving (strong) linearizations satisfies the h yp othesis of Theorem 2.
3) Let P ( λ ) ∈ P n
k ,j b e a singular matrix p olynomial with left minimal indices η 1 , . . . , η ` ( ` > 0)
and righ t minimal indices ε 1 , . . . , ε ` ; note that the num b ers of left and righ t minimal indices
coincide, b ecause the matrix p olynomial is square. T ak e L ( λ ) as an y of the structured
linearizations in [11, 24, 25]. Then, there exists a n um b er q ∈ { 0 , 1 , 2 , . . . , k − 1 } , known
in adv ance for a particular linearization, suc h that L ( λ ) has the left minimal indices
η 1 + q , . . . , η ` + q and righ t minimal indices ε 1 + k − 1 − q , . . . , ε ` + k − 1 − q , see [10, 11].
Th us, ev en for k = 2, the linearization L ( λ ) cannot satisfy the h yp othesis of Theorem 5
and for k > 2 it cannot satisfy the h yp othesis of Theorem 2.
Hence, if k > 2, then a linearization L ( λ ) of P ( λ ) cannot b e a dH p encil with arbitrary
Q and if k = 2 it can only b e a dH p encil with (necessarily non trivial) Q . It remains an
op en question to deriv e additional conditions on the matrix co efficien ts for a p olynomial
P ( λ ) ∈ P n
k ,j in order to guaran tee that its sp ectrum is in the closed left half plane suc h that
all eigen v alues on the imaginary axis (p ossible excluding zero and infinit y) are semisimple.
In the next subsection w e study the case of quadratic matrix p olynomials with dH structure.
6.2 Quadratic matrix p olynomials with dH structure
In the case of quadratic matrix p olynomials with dH structure, i.e., if P ( λ ) = λ 2 A 2 + λA 1 +
A 0 ∈ P n
2 , 1 , where A 2 , A 0 , A 1 + A >
1 ≥ 0, the structured distances to singularit y , to higher
index, and to instabilit y were defined in (26), (27), and (28), respectively . Recalling that for
a matrix Y ∈ R n,n and a v ector u ∈ R n with k u k 2 = 1 the p erturbation matrix ∆ u
Y in (15) is
defined as ∆ u
Y = − uu > Y − Y uu > + uu > Y uu > , w e obtain the follo wing result.
Theorem 29 L et P ( λ ) = λ 2 M − λ ( G − D ) + K ∈ P n
2 , 1 , i.e., M , D , K , G ∈ R n,n with G > = − G
and M , D , K ≥ 0 .
(i) The matrix p olynomial P ( λ ) is singular if and only if al l four matric es M , D , K , G have
a c ommon kernel. In p articular, the structur e d distanc e to singularity d P
sing ( P ( λ )) is
attaine d for a p erturb ation of the form ∆ u
M , ∆ u
D , ∆ u
K , ∆ u
G for some u ∈ R n,n with
k u k 2 = 1 and satisfies
d P
sing ( P ( λ )) = d S
k er ( G, M , D , K ) ,
wher e d S
k er ( G, M , D , K ) is given by The or em 13 . In p articular, the structur e d distanc e
to singularity is b ounde d by
p λ min ( M 2 + D 2 + K 2 − G 2 ) ≤ d P
sing ( P ( λ )) ≤ p 2 · λ min ( M 2 + D 2 + K 2 − G 2 ) . (46)
22

(ii) The matrix p olynomial P ( λ ) is in the closur e of p olynomials in P n
2 , 1 that ar e r e gular
and of index lar ger than one (and thus of index exactly two) if and only if M and D
have a c ommon kernel. In p articular, the structur e d distanc e to higher index d P
hi ( P ( λ ))
is attaine d for a p erturb ation of the form ∆ u
M , ∆ u
D , for some u ∈ R n,n with k u k 2 = 1
and satisfies
d P
hi ( P ( λ )) = d S
k er (0 , M , D ) ,
wher e d S
k er (0 , M , D ) is given by The or em 13 . In p articular, the structur e d distanc e to
higher index is b ounde d by
p λ min ( M 2 + D 2 ) ≤ d P
hi ( P ( λ )) ≤ p 2 · λ min ( M 2 + D 2 ) . (47)
(iii) The matrix p olynomial P ( λ ) is in the closur e of p olynomials in P that ar e unstable if
and only if M and D have a c ommon kernel or D and K have a c ommon kernel. In
p articular, the structur e d distanc e to instability d P
inst ( P ( λ )) is attaine d for a p erturb ation
of the form ∆ u
M , ∆ u
D , ∆ u
K with ∆ u
M = 0 or ∆ u
K = 0 for some u ∈ R n,n with k u k 2 = 1
and satisfies
d P
inst ( P ( λ )) = min  d S
k er (0 , M , D ) , d S
k er (0 , D , K ) 
wher e d S
k er (0 , M , D ) and d S
k er (0 , D , K ) ar e given by The or em 13 . Mor e over, the distanc e
to instability is b ounde d by
√ α ≤ d P
inst ( P ( λ )) ≤ √ 2 · α, (48)
wher e α := min  λ min ( M 2 + D 2 ) , λ min ( D 2 + K 2 )  .
Pr o of . P arts (i) and (ii) immediately follow from Theorem 16. F or part (iii), w e can
apply Theorem 27, if the matrix p olynomial P ( λ ) is singular, regular of index tw o, or if it
is regular and has a Jordan blo c k of size t w o at λ = 0, whic h is equiv alent to the rev ersal
λ 2 K + λ ( D − G ) + M of P ( λ ) ha ving index t w o. In all cases the statemen t follo ws immediately
from (ii) applied to P ( λ ) and its rev ersal, which has the same structure.
6.3 Remo ving the co efficien t Q in dH systems
Due to the m ultiplicativ e structure, in the mo del represen tation (1) the co efficien t Q will
presen t difficulties for the p erturbation analysis and it is an op en question whether the def-
inition of a dH system needs this term at all. In this section we will sho w that the factor Q
can b e remo v ed and the system (3) can b e reduced to a system with Q = I n .
Supp ose first that Q is in v ertible. Then the system (1) is equiv alent to the system
Q > E ˙ x = Q > ( J − R ) Qx. (49)
Then setting ˜
Q = I , ˜
E = Q > E , ˜
J = Q > J Q , ˜
R = Q > RQ , w e see that for the transformed
system the constrain ts (2) are satisfied.
If Q is not in vertible then, using the singular v alue decomp osition, there exist orthogonal
matrices U ∈ R n,n and V ∈ R n,n suc h that
U > QV =  Q 11 0
0 0  , U > E V =  E 11 E 12
E 21 E 22  , U > ( J − R ) U =  L 11 L 12
L 21 L 22 
23

where in all three blo c k matrices the (1 , 1) blo c k is square of size r = rank( Q ) and Q 11 is
in vertible. Since Q > E = E > Q , w e get Q >
1 E 11 = E >
11 Q 1 and E 12 = 0, and the transformed
system is giv en b y a reduced dH system
 E 11 0
E 21 E 22   ˙ x 1
˙ x 2  =  L 11 Q 11 0
L 21 Q 11 0   x 1
x 2  (50)
where x 1 ( t ) ∈ R k , x 2 ( t ) ∈ R n − k .
If the p encil λE − ( J − R ) Q is regular then also the p encil λE − Q is regular and has
index at most one, see [30]. Then E 22 is in v ertible, and therefore the second blo c k-ro w of the
system reads as
˙ x 2 = E − 1
22 ( − E 21 ˙ x 1 + L 21 Q 1 x 1 ) ,
x 1 do es not dep end on x 2 and the v ariable x 2 can b e remo v ed from the system. Then the
reduced system
E 11 ˙ x 1 = L 11 Q 11 x 1 , (51)
satisfies the structured assumption (2) with Q 11 b eing in v ertible, so we can apply the previous
pro cedure to obtain a system as in (49) for the reduced system.
The reduced system (51) ma y ha ve a differen t Jordan structure at the eigen v alue zero than
system (1). Indeed, dH p encils with singular Q may ha v e Jordan blo c ks of size t wo at the
eigen v alue zero (see [30] for examples), while it follo ws from Theorem 2 that the eigen v alue
zero is semisimple if Q is in v ertible. A Jordan blo c k of size 2 at the eigen v alue 0 w ould also
mean that the system is unstable.
W e highligh t that the reduction pro cedure is not advisable if E 22 is ill-conditioned. Also,
note that due to the fact that the pro cedure in volv es nonorthogonal transformations, the
distance to singularit y may c hange considerably during the pro cess. It is an op en problem to
c haracterize the distance to singularit y for a system in the form (1).
When the solution of the second order system (45), or, equiv alen tly , the solution of the
quadratic eigen v alue problem for P ( λ ) = λ 2 M − λ ( G − D ) + K is considered, then the classical
approac h is the linearization of the problem. As remark ed in the pro of of Theorem 27, a
particular linearization of P ( λ ) is of the form λE ( J − R ) Q with
E =  M 0
0 I  , J =  G − I
I 0  , R =  D 0
0 0  , Q =  I 0
0 K  (52)
whic h corresp onds to a system of the form (1). In this case w e can remo v e the matrix Q in a
simpler w ay . Since Q = Q > one case use U = V , and that k er Q = { 0 } ⊕ k er K to reduce the
system to the form
Q 1 := U > QU =  I 0
0 K 1  , U > K U =  K 1 0
0 0  ,
with some symmetric p ositiv e K 1 ∈ R k ,k . As a result w e get a so called trimme d line arization ,
see [8], i.e., a p encil λE 1 − ( J 1 − R 1 ) ∈ R n + k ,n + k , where
E 1 =  M 0
0 K 1  , J 1 =  G − K >
2
K 2 0  , R 1 =  D 0
0 0  , (53)
24

and K 2 =  K 1 0  ∈ R k ,n . Our aim is no w to pro vide results that allo w to compare the
distances for the trimmed linearisation λE 1 − ( J 1 − R 1 ) with the original distances obtained
in Theorem 29. In the distances for the reduced system w e use the matrices
− J 2
1 + E 2
1 + R 2
1 =  M 2 + G > G + K >
2 K 2 + D 2 − GK >
2
K 2 G K 2 K >
2 
and
E 2
1 + R 2
1 =  M 2 + D 2 0
0 K 2 K >
2  .
Hence, λ min ( − J 2
1 + E 2
1 + R 2
1 ) ≤ λ min ( K 2 K >
2 ) = λ min ( K 2
1 ) and if G = 0 the inequalit y b ecomes
an equalit y . W e get immediately the follo wing result.
Prop osition 30 F or the r e duc e d sytem (53) one has the fol lowing statements.
(i) The structur e d distanc e to singularity of the p encil λE 1 − ( J 1 − R 1 ) ∈ L satisfies
d L
sing ( λE 1 − ( J 1 − R 1 )) ≤ q 2 · λ min ( K 2
1 ) , (54)
while for G = 0 we have
λ min ( K 2
1 ) ≤ d L
sing ( λE 1 − ( J 1 − R 1 )) . (55)
(ii) The structur e d distanc e to higher index of the p encil λE 1 − ( J 1 − R 1 ) ∈ L satisfies
p β ≤ d L
hi ( λE 1 − ( J 1 − R 1 )) ≤ p 2 · β , (56)
wher e β = min { λ min ( M 2 + D 2 ) , λ min ( K 2
1 ) } .
7 Conclusions
Distance problems in linear differen tial-algebraic systems with dissipativ e Hamiltonian struc-
ture ha ve been studied. These include the distance to the nearest singular problem, the
distance to the nearest high index problem, and the distance to instabilit y . The c haracteri-
zation of these distances are op en problems for general linear differen tial-algebraic systems,
while w e hav e sho wn that for dissipative Hamiltonian systems and related matrix polynomials,
explicit c haracterizations in terms of common n ull-spaces of several matrices exist.
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