LO W RANK PER TURBA TIONS OF QUA TERNION MA TRICES
CHRISTIAN MEHL ∗ AND ANDR ´
E C. M. RAN †
De dic ate d to the memory of L eib a R o dman, whose work inspir e d us gr e atly.
Abstract. Lo w rank p erturbations of righ t eigenv alues of quaternion matrices are considered.
F or real and complex matrices it is w ell kno wn that under a generic rank- k p erturbation the k largest
Jordan blo c ks of a giv en eigenv alue will disapp ear while additional smaller Jordan blo c ks will remain.
In this pap er, it is sho wn that the same is true for real eigenv alues of quaternion matrices, but for
complex nonreal eigen v alues the situation is different: not only the largest k , but the largest 2 k
Jordan blo c ks of a giv en eigen v alue will disapp ear under generic quaternion perturbations of rank
k . Sp ecial emphasis is also giv en to Hermitian and sk ew-Hermitian quaternion matrices and generic
lo w rank p erturbations that are structure-preserving.
1. In tro duction. In this pap er w e will consider an n × n matrix A with en tries
from the sk ew-field H of the quaternions. Recall from [20] that a n um b er λ ∈ H is
called a righ t eigen v alue if there is a v ector x ∈ H n \ { 0 } suc h that Ax = xλ . Since for
ev ery α ∈ H w e ha ve A ( xα ) = ( xα )( α − 1 λα ), w e see that together with λ also ev ery
similar n um b er α − 1 λα is a righ t eigen v alue. Restricting oneself to one representativ e
of eac h equiv alence class of similar righ t eigen v alues, one can assume without loss of
generalit y that the righ t eigen v alues are in fact complex num b ers with nonnegativ e
imaginary part. This concept then allo ws the computation of a Jordan canonical form
for the matrix A , to b e precise: there exists an in v ertible quaternion matrix S suc h
that
S − 1 AS = J m 1 ( λ 1 ) ⊕ · · · ⊕ J m p ( λ p ) ,
with λ 1 , . . . , λ p ∈ C ha ving nonnegativ e imaginary part for j = 1 , . . . , p . Here,
λ 1 , . . . , λ p are not necessarily pairwise distinct and J m ( λ ) stands for the upp er trian-
gular complex Jordan blo c k of size m × m asso ciated with the eigen v alue λ ∈ C .
The question w e will consider is the follo wing: what happ ens to the Jordan canon-
ical form of A when w e apply a generic additiv e p erturbation of rank k , i.e., when w e
consider the matrix A + U V T for some ( U, V ) from a generic set Ω ⊆ H n × H n ∼
= H 2 n .
F or the complex case this problem was studied in [12], and later in [19, 21, 22]. An
alternativ e treatmen t for the complex case was giv en in [14], and also the real case has
b een studied, see [16]. F urthermore, also the case of complex matrix p encils has been
studied in [9, 8] for the regular case and in [6] for the singular case, while the case
of regular matrix p olynomials w as treated in [7]. The related questions for matrices
with a symmetry structure ha v e b een addressed in a series of pap ers starting with
[14] and con tin ued in [15, 16, 17, 18] and [13, 11] for man y different classes of struc-
tures and the case of structure-preserving rank-one p erturbation. A generalization
to the case of structure-preserving rank- k p erturbations w as then giv en in [5]. Also,
structure-preserving lo w-rank p erturbations of regular matrix p encils with symmetry
structures ha v e b een considered, see [1, 2, 3, 4] for sp ecial p erturbations of rank one
or t w o and [10] for the general case.
T o b e precise ab out the nature of the term ”generic”, we in tro duce the isomor-
phism χ : H n → R 4 n as a particular standard represen tation of H n seen as a real
∗ TU Berlin, Institut f ¨ ur Mathematik, Sekretariat MA 4-5, Straße des 17. Juni 136, 10623 Berlin,
German y , email: [email protected]
† Departmen t of Mathematics, F aculty of Science, V rije Univ ersiteit Amsterdam, De Bo elelaan
1081a, 1081 HV Amsterdam, The Netherlands and Unit for BMI, North W est Univ ersity , Potc hef-
stro om, South Africa, email: [email protected]
1
v ector space as follo ws: If (1 , i , j , k ) is the canonical basis for H o v er R , w e define
χ ( u ) =
u 0
u 1
u 2
u 3
for a v ector u = u 0 + u 1 i + u 2 j + u 3 k ∈ H n , with u i ∈ R n , i = 0 , 1 , 2 , 3. Then a
set Ω ⊆ H n is said to b e generic (or more precisely generic with r esp e ct to the r e al
c omp onents ) if the set χ (Ω) is a generic set in R 4 n , i.e., its complemen t is contained
in a prop er algebraic subset of R 4 n . (Recall that a subset A ⊆ R 4 n is called algebr aic
if it is the set of common zeros of finitely man y real p olynomials in 4 n v ariables, and
it is called pr op er if it is not the full space R 4 n .) Similarly , a set Ω ⊆ C n is said to
b e generic (or more precisely generic with r esp e ct to the r e al and imaginary p arts ) if
it is generic when view ed as the canonical subspace of R 2 n . It is easy to see that if
Ω ⊆ F n is generic and S ∈ F n,n is in v ertible, then the sets S Ω and Ω S are generic as
w ell, where F ∈ { C , H } .
A general result on rank- k p erturbations of complex matrices sa ys that the geo-
metric m ultiplicit y of a fixed eigen v alue can c hange at most b y k if any (not necessarily
b eing generic) rank- k p erturbation is applied, see, e.g., [19] or [14], and the question
arises if this remains true for quaternion matrices. As a first example, consider the
quaternion matrices
A = 1 0
0 1 , B = 1 − k
k 1 , and A + B = 2 − k
k 2 .
Then A is actually a real matrix with eigen v alue 1 with algebraic and geometric
m ultiplicit y t wo and B is a quaternion matrix of rank one. Then one easily c hecks
that
( A + B ) 1
k = 1
k 3 , and ( A + B ) 1
− k = 1
− k 1 ,
so that the eigen v alues of A + B are 3 and 1. This sho ws that for this example the
geometric m ultiplicit y of the eigen v alue 1 do es c hange b y only one from t wo to one as
the reader ma y ha v e exp ected.
Surprisingly , ho w ever, it needs no longer b e the case for quaternion matrices that
a rank- k p erturbation can c hange the geometric m ultiplicit y b y at most k . T o see
this, consider the example
C := i 0
0 i , B = 1 − k
k 1 ,
where B is the same rank-one matrix as ab o v e. Then setting
S := − 1+2 j 1 − k
2 + j − i − j − k
a straigh t forw ard calculation sho ws that
1 + i − k
k 1 + i S = − 1+2 j 2 j − k
2 + j 2 − i − k = S 1 1
0 1 ,
2
or equiv alently S − 1 ( C + B ) S = J 2 (1), so the perturb ed matrix C + B has the eigen-
v alue 1 with geometric m ultiplicit y one and algebraic multiplicit y t wo whic h means
that the geometric m ultiplicit y of the eigen v alue i of the original matrix C has c hanged
b y 2 from t w o to zero. Although w e will see b elo w that the p erturbation B do es not
sho w the generic b eha viour as it generates a Jordan blo c k of size t w o of the newly
created eigen v alue 1, this example sho ws that the effect of quaternion rank one p er-
turbations of quaternion matrices ma y b e significan tly differen t from the analogous
effect observ ed for complex rank-one p erturbations of complex matrices.
The observ ant reader ma y susp ect at this momen t that the differen t b eha vior
of the matrices A and C under a p erturbation with B ma y b e caused b y the giv en
symmetry-structure and its preserv ation or non-preserv ation, resp ectiv ely . Indeed, the
matrices A and B are Hermitian while C is sk ew-Hermitian, so the p erturbation with
B is structure-preserving for A , but not for C . Ho w ev er, the surprising effect in the
second example remains true ev en for p erturbations preserving the sk ew-Hermitian
structure as our third example with the matrices
C = i 0
0 i and D := j − 1
1 j
will sho w. Here, D is sk ew-Hermitian and has rank one. When w e consider
C + D = i + j − 1
1 i + j ,
then a straigh tforw ard computation sho ws that
( C + D ) 1 + √ 2 + k
− (1 + √ 2) i − j = 1 + √ 2 + k
− (1 + √ 2) i − j (1 + √ 2) i ,
( C + D ) − (1 + √ 2) i − j
1 + √ 2 + k = − (1 + √ 2) i − j
1 + √ 2 + k ( √ 2 − 1) i .
This sho ws that the geometric m ultiplicit y of the eigen v alue i of C drops from t w o to
zero ev en under a structure-preserving rank-one p erturbation.
In this pap er w e will sho w that the differen t b eha vior observ ed in the examples
ab o v e is due to the nature of the o ccurring eigen v alues. Indeed, real eigenv alues b eha v e
differen tly than complex eigen v alues: if a generic rank- k p erturbation is applied to a
square quaternion matrix, then the largest k Jordan blo c ks asso ciated with an y real
eigen v alue will disapp ear from the Jordan canonical form while additional smaller
Jordan blo c ks will remain. F or a giv en complex eigen v alue, ho wev er, it will no w b e the
corresp onding largest 2 k Jordan blo c ks that disapp ear while again additional smaller
ones will remain. This effect can b e observ ed for b oth generic rank-one p erturbations
of general quaternion matrices as w ell as for generic structure-preserving rank-one
p erturbations of Hermitian or sk ew-Hermitian quaternion matrices.
The remainder of the pap er is organized as follo ws. In the next section, w e present
one of the main to ols in our in v estigations b y reviewing the w ell-known connection
b et w een quaternion matrices and a sub class of complex matrices with a sp ecial sym-
metry structure. This class will b e denoted b y Q n,n , where the sym b ol Q has b een
c hosen as a reminder of the quaternions. In Section 3, w e generalize some results on
lo w-rank p erturbations from the literature so that they can b e applied to structure-
preserving lo w rank p erturbations in Q n,n . In Section 4, we then discuss the c hanges in
the Jordan structure under generic structure-preserving rank- k p erturbations within
3
Q n,n and translate this result in Section 5 to quaternion matrices. In Section 6 and 7,
w e in v estigate structure-preserving quaternion rank- k p erturbations of Hermitian and
sk ew-Hermitian, resp ectiv ely , and sho w that the same b eha vior as under generic p er-
turbations that ignore the structure can b e observ ed.
2. Reduction to a structured matrix problem. It is well kno wn that the
map ω : H → ω ( H ) ⊆ C 2 , 2 with
ω ( α 1 + i α 2 + j α 3 + k α 4 ) 7→ α 1 + i α 2 α 3 + i α 4
− α 3 + i α 4 α 1 − i α 2
for α i ∈ R , i = 1 , 2 , 3 , 4, is a sk ew-field isomorphism. Its extension (also denoted by
ω ) to matrices will b e an imp ortan t to ol in this pap er: giv en a quaternion matrix
A ∈ H n,m , w e can write A = A 1 + A 2 j , where A 1 and A 2 are complex matrices. Then
ω ( A ) = " A 1 A 2
− A 2 A 1 #
and b y [20, Theorem 5.7.1], the Jordan form of the quaternion matrix A is giv en by
J m 1 ( λ 1 ) ⊕ · · · ⊕ J m p ( λ p ) ,
if and only if the Jordan form of ω ( A ) is giv en b y
J m 1 ( λ 1 ) 0
0 J m 1 ( λ 1 ) ⊕ · · · ⊕ J m p ( λ p ) 0
0 J m p ( λ p ) . (2.1)
Note that the eigen v alues in (2.1) are allo w ed to b e real. In particular, it follows that
eac h real eigen v alue has ev en algebraic and geometric m ultiplicity , and all partial
m ultiplicities o ccur an ev en n um b er of times. In the follo wing it will b e useful to use
a sligh t v arian t of (2.1) that is itself in the range of ω . Applying a blo ck permutation
the matrix in (2.1) is easily seen to b e similar to
J 0
0 J with J = J m 1 ( λ 1 ) ⊕ · · · ⊕ J m p ( λ p ) .
The map ω mapping quaternion matrices to complex matrices has the prop erties
that it is linear (with resp ect to real scalars), m ultiplicativ e, and resp ects the transp ose
op eration [20, Section 3.4]. In particular, w e hav e
ω ( A + U V T ) = ω ( A ) + ω ( U ) ω ( V ) T (2.2)
for A ∈ H n,n and U, V ∈ H n,k . Th us, to study the effect of rank- k p erturbations on
the quaternion matrix A w e can study the effect of structure-preserving p erturbations
of ω ( A ). How ev er, if U = U 1 + U 2 j and V = V 1 + V 2 j with U i , V i ∈ C n,k hav e rank k ,
then it follo ws b y [20, Prop osition 3.2.5(e)] and the prop erties of ω that
ω ( U ) = " U 1 U 2
− U 2 U 1 # and ω ( V ) = " V 1 V 2
− V 2 V 1 #
ha v e rank 2 k . Th us, rank- k p erturbations of quaternion matrices lead to rank-2 k
p erturbations of complex matrices that are structured as in the range of ω .
4
In the follo wing it will b e useful to use an alternativ e c haracterization of this par-
ticular class of structured complex matrices. F or this, w e will in tro duce the follo wing
notation.
Definition 2.1. L et n, k ∈ N and
J := J 2 n := 0 I n
− I n 0 ,
Then we define
Q n,k = X ∈ C 2 n, 2 k J 2 n X = X J 2 k .
It is straigh tforw ard to c heck that a 2 n × 2 k matrix X is in the range of ω if and
only if X ∈ Q n,k . In the follo wing, w e will sometimes switch betw een the sets H n,m
and Q n,m , and in order to mak e it easier for the reader to k eep track in whic h set
w e curren tly are, w e adopt the con ven tion to use “hatted” sym b ols for matrices in
Q n,m . Th us, if A ∈ H n,m then w e denote b
A = ω ( A ) and similarly , if b
B ∈ Q n,m , then
B = ω − 1 ( b
B ) ∈ H n,m .
The next prop osition sho ws that the form ula in (2.2) giv es a parametrization of
matrices in Q n,n that ha v e rank 2 k so that w e are able to identify and describe generic
sets of suc h matrices.
Pr oposition 2.2. L et b
B ∈ Q n,n b e a matrix of r ank 2 k . Then ther e exist two
matric es b
U , b
V ∈ Q n,k of ful l r ank 2 k such that b
B = b
U b
V T .
Pr o of . Since b
B is in the range of ω , there exists a quaternion matrix B ∈ H n,n
suc h that b
B = ω ( B ), and b y the prop erties of ω B m ust ha v e rank k . But then
there exists matrices U, V ∈ H n,k of rank k such that B = U V T b y [20, Prop osition
3.2.5(e)], and hence w e obtain b
B = b
U b
V T with b
U = ω ( U ) , b
V = ω ( V ) ∈ Q n,k ha ving
rank 2 k .
As a side-note, observ e that matrices in the class Q n,n can nev er ha ve odd rank,
so the smallest rank p erturbation of matrices in that class is a p erturbation of rank
t w o.
3. Lo calization results. In this section, w e establish a result that allo ws us
to determine the b eha vior of a p ossibly structured complex matrix under generic
structure-preserving lo w-rank p erturbations b y studying the effect of p erturbations
that lo cally p erturb an arbitrary , but fixed eigen v alue of the matrix. The main theo-
rem is a generalization of [5, Theorem 2.6] and in fact con tains that result as a sp ecial
case. A k ey ingredient for its proof is the following lemma whic h is a generalization
of [17, Lemma 8.1]. W e highligh t that although the lines of the pro ofs of the results
in this section follo w the lines of the pro ofs of the previously obtained results, they
are not immediate. Therefore, a careful revision of eac h single step in the pro of is
necessary to obtain the full generalit y in the main theorem presen ted here. In this
w a y , the result will not only b e applicable in the remainder of this pap er, but also
for an y class of structured matrices and corresp onding structure-preserving lo w-rank
p erturbations that can b e parameterized b y p olynomial functions.
The next lemma is needed for the pro of of the main result and states that newly
created eigen v alues of p erturb ed matrices will generically ha ve m ultiplicities that are
as small as p ossible.
5
Lemma 3.1. L et A ∈ C n,n have the p airwise distinct eigenvalues λ 1 , . . . , λ m ∈ C
with algebr aic multiplicities a 1 , . . . , a m , and let ε > 0 b e such that the discs
D j := µ ∈ C | λ j − µ | < ε 2 /n , j = 1 , . . . , m
ar e p airwise disjoint. F urthermor e, let B : R m → C n,n b e an analytic function with
B (0) = A such that the fol lowing c onditions ar e satisfie d:
1) F or al l u ∈ R m , the algebr aic multiplicity of any eigenvalue of B ( u ) is always
a multiple of ` ∈ N \ { 0 } .
2) Ther e exists a generic set Ω ⊆ R m such that for al l u ∈ Ω the matrix B ( u )
has the eigenvalues λ 1 , . . . , λ m with algebr aic multiplicities e a 1 ,..., e a m , wher e
e a j ≤ a j for j = 1 , . . . , m . (Her e, we al low e a j = 0 in the c ase that λ j no
longer is an eigenvalue of B ( u ) .)
3) F or e ach j = 1 , . . . , m ther e exists u j ∈ R m with k u j k < ε such that the matrix
B ( u ) has exactly ( a j − e a j ) /` p airwise distinct eigenvalues in D j differ ent fr om
λ j and e ach with algebr aic multiplicity exactly ` .
Then ther e exists a generic set Ω 0 ⊆ R m such that for al l u ∈ Ω 0 the eigenvalues of
B ( u ) that ar e differ ent fr om those of A have algebr aic multiplicity exactly ` .
Pr o of . First observ e that there exists a constant K only depending on A such that
for an y u ∈ R m with k u k < K · min { 1 , ε } the matrix B ( u ) has exactly a j eigenv alues
in the disc D j . Indeed, this follo ws from the con tin uity of B and w ell-kno wn results
on matc hing distance of eigen v alues of nearb y matrices, see, e.g., [23, Section IV.1]
and references therein. In the following, w e denote ε 0 = K · min { 1 , ε } .
Next, w e fix λ j and denote by χ ( λ j , u ) the c haracteristic p olynomial (in the
indep enden t v ariable t ) of the restriction of B ( u ) to the sp ectral in v arian t subspace
corresp onding to the eigen v alues of B ( u ) within D j . Then the co efficien ts of χ ( λ j , u )
are analytic functions of the comp onen ts of u , see, e.g., [14, Lemma 2.5] for more
details.
Let q ( λ j , u ) b e the n um b er of distinct eigen v alues of B ( u ) in the disk D j . F ur-
thermore, denote b y S ( p 1 , p 2 ) the Sylv ester resultant matrix of the t w o p olynomials
p 1 ( t ), p 2 ( t ) and recall that S ( p 1 , p 2 ) is a square matrix of size deg ( p 1 ) + deg ( p 2 ) and
that the rank deficiency of S ( p 1 , p 2 ) coincides with the degree of the greatest common
divisor of the p olynomials p 1 ( t ) and p 2 ( t ). W e ha v e
q ( λ j , u ) = 1
` rank S χ ( λ j , u ) , ∂ ` χ ( λ j , u )
( ∂ t ) ` − a j + 1 .
The en tries of S χ ( λ j , u ) , ∂ ` χ ( λ j ,u )
( ∂ t ) ` are scalar m ultiples (which are independent of u )
of the co efficien ts of χ ( λ j , u ), and therefore the set Q ( λ j ) of all u ∈ R m , k u k < ε 0 , for
whic h q ( λ j , u ) is maximal is the complemen t of the set of zeros of an analytic function
of the en tries of u . (In fact, this analytic function can b e c hosen to b e the pro duct
of minors of that order that is equal to the maximal v alue of q ( λ j , u ).) In particular,
Q ( λ j ) is op en and dense in
u ∈ R m k u k < ε 0 .
By h yp othesis, there exists u j ∈ R m suc h that B ( u j ) has exactly 1
` ( a j − e a j ) eigenv alues
with algebraic m ultiplicit y exactly ` in D j differen t from λ j . If u j happ ens not to
b e in Ω, then we ma y sligh tly p erturb u j to obtain a new u 0
j ∈ Ω suc h that B ( u 0
j )
has the eigen v alues λ 1 , . . . , λ m with algebraic m ultiplicities e a 1 ,..., e a m and 1
` ( a j − e a j )
6
eigen v alues with algebraic m ultiplicit y exactly ` in D j differen t from λ j . Suc h c hoice
of u 0
j is p ossible b ecause Ω is generic, the prop ert y of eigenv alues ha ving algebraic
m ultiplicit y exactly ` p ersists under small p erturbations of B ( u j ) b y assumption 1),
and the total n um b er of eigen v alues of B ( u ) within D j , coun ted with m ultiplicities, is
equal to a j , for every u ∈ R m with k u k < ε 0 . Since Ω is op en, clearly there exists δ > 0
suc h that for all u ∈ R m with k u − u j k < δ the matrix B ( u j ) has the eigen v alues
λ 1 , . . . , λ m with algebraic m ultiplicities e a 1 ,..., e a m and 1
` ( a j − e a j ) eigenv alues with
algebraic m ultiplicit y exactly ` in D j differen t from λ j . Since the set of all suc h
v ectors u is op en in R m , it follows from the properties of the set Q ( λ j ) established
ab o v e that in fact w e ha ve
q ( λ j , u ) = 1
` ( a j − e a j ) , for all u ∈ R m , k u − u j k < δ.
So for the op en set
Ω j := Q ( λ j ) ∩ Ω
whic h is dense in u ∈ R m k u k < ε 0 , we ha v e that all eigenv alues of B ( u ) within
D j differen t from λ j ha v e algebraic m ultiplicity exactly ` . No w let
Ω 0 =
m
\
j =1
Ω j ⊆ Ω .
Note that Ω 0 is nonempt y as the in tersection of finitely man y sets that are op en dense
in u ∈ R m k u k < ε 0 .
Finally , let χ ( u ) denote the characteristic polynomial (in the indep enden t v ariable
t ) of B ( u ). Then the num b er of distinct ro ots of χ ( u ) is giv en b y
rank S χ ( u ) , ∂ χ ( u )
∂ t − n + 1
and therefore, the set of all u ∈ Ω on which the n um b er of distinct ro ots of χ ( u )
is maximal, is a generic set. Since Ω 0 constructed ab o v e is nonempt y , this maximal
n um b er is equal to P m
j =1 1
` ( a j − e a j ), i.e., generically all eigen v alues of B ( u ) that are
differen t from λ 1 , . . . , λ m ha v e algebraic m ultiplicity exactly ` .
Next, w e consider the analogue of Theorem 2.6 of [5] which describes the p ossible
c hanges in the Jordan structure of a fixed eigen v alue λ of a matrix from Q n,k under
lo w rank p erturbations, and also presen ts conditions when a generic b eha vior can b e
observ ed.
Theorem 3.2. L et A ∈ C n,n and let the Jor dan c anonic al form of A b e given by
J n 1 ( λ ) ⊕ · · · ⊕ J n m ( λ ) ⊕ ˜
J ,
with n 1 ≥ · · · ≥ n m and λ 6∈ σ ( ˜
J ) . F urthermor e, let P ∈ R n,n [ t 1 , . . . , t r ] b e a matrix
whose entries ar e p olynomials in the indep endent indeterminate variables t 1 , . . . , t r .
Assume that for al l u = ( u 1 , . . . , u r ) ∈ R r we have
(i) rank P ( u ) ≤ κ ;
(ii) the algebr aic multiplicity of any eigenvalue of A + P ( u ) is always a multiple
of ` ∈ N \ { 0 } .
Then the fol lowing statements hold:
7
1. F or e ach ( u 1 , . . . , u r ) ∈ R r ther e exist inte gers η 1 ≥ · · · ≥ η ` such that
(a) the Jor dan c anonic al form of A + P ( u 1 , . . . , u r ) is given by
J η 1 ( λ ) ⊕ · · · ⊕ J η ` ( λ ) ⊕ ˇ
J ,
wher e λ 6∈ σ ( ˇ
J ) ,
(b) ( η 1 , . . . , η ` ) dominates ( n κ +1 , . . . , n m ) ; that is, we have l ≥ m − κ and
η j ≥ n j + κ for j = 1 , . . . , m − κ .
2. Assume that for al l u = ( u 1 , . . . , u r ) ∈ R the algebr aic multiplicity a u of λ as
an eigenvalue of A + P ( u ) satisfies a u ≥ a for some a ∈ N . If ther e exists
u 0 ∈ R r such that a u 0 = a , then the set
Ω = { u ∈ R r | a u = a }
is a generic set.
3. Assume that for any ε > 0 ther e exists u 0 ∈ R r with k u 0 k < ε such that the
Jor dan form of A + P ( u 0 ) is describ e d by
(a) J n κ +1 ( λ ) ⊕ · · · ⊕ J n m ( λ ) ⊕ ˇ
J , λ 6∈ σ ( ˇ
J ) ,
(b) al l eigenvalues that ar e not eigenvalues of A have multiplicity ` pr e cisely.
Then ther e exists a generic set Ω ⊆ R r such that the Jor dan c anonic al form
of A + P ( u ) is describ e d by (a) and (b) for al l u ∈ Ω .
Pr o of . Part (1) is a direct consequence of [9, Lemma 2.1] using the fact that the
rank of P ( u ) is at most κ for an y u ∈ R r .
F or part (2), let Y ( u ) = ( A + P ( u ) − λI n ) n . Then the h yp othesis tells us that
rank Y ( u 0 ) = n − a for some u 0 ∈ R r . Th us w e can apply [14, Lemma 2.1] (or [5,
Lemma 2.2]) to see that the set
Ω := { u ∈ R r | rank Y ( u ) ≥ n − a }
is a generic set. Note that the condition rank Y ( u ) ≥ n − a is equiv alen t to a u ≤ a , and
since the rev erse inequalit y a u ≥ a holds b y assumption it is equiv alen t to a u = a 0 .
Hence, Ω is the desired generic set.
Concerning part (3) observ e that b y part (1) of the theorem, the list of par-
tial m ultiplicities of A + P ( u ) corresp onding to the eigen v alue λ dominates the list
( n κ +1 , . . . , n m ). Hence, the algebraic m ultiplicity a u of A + P ( u ) at λ is at least
a := n κ +1 + · · · + n m . By the h yp othesis there exists a particular u 0 ∈ R r suc h
that a u 0 = n κ +1 + · · · + n m = a . Then b y part (2) the set Ω 1 of all u ∈ R r with
a u = a is generic. Since the only list of partial m ultiplicities that b oth dominates
( n κ +1 , . . . , n m ) and has a u = a 0 is the list ( n κ +1 , . . . , n m ) itself, this sho ws that the
Jordan form describ ed in part (a) is attained b y all matrices in Ω 1 . Moreo v er, since
P is analytic and u 0 can b e c hosen arbitrarily small with A + P ( u 0 ) satisfying the
condition in (b), it follo ws b y Lemma 3.1 that the set Ω 2 of all u ∈ R r satisfying (b)
is also generic. Then the set Ω = Ω 1 ∩ Ω 2 is the desired generic set.
4. Ev en rank p erturbations within Q n,n . W e are now ready to state the first
main result of this pap er, whic h basically sa ys that for eac h eigenv alue of a matrix b
A
in Q n,n under generic p erturbations with matrices of rank 2 k in Q n,n the largest 2 k
partial m ultiplicities disapp ear while the others remain, and that the eigen v alues of
b
A + b
U b
V T whic h are not already eigen v alues of b
A are all simple and non-real.
8
Theorem 4.1. L et b
A ∈ Q n,n , and let the Jor dan c anonic al form of b
A b e given
by A 1 ⊕ A 1 , wher e
A 1 = r 1
M
i =1 J n i, 1 ( λ 1 ) ! ⊕ · · · ⊕ r p
M
i =1 J n i,p ( λ p ) !
⊕ r p +1
M
i =1 J n i,p +1 ( λ p +1 ) ! ⊕ · · · ⊕ r m
M
i =1 J n i,m ( λ m ) !
wher e the eigenvalues λ 1 , . . . , λ m ar e p airwise distinct with λ 1 , . . . , λ p b eing r e al and
λ p +1 , . . . , λ m having p ositive imaginary p art, and wher e the p artial multiplicities ar e
or der e d in de cr e asing or der: n 1 ,j ≥ · · · ≥ n r j ,j for al l j = 1 , . . . , m .
Then, ther e exists a generic set Ω ⊆ Q n,k × Q n,k such that for al l ( b
U , b
V ) ∈ Ω the
Jor dan form of b
A + b
U b
V T is given by C 1 ⊕ C 1 , wher e
C 1 = r 1
M
i = k +1 J n i, 1 ( λ 1 ) ! ⊕ · · · ⊕ r p
M
i = k +1 J n i,p ( λ p ) !
⊕ r p +1
M
i =2 k +1 J n i,p +1 ( λ p +1 ) ! ⊕ · · · ⊕ r m
M
i =2 k +1 J n i,m ( λ m ) ! ⊕ e
J ,
wher e e
J has simple nonr e al eigenvalues with p ositive imaginary p art that ar e differ ent
fr om any of the eigenvalues of A .
Pr o of . Without loss of generality w e can assume that A is already equal to its
Jordan canonical form A 1 ⊕ A 1 . W e then aim to apply Theorem 3.2 for the case
κ = 2 k and ` = 1, and for the function b
P = b
U b
V T whic h is in terpreted as a function
of the r = 8 nk real and imaginary parts of the en tries of U 1 , U 2 , V 1 and V 2 , where
b
U = " U 1 U 2
− U 2 U 1 # and b
V = " V 1 V 2
− V 2 V 1 # .
Hence, it remains to find for eac h eigen v alue λ j and an y e ε > 0 a particular c hoice
of matrices b
U 0 , b
V 0 ∈ Q n,k with k b
U 0 k , k b
V 0 k < e ε suc h that b
A + b
U 0 b
V T
0 satisfies parts
3a) and 3b) of Theorem 3.2. Then Theorem 3.2 yields the existence of a generic
set Ω j ⊆ Q n,k × Q n,k (canonically iden tified with a subset of R 8 nk ) suc h that for
all ( b
U , b
V ) ∈ Ω j the parts 3a) and 3b) of Theorem 3.2 are satisfied. T aking then
the in tersection Ω = Ω 1 ∩ · · · ∩ Ω m yields the desired generic set. Concerning the
matrix e
J , note that since b
A + b
U b
V T is in Q n,n , the set of the new simple eigen v alues
that are not eigen v alues of b
A do es not con tain real eigen v alues (as those w ould ha ve
ev en m ultiplicit y) and is necessarily symmetric with resp ect to the real line. W e can
th us order the new eigen v alues in the Jordan canonical form in suc h a w ay that all
eigen v alues with p ositiv e imaginary part are collected in e
J . In the follo wing we will
consider t w o cases.
Case 1 : k = 1. W e first consider the sub case that λ j is real, that is j ∈ { 1 , . . . , p } .
Let B 1 ∈ C n,n b e the matrix that has zero en tries ev erywhere, except for the p osition
( a j + n 1 ,j , a j + 1) where the en try ε · e i ϕ . Here, ε is a sufficien tly small p ositiv e n um b er,
ϕ satisfies 0 < ϕ < π
n 1 ,j , and w e ha v e
a j =
j − 1
X
s =1
r s
X
i =1
n i,s .
9
Th us, in A 1 + B 1 only a single Jordan blo c k of partial multiplicit y n 1 ,j asso ciated
with the eigen v alue λ j of A 1 is p erturb ed b y the rank-one p erturbation B 1 as
λ j 1
λ j
. . .
. . . 1
ε · e i ϕ λ j
.
The c haracteristic p olynomial χ of this blo c k in the indep enden t v ariable t is giv en
b y ( t − λ j ) n 1 ,j − ε · e i ϕ and th us its ro ots are the v ertices of a regular p olygon on
a circle of radius ε 1 /n 1 ,j with cen ter λ j . Since 0 <ϕ< π
n 1 ,j the set of ro ots of χ
is conjugate-free. In particular, all ro ots of χ are nonreal. F urthermore, c ho osing ε
small enough guaran tees that all ro ots of χ are distinct from eac h of the eigen v alues
of A .
No w let b
B = B 1 ⊕ B 1 . Then b
B ∈ Q n,n has rank t w o, so b y Prop osition 2.2 there
exist rank t w o matrices b
U 0 , b
V 0 ∈ Q n, 1 suc h that b
B = b
U 0 b
V T
0 . Then it is easy to c hec k
that the Jordan canonical form of b
A + b
U 0 b
V T
0 = ( A 1 + B 1 ) ⊕ ( A 1 + B 1 ) corresp onding to
λ j is as desired. In particular, b y the c hoice of ε and ϕ ab o ve and since the eigen v alues
of A 1 + B 1 are the conjugates of those of A 1 + B 1 , all eigen v alues of b
A + b
B that are
not eigen v alues of b
A are simple and nonreal.
Next, consider an eigen v alue λ j with j ≥ p . If there is just one Jordan blo c k
asso ciated with λ j , then we can proceed as for real eigenv alues, where here we can
c ho ose ϕ = 0. Then λ j will not b e an eigen v alue of the p erturb ed matrix. If the
geometric m ultiplicit y of λ j is at least 2, then consider the submatrix
S :=
J n 1 ,j ( λ j ) 0 0 0
0 J n 2 ,j ( λ j ) 0 0
0 0 J n 1 ,j ( λ j ) 0
0 0 0 J n 2 ,j ( λ j )
.
W e aim to find a rank tw o p erturbation of S suc h that all eigen v alues of the p erturb ed
matrix are simple and nonreal. T o ac hieve this, w e use an idea from p ole-placemen t
in con trol theory . Consider the submatrix
S 1 = J n 1 ,j ( λ j ) 0
0 J n 2 ,j ( λ j )
of S . Since λ j is nonreal, this matrix is nonderogatory and th us similar to the com-
panion form of its c haracteristic p olynomial, i.e., there exists a nonsingular T suc h
that
S 1 = T
0 − β 0
1 . . . .
.
.
. . . 0 − β ν − 2
1 − β ν − 1
T − 1 ,
where β 0 , . . . , β ν − 1 are the co efficien ts of the p olynomial
χ = ( t − λ j ) n 1 ,j ( t − λ j ) n 2 ,j = t ν + β ν − 1 t ν − 1 + · · · + β 1 t + β 0 ,
10
and where w e used the abbreviation ν = n 1 ,j + n 2 ,j . No w choose ν v alues µ 1 , . . . , µ ν
suc h that n 1 ,j of them are close to λ j and the remaining n 2 ,j = ν − n 1 ,j are close
to λ j , and suc h that the set { µ 1 , . . . , µ ν } is conjugate-free and do es not in tersect the
sp ectrum of A . Let γ 0 , . . . , γ ν − 1 b e the co efficien ts of the p olynomial
ν
Y
i =1
( t − µ i ) = t ν + γ ν − 1 t ν − 1 + · · · + γ 1 t + γ 0 .
Then setting
e
B := B 11 B 12
B 21 B 22 := T
0 β 0 − γ 0
0 . . . .
.
.
. . . 0 β ν − 2 − γ ν − 2
0 β ν − 1 − γ ν − 1
T − 1
with B ik ∈ C n i,j ,n k,j , i, k ∈ { 1 , 2 } , w e obtain that S 1 + e
B has exactly the eigen v alues
µ 1 , . . . , µ ν ∈ C + . In particular, the eigen v alues of S 1 + e
B are conjugate-free (and
nonreal). Thus, setting
b
B =
B 11 0 0 B 12
0 B 22 B 21 0
0 B 12 B 11 0
B 21 0 0 B 22
,
w e find that b
B ∈ Q ν,ν has rank t w o and the eigenv alues of S + b
B are giv en b y
µ 1 , . . . , µ ν , µ 1 , . . . , µ ν . In particular, the eigenv alues of S + b
B are all simple. By
c ho osing the v alues µ 1 , . . . , µ ν sufficiently close to the v alues λ j and λ j , resp ectiv ely ,
w e can guaran tee that the co efficien ts γ i can b e c hosen to b e arbitrarily close to the
co efficien ts β i for i = 1 , . . . , ν , and th us, b
B can b e c hosen to b e of arbitrarily small
norm.
Case 2 : k > 1. By using the result for the already prov ed case 1, we can find a
sequence of k matrices b
U i b
V >
i with b
U i , b
V i ∈ Q n, 1 b eing of arbitrarily small norm suc h
that in
b
A + b
U 1 b
V >
1 + · · · + b
U k b
V >
k
the c hange in the Jordan canonical form with resp ect to the eigen v alue λ from the
matrix b
A + b
U 1 b
V >
1 + · · · + b
U i − 1 b
V >
i − 1 to b
A + b
U 1 b
V >
1 + · · · + b
U i b
V >
i is that the largest t w o
Jordan blo c k asso ciated with λ disapp ear from the Jordan canonical form while all
smaller ones remain (or λ is no longer an eigen v alue if there w ere at most t wo Jordan
blo c ks left in the previous step), and all newly generated eigen v alues are simple. In
particular, b
A + b
U 1 b
V >
1 + · · · + b
U k b
V >
k then has the Jordan canonical form as claimed
in the theorem. If
b
U i = u i 1 u i 2
− u i 2 u i 1 and b
V i = v i 1 v i 2
− v i 2 v i 1
then c ho osing
b
U = u 11 · · · u k 1 u 12 . . . u k 2
− u 12 · · · − u k 2 u 11 · · · u k 1 ∈ Q n,k
11
and
b
V = v 11 · · · v k 1 v 12 . . . v k 2
− v 12 · · · − v k 2 v 11 · · · v k 1 ∈ Q n,k
giv es the desired example, b ecause b
U b
V > = b
U 1 b
V >
1 + · · · + b
U k b
V >
k as one can easily
c hec k.
5. Rank one p erturbations of quaternion matrices. As a direct application
of the main theorem in the previous section, w e immediately obtain the following the-
orem that describ es the generic c hange in the Jordan structure of a giv en quaternion
matrix under a generic rank- k p erturbation.
Theorem 5.1. L et A b e an n × n quaternion matrix, and let its Jor dan c anonic al
form b e given by
r 1
M
i =1 J n i, 1 ( λ 1 ) ! ⊕ · · · ⊕ r p
M
i =1 J n i,p ( λ p ) !
⊕ r p +1
M
i =1 J n i,p +1 ( λ p +1 ) ! ⊕ · · · ⊕ r m
M
i =1 J n i,m ( λ m ) !
wher e λ 1 , . . . , λ p ar e r e al, and λ p +1 , . . . , λ m ar e non-r e al and in the op en upp er half
plane, and wher e for e ach j = 1 , . . . , m the p artial multiplicities ar e or der e d in de-
cr e asing or der: n 1 ,j ≥ · · · ≥ n r j ,j .
Then ther e exists a generic set Ω ⊆ H n,k × H n,k such that for e ach ( U, V ) ∈ Ω
the Jor dan c anonic al form of A + U V T is given by
r 1
M
i = k +1 J n i, 1 ( λ 1 ) ! ⊕ · · · ⊕ r p
M
i = k +1 J n i,p ( λ p ) !
⊕ r p +1
M
i =2 k +1 J n i,p +1 ( λ p +1 ) ! ⊕ · · · ⊕ r m
M
i =2 k +1 J n i,m ( λ m ) ! ⊕ e
J ,
wher e e
J has simple non-r e al eigenvalues not e qual to any of the eigenvalues of A .
Pr o of . The pro of is based on reduction to the complex structured case treated in
the previous section. The matrix ω ( A ) is in the class Q n,n and for U, V ∈ H n,k we
ha v e ω ( U ) , ω ( V ) ∈ Q n,k . Moreo ver, genericit y of subsets H n,k × H n,k with resp ect to
the four real comp onen ts of eac h matrix pair means exactly the same as genericit y of
the corresp onding subset Q n,k × Q n,k with resp ect to the real and imaginary parts of
eac h matrix pair. Also, w e hav e ω ( A + U V T ) = ω ( A ) + ω ( U ) ω ( V ) T b y (2.2).
Observ e that the Jordan canonical form giv en in this theorem leads to the Jordan
canonical form of the matrix ω ( A ) as giv en in Theorem 4.1. Applying the results of
that theorem, and translating bac k via ω − 1 w e see that for a non-real eigenv alue λ of
A the partial m ultiplicities of A + U V T corresp onding to λ are giv en b y the (2 k + 1)st
and follo wing partial m ultiplicities of A corresp onding to λ (if an y), while for a real
eigen v alue λ of A the partial m ultiplicities of A + U V T corresp onding to λ are giv en
b y the ( k + 1)st and follo wing partial multiplicities of A corresponding to λ (if any).
So non-real eigen v alues lose the largest 2 k partial m ultiplicities, but real eigen v alues
only the largest k ones. In addition, eigen v alues of A + U V T whic h are not eigen v alues
of A are simple and non-real.
12
6. Rank- k p erturbations of Hermitian quaternion matrices. In this sec-
tion, w e will fo cus on Hermitian quaternion matrices, i.e., matrices A ∈ H n,n satisfying
A ∗ = A . In that case, the corresp onding matrix ω ( A ) ∈ Q n,n is a complex Hermitian
matrix, and consequen tly all its eigen v alues are real, and all its partial m ultiplicities
are equal to one. Since it is a matrix in Q n,n , the geometric m ultiplicit y of eac h
eigen v alue of ω ( A ) is ev en.
While the result on the generic b eha vior of Hermitian matrices in H n,n under
arbitrary p erturbations still follo ws from Theorem 5.1, it is a natural question to ask
whether this remains true under structur e-pr eserving transformations. Observ e that
a rank- k Hermitian quaternion p erturbation of A tak es the form A + B with B ∈ H n,n
b eing Hermitian and of rank k . Th us, in Q n,n w e should b e considering ω ( A ) + ω ( B )
with ω ( B ) b eing Hermitian of rank 2 k .
Note that a-priori this is a restriction on the t yp e of rank-2 k Hermitian p ertur-
bations w e are allo w ed to make to ω ( A ), b ecause our p erturbation matrix does not
only ha v e to b e Hermitian, but also to b e in the range of ω . Indeed, for a 2 n × 2 n
Hermitian matrix of rank 2 k it is p ossible that the n um b er of p ositiv e (or the n um b er
of negativ e) eigen v alues is o dd, but a matrix in Q n,n alw ays has eigen v alues with ev en
geometric m ultiplicities. W e therefore start our in v estigation by c haracterizing the set
of Hermitian matrices of rank 2 k that are in the range of ω . W e do this b y using the
follo wing results that generalize the w ell-kno wn results on the sp ectral decomp osition
and Sylv ester’s La w of Inertia to the case of Hermitian quaternion matrices.
Pr oposition 6.1 ( [20, Theorem 5.3.6.(c) and Theorem 4.1.6.(a)] ).
L et A ∈ H n,n b e Hermitian. Then the fol lowing statements hold.
1. Ther e exists a unitary matrix Q ∈ H n,n such that
Q ∗ AQ = diag( α 1 , . . . , α n ) ,
wher e α 1 , . . . , α n ∈ R .
2. Ther e exists an invertible matrix S ∈ H n,n and uniquely define d inte gers π , ν
such that
S ∗ AS =
I π 0 0
0 − I ν 0
0 0 0
. (6.1)
Note that 1) confirms our observ ation at the b eginning of this section that the
eigen v alues of a Hermitian quaternion matrix are all real and semisimple. P art 2)
immediately yields a c haracterization of Hermitian quaternion matrices of rank k .
Cor ollar y 6.2. L et A ∈ H n,n b e a Hermitian quaternion matrix of r ank k .
Then ther e exists an inte ger π and a matrix U ∈ H n,k of ful l r ank such that
A = U Σ U ∗ , wher e Σ = I π 0
0 − I k − π . (6.2)
Pr o of . This follo ws immediately from part 2) of Prop osition 6.1 b y noting that A
is of rank k if and only if π + ν = k in (6.1). The result then follo ws from letting U
b e the part of S ∗ that consists of its first k columns.
Cor ollar y 6.3. L et b
A ∈ Q n,n b e Hermitian and of r ank 2 k . Then ther e exists a
matrix b
U ∈ Q n,k of ful l r ank 2 k and a diagonal matrix b
Σ ∈ Q n,n satisfying b
Σ 2 = I 2 n
such that b
A = b
U b
Σ b
U ∗ .
13
Pr o of . The result follo ws immediately by using Corollary 6.2 on A := ω − 1 ( b
A ) to
obtain a decomp osition A = U Σ U ∗ as in (6.2). Then applying ω yields the desired
decomp osition with b
U = ω ( U ) and b
Σ = ω (Σ) = Σ ⊕ Σ.
W e now obtain the follo wing result on generic rank-2 k p erturbations of Hermitian
matrices in Q n,n .
Theorem 6.4. L et b
A ∈ Q n,n b e Hermitian, and let λ 1 , . . . , λ p b e the p airwise
distinct (ne c essarily r e al) eigenvalues of b
A , with multiplicities 2 r 1 ,..., 2 r p (wher e ne c-
essarily the algebr aic multiplicities c oincide with the ge ometric multiplicities). F ur-
thermor e, let b
Σ ∈ Q k ,k b e diagonal such that b
Σ 2 = I 2 k . Then ther e exists a generic
set Ω ⊆ Q n,k such that for al l b
U ∈ Ω the fol lowing statements hold:
1. F or al l j ∈ { 1 , . . . , p } the eigenvalue λ j of b
A + b
U b
Σ b
U ∗ has multiplicity 2 r j − 2 k
if r j > k , and if r j ≤ k , then λ j is not an eigenvalue of b
A + b
U b
Σ b
U ∗ .
2. A l l eigenvalues of b
A + b
U b
Σ b
U ∗ which ar e not eigenvalues of b
A have multiplicity
pr e cisely two.
Pr o of . W e will apply Theorem 3.2 for the case κ = 2 k and ` = 2, and for the
function P = b
U b
Σ b
U ∗ whic h is a p olynomial in the 4 nk real and imaginary part of the
en tries of U 1 , U 2 ∈ C n,k when w e write
b
U = " U 1 U 2
− U 2 U 1 # .
Note that w e are indeed in the case ` = 2, b ecause eac h eigenv alue of a Hermitian
matrix in Q n,k has ev en m ultiplicit y . Th us, it remains to find for eac h eigen v alue λ j
a particular matrix b
U 0 ∈ Q n,k suc h that b
A + b
U 0 b
Σ b
U ∗
0 satisfies parts 3a) and 3b) of
Theorem 3.2 and suc h that the norm of b
U 0 can b e c hosen to b e arbitrarily small.
T o this end w e may assume that k = 1, b ecause as in the pro of of Theorem 4.1 an
example in the case k > 1 can b e constructed via k consecutiv e rank-2 p erturbations
from Q n,k . F urthermore, w e may assume without loss of generalit y that b
A has the
form
b
A = diag( α 1 , . . . , α n , α 1 , . . . , α n ) ,
where α 1 = λ j . Then choosing the v alue c ∈ R \ { 0 } suc h that α 1 + c and α 1 − c
are differen t from all the other eigen v alues of b
A , setting u 0 := ce 1 + ce n +1 ∈ Q n, 1 ,
and considering b
A + u 0 b
Σ u ∗
0 = b
A ± u 0 u ∗
0 yields the desired example as c can clearly b e
c hosen suc h that k u 0 k has arbitrarily small norm.
An immediate consequence is the follo wing result on rank- k p erturbations of Her-
mitian quaternion matrices whic h will b e pro v ed completely analogously to Theo-
rem 5.1.
Theorem 6.5. L et A ∈ H n,n b e a Hermitian and let λ 1 , . . . , λ p its p airwise
distinct (ne c essarily r e al) eigenvalues with multiplicities r 1 , . . . , r p (wher e algebr aic
and ge ometric multiplicities c oincide). F urthermor e, let Σ = I π ⊕ ( − I k − π ) . Then
ther e exists a generic set Ω ⊆ H n,k such that for al l U ∈ Ω the fol lowing statements
hold:
1. F or al l j ∈ { 1 , . . . , p } the eigenvalue λ j of A + U Σ U ∗ has multiplicity r j − k
if r j > k , and if r j ≤ k , then λ j is not an eigenvalue of A + U Σ U ∗ .
2. A l l eigenvalues of A + U Σ U ∗ which ar e not eigenvalues of b
A ar e simple.
14
7. Rank- k p erturbations of sk ew-Hermitian quaternion matrices. In
this section, we will focus on skew-Hermitian quaternion matrices, i.e., quaternion ma-
trices A ∈ H n,n satisfying A ∗ = − A . Again, the corresp onding matrix ω ( A ) ∈ Q n,n
will ha v e the corresp onding structure, i.e., it will b e sk ew-Hermitian. It is imp ortan t to
note that a common tric k that is used in complex matrix algebra is no longer a v ailable
when dealing with the quaternions: while a complex Hermitian matrix becomes skew-
Hermitian when it is m ultiplied b y the imaginary unit i and vice v ersa, this need not
b e the case for a Hermitian matrix A ∈ H n,n , b ecause w e obtain ( i A ) ∗ = A ∗ i ∗ = − A i
and the matrix A i ma y b e differen t from i A as the follo wing example sho ws.
Example 7.1. Consider the matrix
A = 1 j
− j 1 .
Then A ∗ = A is Hermitian, but i A is not sk ew-Hermitian as
( i A ) ∗ = i k
− k i ∗
= − i k
− k − i 6 = − i − k
k − i = − i A.
When Theorem 6.1 is adapted to the sk ew-Hermitian case one should ha v e in
mind that b y the previous example the transition from Hermitian matrices to sk ew-
Hermitian matrices is not a trivial task. Nev ertheless, observ e that part (a) in the
follo wing theorem lo oks exactly lik e the corresp onding results on complex matrices
that can b e obtained from the corresp onding result on Hermitian matrices via the
“m ultiplying with i ”-tric k. On the other hand, comparing the parts (b) w e see that
in the sk ew-Hermitian case “ Sylvester’s L aw of Inertia ” turns out to b e substan tially
differen t from the corresp onding result in the case of Hermitian quaternion matrices.
Pr oposition 7.2 ( [20, Theorem 5.3.6.(d) and Theorem 4.1.6.(b)] ).
L et A ∈ H n,n b e skew-Hermitian. Then the fol lowing statements hold.
1. Ther e exists a unitary matrix Q ∈ H n,n such that
Q ∗ AQ = diag( i α 1 ,..., i α n ) ,
wher e α 1 , . . . , α n ∈ R .
2. Ther e exists an invertible matrix S ∈ H n,n and a uniquely define d inte ger r
such that
S ∗ AS = i I r 0
0 0 . (7.1)
As a corollary of Theorem 7.2, w e immediately obtain the follo wing c haracteriza-
tions of sk ew-Hermitian matrices of rank k in H n,n or rank 2 k in Q n,n .
Cor ollar y 7.3. L et A ∈ H n,n b e a skew-Hermitian quaternion matrix of r ank
k . Then ther e exists a matrix U ∈ H n,k of ful l r ank such that A = U ( i I k ) U ∗ .
Cor ollar y 7.4. L et b
A ∈ Q n,n b e a skew-Hermitian quaternion matrix of r ank
2 k . Then ther e exists a matrix b
U ∈ Q n,k of ful l r ank 2 k such that
b
A = b
U ω ( i I k ) b
U ∗ = b
U i I k 0
0 − i I k b
U ∗ .
15
It is easily seen that the “m ultiplying with i ”-tric k will also not work in the set
Q n,n as this set is only closed under scalar m ultiplication with real n umbers. W e
therefore need an analogue of Theorem 6.4 for the case of sk ew-Hermitian matrices.
Theorem 7.5. L et b
A ∈ Q n,n b e skew-Hermitian, and let λ 1 , . . . , λ p b e the
p airwise distinct (ne c essarily pur ely imaginary) eigenvalues of b
A , with multiplicities
r 1 , . . . , r p (wher e ne c essarily the algebr aic multiplicities c oincide with the ge ometric
multiplicities, and r j is even if λ j = 0 ). Then ther e exists a generic set Ω ⊆ Q n,k
such that for al l b
U ∈ Ω the fol lowing statements hold for al l j ∈ { 1 , . . . , p } :
1. If λ j 6 = 0 , then λ j is an eigenvalue of b
A + b
U ( i I k ) b
U ∗ with multiplicity r j − 2 k
if r j > k , and if r j ≤ k , then λ j is not an eigenvalue of b
A + b
U ( i I k ) b
U ∗ .
2. If λ j = 0 , then λ j is an eigenvalue of b
A + b
U ( i I k ) b
U ∗ with (ne c essarily even)
multiplicity r j − 2 k if r j > 2 k , and if r j ≤ 2 k , then λ j is not an eigenvalue
of b
A + b
U ( i I k ) b
U ∗ .
3. A l l eigenvalues of b
A + b
U ( i I k ) b
U ∗ which ar e not eigenvalues of b
A ar e nonzer o
and simple.
Pr o of . W e will apply Theorem 3.2 for the case κ = 2 k and ` = 1, and for the
function P = b
U ( i I k ) b
U ∗ whic h is a p olynomial in the 4 nk real and imaginary part of
the en tries of U 1 , U 2 ∈ C n,k when w e write
b
U = " U 1 U 2
− U 2 U 1 # .
Indeed, note that in con trast to the Hermitian case we ha ve ` = 1 instead of ` = 2.
The remainder of the pro of uses the same strategy as the pro of of Theorem 6.4. Th us,
w e ma y again assume that k = 1 and that A is diagonal, i.e.,
A = diag( i α 1 ,..., i α n , − i α 1 ,..., − i α n ) ,
where i α 1 = · · · = i α r j = λ j if λ j 6 = 0, or i α 1 = · · · = i α r j / 2 = 0 if λ j = 0. It remains to
find one particular matrix b
U ∈ Q n, 1 suc h that its norm can b e c hosen to b e arbitrarily
small and suc h that 3a) and 3b) of Theorem 3.2 are satisfied. Constructing suc h an
example is the part where the pro of of this theorem will differ substan tially from the
corresp onding part of the pro of of Theorem 6.4. W e will distinguish t w o cases.
Case 1: λ j = 0. In this case, let
b
U = ce 1 i ce 1
i ce 1 ce 1 ∈ Q n, 1 ,
where c ∈ R . Then we obtain
b
P := b
U i 0
0 − i b
U ∗ = 0 2 c 2 e 1 e T
1
− 2 c 2 e 1 e T
1 0 ,
and the eigen v alues of b
A + b
P are giv en b y the v alues ± i α 2 ,..., ± i α n and in addition
b y the eigen v alues of
0 2 c 2
− 2 c 2 0
whic h are ± i 2 c 2 . Th us, if the sufficien tly small c is c hosen such that 2 c 2 is differ-
en t from ± α 2 ,..., ± α n then w e ha v e found our example such that 3a) and 3b) of
Theorem 3.2 are satisfied.
16
Case 2: λ j 6 = 0. If r j = 1, then c ho osing the same p erturbation as ab o ve will pro-
duce a p erturb ed matrix with the eigen v alues ± i α 2 ,..., i α n and ± i p α 2
1 + 4 c 4 whic h
sho ws that λ j is not an eigen v alue of the p erturb ed matrix. F urthermore, c ho osing c
appropriately guaran tees that the newly generated eigen v alues are all simple. Th us,
let r j > 1 whic h implies i α 2 = λ j . No w c ho ose c > 0 sufficien tly small suc h that in
particular w e ha v e α 2
1 − c 2 > 0 and set
b
P = i ce 1 − i ce 2 ce 2 ce 1
− ce 2 − ce 1 − i ce 1 i ce 2 .
Then w e ha v e b
P ∈ Q n,n and in addition b
P is sk ew-Hermitian and of rank 2. Note
that in b
A + b
P only the 4 × 4 submatrix is p erturb ed that consists of the ro ws and
columns with indices 1 , 2 , n + 1 , n + 2 and whic h is giv en b y
i ( α 1 + c ) 0 0 c
0 i ( α 1 − c ) c 0
0 − c − i ( α 1 + c ) 0
− c 0 0 − i ( α 1 − c )
.
Since the eigen v alues of this submatrix are the four pairwise distinct complex n um b ers
i c ± p α 2
1 − c 2 and − i c ± p α 2
1 − c 2 whic h are clearly also m utually distinct from the
v alues ± i α 3 ,..., ± i α n if c had b een c hosen sufficiently small, w e ha v e constructed our
desired example that can also b e constructed to b e of arbitrarily small norm.
Note that the additional statemen t that the newly generated eigen v alues are
nonzero is implied b y their simplicit y , since the eigen v alue zero m ust ha v e even m ul-
tiplicit y as a real eigen v alue.
As a direct consequence of Theorem 7.5, w e obtain the follo wing analogue of
Theorem 6.5 and its pro of is again analogous to the one of Theorem 5.1.
Theorem 7.6. L et A ∈ H n,n b e a skew-Hermitian quaternion matrix and let
λ 1 , . . . , λ p its p airwise distinct (ne c essarily pur ely imaginary) eigenvalues with multi-
plicities r 1 , . . . , r p . Then ther e exists a generic set Ω ⊆ H n,k such that for al l U ∈ Ω
the fol lowing statements hold for al l j ∈ { 1 , . . . , p } :
1. If λ j 6 = 0 , then λ j is an eigenvalue of A + U ( i I k ) U ∗ with multiplicity r j − 2 k
if r j > 2 k , and if r j ≤ 2 k , then λ j is not an eigenvalue of A + U ( i I k ) U ∗ .
2. If λ j = 0 , then λ j is an eigenvalue of A + U ( i I k ) U ∗ with multiplicity r j − k
if r j > k , and if r j ≤ k , then λ j is not an eigenvalue of A + U ( i I k ) U ∗ .
3. A l l eigenvalues of A + U ( i I k ) U ∗ which ar e not eigenvalues of b
A ar e nonzer o
and simple.
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