Real and Complex Hamiltonian Square Ro ots of

SkewHamiltonian Matrices

y z

HeikeFab ender D Steven Mackey Niloufer Mackey

x

Hongguo Xu

January

DedicatedtoProfessor Ludwig Elsner

on the occasion of his th birthday

Abstract

We present a constructive existence pro of that every real skewHamiltonian matrix

W has a real Hamiltonian square ro ot The key step in this construction shows how

one may bring anysuch W into a real quasiJordan canonical form via symplectic

similarityWe show further that every W has innitely many real Hamiltonian square

ro ots and givealower b ound on the dimension of the set of all such square ro ots

Extensions to complex matrices are also presented

This report is an updated version of the paper Hamiltonian squareroots of skew

Hamiltonian matrices that appearedinLinear Algebra its Applicationsv

pp

AMS sub ject classication A A A F B

Intro duction

Any matrix X suchthatX A is said to b e a square ro ot of the matrix AF or general

nn

complex matrices A C there exists a welldevelop ed although somewhat complicated

theory of matrix square ro ots and a numb er of algorithms for their eective

computation Similarly for the theory and computation of real square ro ots for real

matrices By contrast structured squareroot problems where b oth the matrix A

Fachb ereich Mathematik und Informatik Zentrum fur Technomathematik Universitat Bremen D

Bremen FRG email heikemathunibremende

y

Department of Mathematics Computer Science Kalamazo o College Academy Street Kalamazo o

MI USA

z

Department of Mathematics Statistics Western Michigan University Kalamazo o MI USA

email nilmackeywmichedu This work was supp orted in part by NSF Grant CCR

x

Fakultat fur Mathematik TU ChemnitzZwickau D Chemnitz FRG email

hongguoxumathematiktuchemnitzde This work was supp orted in part by the Chinese National

Natural Science Foundation and by Deutsche Forschungsgemeinschaft Research Grant Me

and its square ro ot X are required to have some extra not necessarily the same sp ecied

structure have b een comparatively less studied Some notable exceptions include p ositive

semidenite square ro ots of p ositive semidenite matrices M matrix square

ro ots of M matrices coninvolutory square ro ots of coninvolutory matrices and

skewsymmetric square ro ots of symmetric matrices In this pap er weinvestigate another

such structured square ro ot problem that of nding real Hamiltonian square ro ots of real

skewHamiltonian matrices

A real n n matrix H of the form

E F

H

T

G E

nn

T T

is said to b e Hamiltonian if E F G IR with F F and G G Equivalently one

maycharacterize the set H of all n n Hamiltonian matrices by

nn

T

H fH IR j JH JHg

I

where J and I is the n n identity matrix Complementary to H is the set

I

nn T

W fW IR j JW JW g

of all skewHamiltonian matrices Matrices in W are exactly those with blo ck structure

A B

W

T

C A

nn

T T

where A B C IR with B B and C C Another useful waytolookat

Hamiltonian and skewHamiltonian matrices is from the p oint of view of bilinear forms

k

Asso ciated with any nondegenerate bilinear form bx y onIR one has the following sets of

matrices

k k k

Ab fS IR j bSx Sy bx y x y IR g

k k k

Lb fH IR j bHxy bx H y x y IR g

k k k

J b fW IR j bWxy bx W y x y IR g

These are resp ectively the automorphism group Lie algebra and Jordan algebra of the

form bItisnow easy to see that H is just the Lie algebra L band W the Jordan algebra

n

T I

J b of the bilinear form bx y x Jy dened on IR by the matrix J

I

The eigenproblem for Hamiltonian matrices arises in a numb er of imp ortant applications

and many algorithms for computing their eigenvalues and invariant subspaces have b een

describ ed in the literature see for references In Van Loan prop osed a metho d

for calculating the eigenvalues of Hamiltonian matrices by rst squaring them Thus he was

led to consider the set

nn

H fN IR j N H H Hg

of all squaredHamiltonian matrices The calculation

n

bH x y bHxHybx H y x y IR

shows immediatelythat H W Indeed the same argumentshows that L b Jbfor

any bilinear form b Almost all the algorithms prop osed byVan Loan in dep end only

on the skewHamiltonian blo ck structure of matrices in H and hence apply equally well to

every matrix in W It is then natural to wonder whether the sets H and W might actually

b e the same

In this pap er weshow that indeed H W or in other words every real skew

Hamiltonian matrix has a real Hamiltonian square ro ot The pro of o ccupies the next three

sections after outlining the strategy of the pro of in Section we fo cus in Sections and

on the main technical result of this pap er a symplectic canonical form for real skew

Hamiltonian matrices Then in Section we consider the square ro ot sets themselves for a

general W W what can b e said ab out the size and top ological nature of the set of all the

real Hamiltonian square ro ots of W We close in Section with results on related structured

square ro ot problems involving complex Hamiltonian and skewHamiltonian matrices

The Generic Case

We b egin by giving a short pro of that almost all real skewHamiltonian matrices ie all

matrices in an op en dense subset of W have a real Hamiltonian square ro ot This preliminary

result serves to make the general case more plausible and at the same time allows us to

intro duce the basic elements and strategy of the general pro of in a setting where there are

no technical details to obscure the main line of the argument

An imp ortantway to exploit the structure of Hamiltonian and sk ewHamiltonian matri

ces is to use only structurepreserving similarities To that end consider the set S of real

symplectic matrices dened by

nn T

S fS IR j S JS J g

Equivalently S is the automorphism group of the bilinear form dened by J It is well

known and easy to show from either denition that S forms a multiplicative group and that

symplectic similarities preserve Hamiltonian squaredHamiltonian and skewHamiltonian

structure for any S S

H H S HS H

N H S NS H

W W S WS W

The rst simplifying reduction we use was intro duced byVan Loan in There he

showed that anyskewHamiltonian W can b e brought to blo ckupp ertriangular form byan

orthogonalsymplectic similarity That is for any W W one can explicitly compute an

orthogonalsymplectic Q such that

U R

nn T

where U R IR Q WQ

T

U

Van Loan actually shows that one can attain an upp er Hessenb erg U with an orthogonal

symplectic similarity however this extra structure will play no role in this pap er

Now supp ose we could somehowcontinue this reduction by not necessarily orthogonal

symplectic similarities all the waytoblockdiagonal form Then the following prop osition

shows that wewould b e done

Prop osition Suppose for W W there exists some S S such that

A

S WS

T

A

nn

with A IR Then W hasareal Hamiltonian squareroot

nn

Pro of Every A IR can b e expressed as a pro duct A FG of two real symmetric

matrices F and G Consequently anyblockdiagonal skewHamiltonian matrix

A

F

has a Hamiltonian square ro ot of the form Then

T

G

A

F F A

S S S S S W S

T

G A G

F

S expresses W as the square of the Hamiltonian matrix S

G

Is there any reason to b elieve that such a symplectic blo ckdiagonalization can b e achieved

for every skewHamiltonian matrix In the sp ecial case of matrices it has b een shown

using quaternions that every skewHamiltonian can be blockdiagonalized in the sense

of Prop osition and this can even b e done byan orthogonalsymplectic similarity Thus

every skewHamiltonian has a Hamiltonian square ro ot For larger matrices it is still not

clear whether blo ckdiagonalization is always p ossible via orthogonalsymplectic similarity

so we turn next to see what can b e achieved with nonorthogonal symplectic similarities

To continue moving forward from V an Loans blo ckupp ertriangular form towards blo ck

diagonal form we try using similarities byblockupp ertriangular symplectics One can verify

nn

VX

directly from the denition that a blo ckupp ertriangular matrix with V X Y IR

Y

T

is symplectic i V is invertible Y V and V X is symmetric The twosimplesttyp es

V

of blo ckupp ertriangular symplectics then are the blo ckdiagonal symplectics and

T

V

nn

T

IX

with I IR and X X We will see that one can go quite the symplectic shears

I

a long way using just these two sp ecial typ es of nonorthogonal symplectic matrices

Now consider the set M of all n n skewHamiltonian matrices whose eigenvalues

eachhavemultiplicity exactly tw o From the Van Loan reduction it is clear that any

eigenvalue of a skewHamiltonian matrix must have even multiplicityso M consists precisely

of those matrices in W whose eigenvalues are of minimal multiplicityThus MW can

nn

b e viewed as the natural skewHamiltonian analog of the subset of matrices in IR with

distinct eigenvalues it should then not b e surprising that M is a dense op en subset of W

with complement WnM of measure zero In this sense wemay regard M as the generic

skewHamiltonian matrices The next prop osition shows that the simple to ols intro duced so

far are already sucient to symplectically blo ckdiagonalize anyskewHamiltonian matrix

in M

Either one of the matrices F or G may also b e chosen to b e nonsingular This extra prop ertyisnot

needed here but plays an imp ortant role later in the pro of of Theorem



Every blo ckupp ertriangular symplectic can b e uniquely expressed as the pro duct of a blo ckdiagonal

symplectic and a symplectic shear although wemake no use of this fact here

Prop osition For any W M there exists an S S such that

A

S WS

T

A

nn

with A IR

Pro of A sequence of three symplectic similarities reduces any W Mto blo ckdiagonal

UR

form First do Van Loans reduction constructing S S so that S WS

T

U

nn

The assumption W M means that U IR has n distinct eigenvalues Next p erform

nn

V

a similarityby a blo ckdiagonal symplectic S cho osing V IR so that

T

V

V UV A is in real Jordan form This gives

A K

S S WS S

T

A

T

with K V RV The blo ckdiagonalization of W is completed by similarity with a

IX

symplectic shear S Wehave

I

T

A AX XA K

S S S WS S S

T

A

All that remains then is to show that for anyskewsymmetric K one can always nd a

symmetric solution X to the Sylv ester equation

T

AX XA K

A

Using such a solution X in the shear S wewillhave S WS with S S S S

T

A

and the prop osition will b e proved

In solving we can make use of the following wellknown and fundamental fact ab out

k k k

Sylvester equations of the form AX XB Y where A IR B IR and Y IR

whenever the sp ectra of A and B are disjoint then the equation AX XB Y has a

k k

for any Y IR see Prop osition in x To bring this result unique solution X IR

into playtosolve partition A X and K into blo cks compatible with the blo ckdiagonal

structure of A Since A is the real Jordan form of a matrix with distinct eigenvalues wecan

write A A A A where each A is or and any diagonal

mm ii

a b

blo ck A has the form with b

ii

ba

th T

With X and K partitioned conformally with A observe that the ij blo ckofAX XA

th

dep ends only on the ij blo ckofX

T T

AX XA A X X A i j m

ij ii ij ij jj

T

Thus the equation AX XA K decomp oses blo ckwise into m independent subproblems

T

A X X A K i j m

ii ij ij jj ij

Among the diagonalblo ck subproblems i j there are two cases to consider Whenever

A is equation collapses to a scalar equation and any real X is a solution When

ii ii

a b

A is equation can b e solved by a simple computation Let A with

ii ii

ba

xy

k

b X Then and K

yz

ii ii

k

x z

T

A X X A b

ii ii ii ii

x z

so x kb together with y z provides one of many p ossible symmetric solutions X

ii

For each of the odiagonalblo ck subproblems i j the ab ove fundamental fact guarantees

the existence of a unique solution X to Taking transp ose of b oth sides of shows

ij

T

andthus t together compatibly to form a that these blo ckwise solutions satisfy X X

ij ji

symmetric solution X for

remarks

This pro of highlights the imp ortance of the solvabilityofvarious typ es of Sylvester

equations esp ecially

T

AX XA Y

for the symplectic blo ckdiagonalization problem By extending the argument used

ab ove we will see in the next section howtocharacterize the set of matrices A for

which has a symmetric solution X for every skewsymmetric Y As one might

guess from Prop osition among such As are all matrices with distinct eigenvalues

The counterexample A I suggests that multiple eigenvalues cause diculties but

that is not always the case It turns out that the problem is not multiple eigenvalues

p er se but rather multiple Jordan blo cks see Prop osition in x

In light of the rst remark wecannow see that the second step of the ab ove reduction

to blo ckdiagonal form similaritybytheblockdiagonal symplectic S is unnecessary

UR

For matrices in M one can always go directly from the Van Loan reduced form

T

U

U

IZ

to blo ckdiagonal form Sucha via similarityby some symplectic shear

T

I

U

T

similarity leads to the equation UZ ZU R where U has distinct eigenvalues

so it will have a symmetric solution Z for anyskewsymmetric R With V and X

T

dened as in the pro of of Prop osition the matrix Z VXV is one such solution

although there are many others in fact there is a whole ndimensional ane subspace

of symmetric solutions

Now that weknowthat H contains an op en dense subset of W it is natural to consider

trying the usual kind of analytic argument to complete the pro of that H W That is

we could approximate an arbitrary skewHamiltonian W by a sequence W W with

i

by Prop osition and then try to W Mpick Hamiltonian square ro ots H for each W

i i i

show that the set fH g has some limit p oint H Any such H would b e a Hamiltonian square

i

ro ot of W Noweven though each W M has innitely many Hamiltonian square ro ots it

i

is not immediately evident that one can always cho ose the H so as to guarantee the existence

i

of any limit p oints at all for fH g Instead of pursuing this analytic line of attack wewill

i

continue with a more algebraic approach showing that the symplectic blo ckdiagonalization

result of Prop osition can b e extended to all of W Indeed wewillprove the following

canonical form result whichmay itself b e of some indep endentinterest

Theorem Every real skewHamiltonian matrix can bebrought into skewHamiltonian

Jordan form via symplectic similarity That is for any W W there exists an S S such

that

A

S WS

T

A

nn

where A IR is in real Jordan form The matrix A is unique up to a permutation of

real Jordan blocks

As an immediate corollary we then have

Theorem Every real skewHamiltonian matrix has a real Hamiltonian squareroot In

other words H W

The main goal of the next two sections of the pap er is to prove Theorem Unfortunately

the p otential presence of nontrivial Jordan structure in the general skewHamiltonian ma

trix intro duces diculties which cannot b e handled using only symplectic shears although

they still have an imp ortant role to play We b egin with some technical results concern

ing the detection and manipulation of Jordan structure and further results ab out Sylvester

equations

Auxiliary Results

Sylv ester Equations

In the pro of of Prop osition wehave seen that the eect of similarityby a symplectic shear

on a blo ckupp ertriangular skewHamiltonian matrix is simply to replace the blo ck K

T

by the expression AX XA K

T

I X A K I X A AX XA K

T T

I A I A

It is imp ortant for the symplectic reduction of general skewHamiltonian matrices to struc

tured Jordan form to nd out how far it is p ossible to simplify various typ es of such matrix

expressions These simplication questions can b e concisely expressed in terms of the corre

k k

sp onding Sylvester op erators so let us intro duce the following notation Supp ose A F

and B F are xed but arbitrary square matrices with entries in the eld F Then we

denote by SylA B the linear Sylvester op erator

k k

SylA B F F

X AX XB

In this section wecharacterize the range of several typ es of such op erators b eginning with

awellknown result referred to earlier in the pro of of Prop osition Knowing the range of

an op erator SylA B enables us immediately to see howmuch it is p ossible to simplify the

Sylvester expression AX XB Y for an arbitrary Y

Although the pro of of Theorem involves only real Sylvester op erators and the simpli

cation of their asso ciated Sylvester expressions it is often convenienttoprove results rst

for complex op erators and then derive the real case from the complex case Much of this

section follows that pattern In order to pro ceed directly to our main result in x the pro ofs

of Prop ositions and will b e deferred to an app endix WeuseF here to denote C or IR

k k

Prop osition Let A F and B F Then the operator SylA B is nonsingular

i the spectra A and B are disjoint subsets of C

can b e found in many places eg Pro of Pro ofs of this result for F C

When A and B are real T SylA B maybeviewed either as a real op erator T

R

or as a complex op erator T Since T is the complexication of T wehave

C C R

dim ker T dim ker T Thus T is nonsingular i T is nonsingular

C C R R R C

Next wecharacterize the range of one of the simplest typ es of singular Sylvester op erator

with AB Let N denote the k k nilp otent matrix and

k

M I N denote the k k Jordan blo ck corresp onding to the eigenvalue We

k k k

th k

also need the notion of the m antidiagonal of a matrix Y F bywhich is meant the

set of all entries Y such that i j m Note that a k matrix Y has a total of

ij

th k

k antidiagonals Since the collection of the m antidiagonals of Y F with

m k k k d plays a particularly imp ortant role in the following

result we refer to this collection as the last d antidiagonals of Y

T

erator SylA B where A M and B M are Prop osition Consider the op

k

Jordan blocks corresponding to the same eigenvalue F Let d mink Then the

T k

range of SylA B consists of al l Y F such that the sum of the entries along each of

T

the last d antidiagonals of Y is zero Thus dim range SylA B k d

F

Many of the Sylvester expressions AX XB Y arising in the pro of of Theorem

require simplication with a symmetric X not just with an arbitrary unstructured X This

IX

is b ecause shears are symplectic i X is symmetric W e address this situation in the

I

next prop osition But rst a little more notation let FSymn and FSkewn denote the

nn T T

sets of all matrices X F such that X X and X X resp ectively Also recall

nn

that a matrix A F is said to b e nonderogatory if the complex Jordan form of A

has exactly one Jordan blo ck for eacheigenvalue

nn T

Prop osition For A F consider the operator SylA A with domain and codomain

restrictedtoFSymn and FSkewnrespectively That is consider

T FSymn FSkew n

A

T

XA X AX

Then T is onto A is nonderogatory

A

The nal result we need is the real analog of Prop osition for complexconjugate eigen

k

T

value pairs Our goal is to characterize the range of real op erators SylA B IR

k

IR in completely real terms when A and B are real Jordan blo cks corresp onding to

the same complexconjugate eigenvalue pair Toachieve this we need some preliminary

denitions and simple facts ab out real matrices and their relations to complex

matrices

Consider the centralizer C and anticentralizer A of J dened by

a b

j a b IRg C fX IR j JX XJ g f

ba

cd

and A fX I R j JX XJ g f j c d IRg

d c

Then the following lemma is straightforward to prove

Lemma

IR C A

p

Every matrix in C is diagonalized by similarity with the unitary matrix

ii

aib

H a b

That is

aib

ba

H

cd

Every matrix in A is antidiagonalized by similarity with That is

d c

cid

cid

uv

g is a real subalgebraof C The set U f and the map j u v C

v u

IR U

H

X X

is an algebra isomorphism

With these facts in hand wecannowprove the following prop osition

k k

T

Prop osition Consider the operator SylA B where A IR and B IR are

both real Jordan blocks corresponding to the same complexconjugate eigenvalue pair a ib

That is both A and B are of the form

I

a b

where with b

b a

I

T

The range of SylA B can be characterized as fol lows Let d mink Partition

Y Y

k

Y IR into blocks Y IR so that Y Let the set of al l

ij

Y Y

k k

th

blocks Y such that blockantidiagonal of Y Then the range of i j m becal led the m

ij

k

T

SylA B consists of al l Y IR such that the sum of the A components of the blocks

along each of the last d blockantidiagonals of Y is

From the ab ovecharacterizations of the ranges of Sylvester op erators we can now see

exactly howmuch it is p ossible to simplify the four typ es of Sylvester expression AX XB Y

app earing in the pro of of Theorem Each simplication result is an immediate consequence

of the indicated prop osition

Prop osition

k k k

a Suppose A IR and B IR have disjoint spectra Then for any Y IR there

k

exists a unique X IR such that AX XB Y Proposition

k k

b Suppose A IR isareal Jordan block corresponding to either a real eigenvalue or a

k k

thereexist complexconjugate eigenvalue pair Then for any skewsymmetric Y IR

k k

T

innitely many symmetric X IR such that AX XA Y Proposition

k k

c Suppose A IR and B IR are Jordan blocks corresponding to the same real

k

eigenvalue Let d mink Then for any Y IR there exist innitely many

k

T

X IR such that AX XB Y is zero everywhere except possibly in the last d

entries of the bottom row Proposition

k k

d Suppose A IR and B IR arereal Jordan blocks corresponding to the same

k

complexconjugate eigenvalue pair Let d mink Then for any Y IR there

k

T

exist innitely many X IR such that AX XB Y is zero everywhere except

possibly in the last d blocks of the bottom row These last d blocks are

al l elements of A Proposition

Jordan Structure

An imp ortant step in the pro of of Theorem concerns certain blo ckdiagonal matrices B

e

ks and p erturbations B B C that dier from each other only in a single column of blo c

p

e

We need to compare the maximum Jordan blo cksizeofsuch pairs B and B The results in

this section address this question

nn

with eigenvalue itiswellknown that the Jordan structure For a matrix A C

of A corresp onding to can b e deduced from the ranks of the p owers of A I That is if

k

r rankA I then the numb er and sizes of all the Jordan blo cks asso ciated with

k

are completely determined by the sequence of numb ers r r r For our purp oses we

n

need only the following basic result

nn

Prop osition Suppose A C is a matrix with exactly one eigenvalue Letting r

k

k

rankA I there is an integer s with s n such that n r r r

s

r r The largest Jordan block of A has size s s

s n

Next consider matrices of the form

B F

B

B

p

B C

p

B

p

B

p

B

q

where B diag B B B C is zero everywhere except p ossibly in the odiagonal

q p

th

blo cks of the p column of blo cks and stands for an arbitrary matrix of the appropriate

size Observethatifwe x the sizes of the diagonal blo cks and the column p then the set

of matrices of the form is closed under multiplication Wehave the following two results

for certain sp ecial matrices of this form

nn

is a matrix of the form satisfying the fol lowing Prop osition Suppose A C

conditions

n n

k k

Each B C with k p is a Jordan block M corresponding to the same

k n

k

n n

p p

eigenvalue B C is the transpose of a Jordan block corresponding to ie

p

T

B M

p n

p

B is the largest block on the diagonal of A so that n n for al l k

k

The nonzero odiagonal blocks are not in the rst column ie p The top

n n

th

p

most block of the p column of blocks F C isoftheform where f

f

n

p

is nonzero f f C

n

p

Then is the only eigenvalue of A and in the Jordan canonical form of A thereisatleast

one Jordan block with size bigger than n n Hence the largest Jordan block of A is strictly

bigger than the largest Jordan block of B

Pro of That is the only eigenvalue of A follows immediately from partitioning A into

A A



blo ckupp ertriangular form A where A diagB B B In order

p

A



to establish the claim ab out Jordan blo ck size it suces by Prop osition to show that

n n

or equivalently that A I Thus we consider p owers of rankA I

T

N N C N A I diagN

n n p n n

p q



k

All p owers A I are of the form and for convenience we designate the topmost blo ck

th k k

in the p column of blo cks of A I by F Then it is easy to see inductively that

k

k

N

F

n

k

T k

N

A I

n

p

k

N

n

q

k

where F satises the recurrence

k k k T

F F F N F F N

n n

p

From this recurrence we deduce that

f f

n

p

n

F

f

n

p

n n n

p p

n

isaHankel matrix with f as the top row Since f is nonzero so is F and hence also

n

A I

To complete this section we establish a real analog of Prop osition for matrices of the

form where each B is a real Jordan blo ck corresp onding to the same complexconjugate

k

eigenvalue pair We employ the same sort of strategy as in the pro of of Prop osition

rst convert the real Jordan blo cks to complex Jordan blo cks by an appropriate similarity

then apply Prop osition to the resulting complex matrix and nally translate backinto

completely real terms

nn

Prop osition Suppose L IR is a matrix of the form satisfying the fol lowing

conditions

n n

k k

with k p is a re Each B IR al Jordan block corresponding to the complex

k

n n

p p

is the transpose of a a ib a ib B IR conjugate eigenvalue pair

p

T

real Jordan block corresponding to In other words B and B with k p have

p

k

the form as describedinProposition

n n

is the largest block on the diagonal of Lson n for al l k B IR

k

The nonzero odiagonal blocks are not in the rst column ie p When each

th

block in the p column of C is partitionedinto subblocks then every such

p

th

subblock is an element of A The topmost block of the p column of blocks

n n n

p p

has the form F IR is nonzero and where g g g IR

n

g

p

g A for i n

i p

Then and are the only eigenvalues of L and in the real Jordan canonical form of L

thereisatleast one real Jordan block with size bigger than n n Hence the largest real

Jordan block of L is strictly bigger than the largest real Jordan block of B

Pro of Recall the unitary matrix P dened in the pro of of Prop osition

n n n

From the discussion there of the eect of similarityby on real matrices we see that

n

nn

UV

H

b

L L will b e of the form with U V C More sp ecically

n n

V U

T

M M U diag M M

n n n n

q p



where M M denotes the n n Jordan blo ckfor a ib The matrix V

n n k k

k k

partitioned conformally with U has nonzero entries only in the odiagonal blo cks of the

th

p column of blo cks ie

b

F

V

h i

n n

th

p

b b b

b

of this p column has the form where f f f The topmost blo ck F C

b

n

p

f

n

p

C is nonzero

A nal p ermutation similarity shifts these nonzero blo cks of V into U and thus blo ck

CS

diagonalizes L Letting P with

SC

C diag I I I I and

n n n n n

p q

p p

S diag I

n n n n n

p q

p p

A

T

b

wehave P LP where

A

b

M F

n

M

n



A

T

M

n

p

M

n

q

is exactly the typ e of matrix considered in Prop osition Thus A and A have only the

resp ectively and each has at least one Jordan blo ck bigger than n n eigenvalues and

Consequently L has only the eigenvalues and and at least one real Jordan blo ckbigger

than n n

SkewHamiltonian Jordan Form

The results of x provide us with all the technical to ols needed to show that every real skew

Hamiltonian matrix can b e symplectically broughtinto structured real Jordan form This is

the content of Theorem whichwe recall now and prove

Theorem For any W W there exists an S S such that

A

S WS

T

A

nn

where A IR is in real Jordan form The matrix A is unique up to a permutation of

real Jordan blocks

Pro of Begin as in Prop osition with Van Loans reduction constructing an orthogonal

UR

symplectic S S so that S WS Next p erform a similarity with a blo ck

T

U

nn

V

diagonal symplectic S whereV IR is chosen so that V UV D is in

T

V

DK

T

WS S where K V RV Wewill S real Jordan form Then wehave S

T

D

assume that the real Jordan blo cks of D corresp onding to the same real eigenvalue or to

the same complexconjugate eigenvalue pair have all b een group ed together into blo cks B

i

that is we can write D diagB B B where

i the sp ectrum of each B is either a single real numb er or a single complexconjugate

i

pair and

ii distinct blo cks B and B have disjoint sp ectra

i j

IX

With a symplectic shear we can nowblockdiagonalize K Recall that the eect

I

of similarityby a symplectic shear on a blo ckupp ertriangular skewHamiltonian matrix

DK

T

is simply to replace K by the expression DX XD K Thus we wish to nd

T

D

T

a symmetric X so that DX XD K is a blo ckdiagonal matrix conformal with D To

build suchan X start by partitioning K and X into blo cks conformal with the direct

sum decomp osition D B B B The blo ckdiagonal nature of D means that

T

DX XD K may b e handled blo ckwise

T T

DX XD K B X X B K i j

ij i ij ij j ij

Since B and B have disjoint sp ectra for any i j weknow from Prop osition a that

i j

T

e e e

there is a unique X such that B X X B K Transp osing this equation and

ij

i ij ij j ij

T T

e e e

X X Letting X invoking the skewsymmetry of K ie K K shows that

ii

ji ij ij ji

e e

for i we see that the blo cks X t together to form a symmetric matrix X The

ij

e

I X

corresp onding symplectic shear S gives us

I

D K

diag

S S S WS S S

T

D

where K diag K K K The problem of symplectically blo ckdiagonalizing an

diag

arbitrary real skewHamiltonian is thus reduced to that of symplectically blo ckdiagonalizing

h i

B K

i ii

ewHamiltonian matrices whose degenerate skewHamiltonian matrices that is sk

T

B

i

sp ectrum consists either of a single real number typ e or a single complexconjugate pair

typ e

Up to this p oint the pro of of Theorem is essentially the same as the pro of of Prop osi

tion In the generic class M of skewHamiltonians considered in Prop osition however

the B were only or blo cks and the corresp onding degenerate subproblems could b e

i

handled directly and explicitly in an elementary manner It is in blo ckdiagonalizing larger

degenerate subproblems that the chief technical diculty of the general case lies Symplectic

shears alone cannot in general b e sucient for this task b ecause the real Jordan structure

i h i h

B B K

i i ii

may dier from that of This is where the results of xcomeinto play of

T T

B B

i i

To complete the pro of of Theorem we describ e an iterative pro cedure terminating in a

nite numb er of steps which brings any degenerate skewHamiltonian matrix into structured

skewHamiltonian Jordan form We concentrate on the typ e case matrices with a single

It is easy to see that the following argument complexconjugate eigenvalue pair

will also work for the typ e case simply by replacing Prop osition d with Prop osition c

BK

and Prop osition with Prop osition Let us supp ose then that is a degenerate

T

B

skewHamiltonian matrix where B diagA A A is in real Jordan form Each A

q k

n n

k k

IR is a real Jordan blo ck of the form and A is the largest suchblock

We b egin with the termination case for this pro cedure When B has only one real Jordan

blo ck then blo ckdiagonalization can b e achieved in one step By Prop osition b there exists

T

a symmetric X suchthat BX XB K Thus similarityby the symplectic shear

B BK

IX

andwe are done into using this X transforms

T T

I

B B

Now supp ose that B has more than one Jordan blo c k We dene a twostep reduction

BK

pro cess to simplify not necessarily all the waytoblockdiagonal form but at least

T

B

bringing it closer to structured Jordan form

STEP Simplication of K

IX

Here we simplify K as much as p ossible using only a symplectic shear

I

Begin by partitioning K and X conformally with the real Jordan decomp osition

of B Now simplify K blo ckwise replacing eachblock K by the expression

ij

T

A X X A K Y where X is chosen to pro duce a Y with as many

ij ij

i ij ij j ij ij

zero es as p ossible By Prop osition b eachblock K on the diagonal of K can

ii

b e zero ed out completely In general the odiagonal blo cks K i j cannot

ij

b e completely zero ed out in this way but Prop osition d shows what we can

b e sure of achieving For blo cks K ab ove the diagonal i jcho ose X so

ij ij

that all entries are zero ed out except p ossibly for the last d blo cks of

the b ottom row In other words each Y with i j has the form Y

ij ij

g

n

j

where g g g IR and g A for i n For blo cks K

n i j ji

j

T

so that X will b e symmetric and b elow the diagonal jiwecho ose X X

ij ji

T

Y Y This zero es out all entries of each K ji except p ossibly for the

ji

ji ij

b ottom d blo cks of the last column Note that these blo cks are

BK

also elements of A since any g A is symmetricThus we simplify to

T

i

B

BY

IX

where Y has the form via similarityby the symplectic shear

T

I

B

Y Y

q

T

Y

Y

Y

q q

T T

Y Y

q q q

STEP Transfer of Jordan structure

If all the blo cks Y Y in the rst rowofY and hence also all the blo cks

q

in the rst column of Y are zero then we can deate to a smaller degenerate

h i

e e

B K

e

skewHamiltonian where B diagA A and

q

T

e

B

Y Y

q

T

Y

e

K

T

Y

q

Otherwise there is some blo ck Y in the rst rowof Y that is nonzero By

p

BY

a p ermutationlike symplectic similarityon we can shift Y indeed the

T

p

B

th

C S

where Let Q whole p column of blo cks from Y into B

SC

C diag I I I I and

n n n n n

p q

p p

I S diag

n n n n n

q p

p p

i h

e

BY

T

L Y

where Q Then wehave Q

T

T

B

L

A Y

p

A Y

p

L

T

A

p

T

Y A

pq q

is exactly the typ e of matrix considered in Prop osition Consequently L has

only the eigenvalues and and the largest real Jordan blo ckof L is strictly

bigger than the largest real Jordan blo ckof B Roughly sp eaking similarity

by Q has the eect of transferring some Jordan structure from Y into B

To complete Step p erform a similarity with the blo ckdiagonal symplectic

Z

e

cho osing Z so that Z LZ B is in real Jordan form with the T

T

Z

largest real Jordan blo ckinthe p osition

The result of this twostep reduction pro cess then is a matrix

e e

e

B K

L Y

T T

T

T

e

L

B

BK

of the same form as the input to the twostep reduction pro cess but with the crucial

T

B

e

dierence that the largest real Jordan blo ckof B is strictly bigger than the largest real

Jordan blo ckofB

i h

e e

B K

After nitely many iterations we Now rep eat this twostep reduction pro cess on

T

e

B

can either deate to a smaller degenerate skewHamiltonian or we reach a stage where the

e

largest real Jordan blo ck has grown in size to ll all of B Onany deated problem weagain

iterate the twostep reduction after nitely many iterations we can either deate once more

e

or the largest real Jordan blo ck will havegrown to ll all of B Only nitely many such

e

deations can o ccur and ultimately wemust reach the termination case a B with only one

real Jordan blo ck Blo ckdiagonalization is achieved in one nal step as describ ed ab ove

A

Thus wehaveshown that there exists a symplectic S such that S WS with

T

A

nn

A

A IR in real Jordan form But anymatrixA is similar to its transp ose so must

A

b e the usual real Jordan canonical form of W The uniqueness of this Jordan form then

immediately implies the uniqueness of Auptoapermutation of Jordan blo cks

Innitely Many Square Ro ots

With the completion of the pro of of Theorem weknowthatevery real skewHamiltonian

matrix W has at least one real Hamiltonian square ro ot Let us next consider the set

p

H

W fH HjH W g of all the Hamiltonian square ro ots of W and what can b e

said ab out the size and top ological nature of this set for various W W A closer lo ok at

the pro of of Theorem shows that there are innitely many distinct symplectic similarities

bringing any given W W into structured real Jordan form Hence it is quite reasonable to

exp ect that every W W actually has innitely many distinct Hamiltonian square ro ots

Indeed by sharp ening the previous arguments we can obtain a uniform lower b ound on the

p

H

size of the W sets First we need one more result ab out Sylvester op erators strengthening

a theorem of Taussky and Zassenhaus The pro of will b e deferred to the app endix

nn

Prop osition Let A IR andconsider as in Proposition the restricted domain

Sylvester operator

T IRSymn IRSkewn

A

T

X AX XA

nn

Denote the set of al l nonsingular matrices in ker T by Invker T Then for any A IR

A A

Invker T is a dense open submanifold of ker T thus dim Invker T dimker T

A A A A

With this result in hand we can now establish the following lower b ound on the size of

Hamiltonian square ro ot sets

Theorem Every n n real skewHamiltonian matrix W has at least a nparameter

family of real Hamiltonian squareroots

A

Pro of Pickany xed S S suchthatS WS is blo ckdiagonal and factor

T

A

A as a pro duct A FG of n n symmetric matrices F and G Without loss of generality

wemay also assume that G is nonsingular Since in general there are many such

factorizations of A let us intro duce the set

G fG IRSymn j G is nonsingular and F AG is symmetricg

F

Then as wehave previously seen H S is a Hamiltonian square ro ot of W for S

G

G

any G G

p

H

IX

To construct even more elements of W from H consider symplectic shears T

G X

I

such that

A A

T T

T T

X

X

A A

Dening X fX IRSymn j T satises g it is easy to see that X is just the subspace

X

ker T where T is the op erator considered in Prop osition Now insert a similarityby

A A

any such T into H to dene

X G

F

H ST T S

GX X

X G

H W for every G X GXTo see that these Clearly H is Hamiltonian and

GX

GX

matrices H are all distinct let G X and G X b e ordered pairs from GXThen

GX

F F



S S ST T ST T H H

X X G X G X

G G X X

  

 

F F



T T T T

X X

G X G X



 

X G F X G X X G F X G X

     

G G X G G X

  

G G and X X since G G is nonsingular

Thus H and H are distinct whenever the pairs G X and G X are distinct

G X G X

 

p

H

W contains a family fH g parametrized by GX All that remains is to b ound the so

GX

sizes of G and X

Since X ker T wehave dim Xn just from consideration of the dimensions of the

A

domain and co domain of T Bycontrast the set G is not a subspace so a lower b ound on

A

its dimension requires a bit more discussion Observe that for any nonsingular symmetric

G

T T

AG AG AG G A G ker T

A

Thus G G i G Invker T Nowby Prop osition Invker T is a submanifold

A A

with the same dimension as k er T But matrix inversion is a dieomorphism of GL IR

A n

which maps G bijectively to Invker T so G must also b e a submanifold with dim G

A

dim Invker T dim ker T Putting this all together wehave

A A

p

H

dim W dimGX dimG dimX dimker T n

A

The family fH g of Hamiltonian square ro ots constructed in Theorem do es not always

GX

have dimension n In fact since dim ker T canbemuch larger than n it is p ossible for

A

p

H

dim W to b e much larger than n The most extreme example of this o ccurs for W I

n

p

H

n n so that dim I n nHowever it is muchmoretypical where dim ker T

n

I

n

p

H

that the lower b ound dim W n is actually attained The next prop osition makes this

p

H

precise using a standard argument from dierential top ology to show that W is exactly

ndimensional for all but a measure zero subset of exceptional cases

Prop osition For almost al l n n real skewHamiltonian matrices W ie for al l but

p

H

ameasurezero subset of W the set W is a smooth ndimensional submanifold of H

Pro of Consider the squaring map

f HW

H H

p

H

Clearly f is smo oth and the preimages f W are exactly the square ro ot sets W Now

for smo oth maps the Preimage Theorem says that any nonempty preimage of a regular

value is a smo oth submanifold of the domain and the dimension of this submanifold is the

dierence of the dimensions of the domain and co domain But the squaring map is onto

by Theorem so every preimage is nonemptyAndby Sards Theorem almost every

p oint in the co domain of a smo oth map is a regular value Thus we see that for almost every

p

H

skewHamiltonian matrix W the set f W W is a submanifold of H with

p

H

dim W dimHdim W n n n nn

remarks

Sards theorem is completely nonconstructive and in general gives no information

ab out whichvalues of a smo oth map are regular However the squaring map f is

simple enough that it is p ossible to explicitly characterize its regular values and thus

p

H

give an explicit sucient condition for W to b e a ndimensional submanifold The

rst step toward achieving this is to describ e the set of regular points of f ie to

nd those H H where the Frechet derivativedf is onto But df is precisely

H H

the map SylH H HWsowe are back to the problem of deciding when

certain Sylvester op erators are onto By an appropriate change of co ordinates as

in the App endix one can transform this problem to the equivalent question of the

T

surjectivityof SylH H IRSym IRSkew exactly the situation considered in

Prop osition Thus wemay conclude that H His a regular p ointof f i H is

nonderogatory Now the regular values of f are by denition the matrices W W

such that every H f W is a regular p oin t so one might guess the regular values

to b e exactly those W W that are squares of nonderogatory H H equivalently

those W W with the minimal number two of Jordan blo cks for each eigenvalue

This is almost but not quite correct It is p ossible to showthatW W is a regular

value of f i W has exactly two Jordan blo cks for each eigenvalue and the multiplicity

of the eigenvalue zero is not two

The squaring map f is not just smo oth its a quadratic p olynomial map in the entries

p

H

of H Thus every preimage f W W is an algebraic variety with dim n

Prop osition says that most of these varieties are actually smo oth submanifolds with

dim n

p

H

W enables us The existence of the family fH j G X G X g contained in any

GX

p

H

to conclude that every W is unboundedTo see this observe that G is unb ounded

since G G kG G for all k But similaritybyanyxedS S is a nonsingular

AG

linear op erator on Hso fH g S f j G G gS must also b e unb ounded

G

G



The Preimage Theorem is also known as the Regular Value Theorem

p

H

Matrices in W need not b e symplectically similar or even have the same Jordan

form Examples of b oth situations can already b e seen in the case J

p

H

I that are not symplectically similar although they do and J are elements of

p

H

have the same Jordan form Square ro ot sets W can contain matrices with distinct

p

H

Jordan forms only if W is singular N and are b oth elementsof

r

h i

N



H

However not every singular W exhibits this b ehavior every elementof

T

N



must have Jordan form N

Using the results of this section it is p ossible to construct explicit nparameter families

of Hamiltonian square ro ots at least for small n As an illustration consider W

i h

N



that One easily nds for A N

T

N



c d b

X ker T and G b

A

d b a

b

Assembling these ingredients as in the pro of of Theorem yields the parameter family

bd ad bc b bcd ad bd

B C

bd bd

B C

H H a b c d b

GX

A

b bd

b

b a bc ad bd

A direct computation shows that H a b c d W for every a b c d IRwithb

p

H

W for any skewHamiltonian matrix It is not dicult to explicitly calculate

k

W W f j k IR g and also to see how these square ro ot sets t together

k

to partition the dimensional space H of all Hamiltonian matrices This is

shown in Figure which also nicely illustrates the rst four remarks In this gure we

have identied H with IR using the isometry

H IR

p

ab

a b c b c

c a

p p

H

H

Each The cone is exactly the set of all nilp otent matrices ie kI with

p

jk j J By contrast k isatwosheeted hyp erb oloid intersecting the J axis at

p

H

every kI with kisahyp erb oloid of one sheet

It is also interesting to note the relation of real similarity and symplectic similarity

classes in H to these square ro ot sets Although it is not true for larger Hamiltonian

p

H

matrices in H every W is a nite union of similarity classes For example every

p

H

hyp erb oloidal W is just the intersection of some real similarity class in IR with

H Every onesheeted hyp erb oloid is also a symplectic similarity class on the other

hand each sheet of a twosheeted hyp erb oloid is a distinct symplectic similarity class

By contrast the cone C of nilp otents is the union of three symplectic similarity

classes the zero matrix together with the two connected comp onents C and C

of C n Although matrices in C are not symplectically similar to those in C they

are real similar so that C C constitutes a single real similarity class in H Thus

C is the union of two real similarity classes α J

J

0

- J

Figure Hamiltonian square ro ot sets

p

H

none of the Hamiltonian square ro ots of any With the trivial exception of

skewHamiltonian W is a p olynomial in W The basic reason for this is the eigenvalue

is an eigenvalue of H H structure of real Hamiltonian matrices whenever C

then so is But distinct eigenvalues of a p olynomial square ro ot can not have

the same square so the only wayfor H H to b e a p olynomial square ro ot of W is

to have only the eigenvalue zero ie H and W must b e nilp otent Now it is easy to

see from the Jordan form that we can only have H pW and H W if all of the

Jordan blo cks are that is H W

The characterization of the set G given in Theorem and Prop osition constitutes

an alternate pro of of the twosymmetrics factorization theorem for real n n matrices

An imp ortant feature of this pro of is that it go es b eyond the mere existence of the

factorization to provide systematic although not complete information ab out the set

of all such factorizations

Complex Structured Square Ro ots

The notion of Hamiltonian and skewHamiltonian structure extends to complex matrices

so it is natural to consider the question of the existence of structured square ro ots in these

complex classes In this section wesurvey the various p ossibilities b eginning with some

denitions and simple prop erties

Recall from xandx that S HandW can b e viewed as the automorphism group Lie

n

T

Jy dened on IR of the bilinear form bx y x algebra and Jordan algebra resp ectively

I

by the n n matrix J This realbilinear form bx y has unique extensions b oth

I

n

Toeach of these to a complexsesquilinear form and to a complexbilinear form on C

n

forms on C there is an asso ciated automorphism group Lie algebra and Jordan algebra

of complex n n matrices Thus there are two natural but distinct ways of extending

the notions of real symplectic Hamiltonian and skewHamiltonian structure to complex

matrices leading to the following denitions

nn

j S JS J g S fS C

C

nn

fH C H j JH JHg

C

nn

j JW JW g W fW C

C

T

where denotes either transp ose or conjugatetransp ose The complexbilinear

extension of b leads to being T while the sesquilinear extension of b results in b eing

We remark that the use of conjugatetransp ose to dene complex symplectic Hamiltonian

and skewHamiltonian matrices seems to b e standard in control theory but using

transp ose app ears to b e more typical in the study of Lie groups representation theoryand

dynamical systems The terms J orthogonal J symmetric and J skewsymmetric

T T T

for S H and W andJ unitary J Hermitian and J skewHermitian for S H and W

C C C C C C

are also commonly used

analogous to the and W Characterizations of the blo ck structure of matrices in H

C C

AB

ones given for H and W in x are easily obtained For example W W i W



C

CA

nn nn

where A C is arbitrary but B C C arebothskewHermitian The following

simple prop erties of these classes of complex matrices are easy to check

Lemma

and WW a HH

C C

W W W b H W W Wand H

C C C C

and S WS W S HS H W W H H c S S

C C C C C

nn

T T

d H and W arecomplex subspaces of C while H and W are only real sub

C C C C

spaces But we have i H W and i W H

C C C C

From part b of this lemma it is clear that it only makes sense to lo ok for structured square

Prop ositions of matrices in W and W or W ro ots ie square ro ots in H W H

C C C

and settle the existence question for all p ossible cases

Prop osition The fol lowing table summarizes the existenceofreal and complex Hamil

tonian and skewHamiltonian squareroots when is taken to beconjugatetranspose

p p p p

W H W H W W W W

C C

W W a Always b Always c Sometimes d Always

W W e Sometimes f Sometimes g Sometimes h Sometimes

C

Pro of

a This is Theorem

b Trivially true since HH

C

c The following three conditions on matrices in W are equivalent

i W W has a U W such that U W

A

ii W W is real symplectically similar to some blo ckdiagonal matrix

T

A

nn

suchthat A IR has a real square ro ot

A

that is real symplectically similar to iii For every blo ckdiagonal matrix

T

A

nn

W W the matrix A IR has a real square ro ot

By Theorem wemay symplecticallyblockdiagonalize U so that i ii

nn

B

U S S with B IR Then

T

B



A

B

S S W U S S with A B

T

T

A

B

A

ii iii Supp ose W is symplectically similar to with A B for some

T

A

h i

b

nn

A

B IR and is any other blo ckdiagonal matrix in W symplectically similar

T

b

A

h i

b

A

A

to W Now b oth and can b e broughtinto skewHamiltonian Jordan

T

T

b

A

A

V

form via similarity with blo ckdiagonal symplectics of the form But since

T

V

the skewHamiltonian Jordan form of W is essentially unique we can conclude that A

b b

and A must b e real similar to each other Thus A also has a real square ro ot

A

S for some S S and By Theorem weknowthatW S iii i

T

A

nn nn

A IR and from condition iii wehave A B for some B IR Thus

B

S W W U for U S

T

B

Condition iii implies that W is a matrix in W with no square ro ot in W

d There are two simple ways to see this

p

H

F

W as in Prop osition That is let H S S First construct some H

G

iF

f f

H b e suchthat H W ThenW S S W and W W

iG C

p

H

e f e

Alternately use Lemma d Pickany H and W Then W i H is in W

C

f

W W

e Clearly this is only true for W W W

C



Conditions for the existence of real square ro ots may b e found in and

f There are many matrices in W with no square ro ot in H This fact can b e established

C C

by examining the p ossible arrangements in the complex plane of the eigenvalues of

p ossess Itiswell known that the sp ectra of matrices in H and H matrices in W

C C C

a reection symmetry whenever C is an eigenvalue of H H thenso

C

is and b oth have the same multiplicity indeed even the same Jordan structure

A general H H then has an even numb er of eigenvalues counting multiplicity

C

group ed in pairs symmetric with resp ect to the imaginary axis and the rest of its

eigenvalues distributed arbitrarily on the imaginary axis An analogous description for

matrices in W may also b e given Any W W can b e expressed as W iH for

C C

some H H so W iH Thus a general W W hasaneven number of its

C C

eigenvalues group ed in complexconjugate pairs with the remaining ones spread out on

the real axis without restriction Now consider the square of any H H From the

C

ie either ab ovewe see that the eigenvalues of H are just like those of matrices in W

C

real or in complexconjugate pairs except for one additional restriction any positive

i

eigenvalue of H must have even multiplicityThus the matrix Z W

i

C

with simple eigenvalues and cannot b e the square of any H H

C

g Clearly a square ro ot in W can exist only if W is real and the condition describ ed in

part c is satised

i

h The same Z as used in part f also provides an example of a matrix in W

i

C

with no square ro ot in W To see why this is so consider the square of a general

C

W W Since W can b e written as W iH for some H H wehave W H

C C

thus any negative eigenvalue of W must have even multiplicity Consequently Z with

a simple eigenvalue at cannot b e the square of any W W

C

The alert reader will have noticed that no complex analog of Theorem for W played

C

any part in the discussion of Prop osition The reason is simple no such result is true for

i

general matrices in W Consider for example the matrix U W SinceU has

C C

A

Jordan form it cannot b e brought in to the form byany similarity let alone



A

by similarity with some matrix from S

C

T

By contrast the complex analog of Theorem do es hold for the class W Indeed

C

with only minor changes the very same pro of given in this pap er for real skewHamiltonian

T

matrices also yields the following theorem The existence of this result for W but not for

C

accounts for much of the dierence b etween Prop ositions and W

C

T T

Theorem For any W W there exists an S S such that

C C

A

S WS

T

A

nn

is in Jordan canonical form The matrix A is unique up to a permutation where A C

of Jordan blocks

Prop osition The fol lowing table summarizes the existenceofreal and complex Hamil

tonian and skewHamiltonian squareroots when is taken to betranspose

p p p p

T T

W H W H W W W W

C C

W W a Always b Always c Sometimes d Sometimes

T

W W e Sometimes f Always g Sometimes h Sometimes

C

Pro of

ac These are the same as parts a and c of Prop osition

T

b Trivially true since HH

C

d This case is covered by the discussion in part h b elow

T

e Clearly this is only true for W W W

C

f The argument used in Prop osition and Theorem to showthatevery real skew

Hamiltonian matrix has a real Hamiltonian square ro ot works equally well here to show

T T T

that every W W has a square ro ot in H By Theorem there is an S S

C C C

nn

A

Butany complex A can b e factored as such that S WS whereA C

T

A

nn

the pro duct A FG of two complexsymmetric matrices F G C so that

A

T T

F F F

H and Thus S S H is a square ro ot of W

T

C C

G G G

A

g Clearly a square ro ot in W can exist only if W is real and the conditions describ ed in

partc of Prop osition are satised

h Using Theorem the argument in Prop osition c can b e trivially mo died to show

T

that the following three conditions on matrices in W are equivalent

C

T T

i W W has a U W suchthat U W

C C

A

T T

ii W W is similar via a matrix in S tosome blo ckdiagonal matrix

T

C C

A

nn

hthat A C has a complex square ro ot suc

A

T

iii For every blo ckdiagonal matrix via some that is similar to W W

T

C

A

nn

T

matrix in S the matrix A C has a complex square ro ot

C

h i

N



The equivalence of these conditions implies that the matrix W W

T

N



T

seen earlier in remarks and of x has no square ro ot in W although it do es

C

have innitely many square ro ots in W

C

Conclusions

This pap er has addressed the theoretical asp ects of the HamiltonianskewHamiltonian struc

tured square ro ot problem Wehave settled the existence question every real skew

Hamiltonian matrix has a real Hamiltonian square ro ot Furthermore wehaveshown that

p

H

W of all real Hamiltonian square ro ots for anyn n real skewHamiltonian W the set

p

H

of W is an unb ounded algebraic variety with dimension at least nInfact W is a smo oth



Conditions for the existence of complex square ro ots may b e found in and

manifold of dimension exactly n for almost every W The existence question for various

typ es of complex structured square ro ots of complex Hamiltonian and skewHamiltonian

matrices has also b een resolved

We emphasize the main technical result of this pap er whichmay b e of signicantinde

p endentinterest every real skewHamiltonian matrix may b e broughtinto structured real

Jordan canonical form via real symplectic similarity It is natural to ask whether there is an

analogous structured canonical form for real Hamiltonian matrices This question has b een

recently settled but the canonical form is considerably more complicated and is usually

not blo ck triangular

Finally the problem of nding go o d numerical metho ds to compute Hamiltonian square

ro ots for general skewHamiltonian matrices remains op en Clearly a Schurlike metho d

involving Van Loans reduction Sylvester equations and matrix inversion can b e develop ed

see Remark following Prop osition and the pro of of Theorem but such a metho d

can only b e applied in the generic case Alternatively one might consider iterative metho ds

as in Unfortunately all current iterative metho ds compute only square ro ots that are

p olynomials in the original matrix and no nonzero Hamiltonian square ro ot of any W W

is a p olynomial in W Consequently the outlo ok for nding any structurepreserving matrix

iteration to compute Hamiltonian square ro ots app ears less than promising We are currently

exploring ways to overcome these diculties

Acknowledgements We thank Nick Higham Volker Mehrmann Charlie Van Loan

and YiehHei Wan for helpful discussions and comments on various asp ects of this pap er

App endix

In this app endix we present pro ofs for Prop ositions and from x and for Prop osition

from x In each of these pro ofs wemake use of the following wellknown technique of

k k

change of co ordinates b etween Sylvester op erators Let A F and B F be

k k

xed but arbitrary Then for anyinvertible matrices U F and Z F wehave

AX XB Y U AX XBZ UY Z

UAU UXZ UXZZ BZ UY Z

In other words the following diagram commutes

SylAB

k k

F F

X UXZ Y UY Z

y y

SylUAU Z BZ

k k

F F

Thus the kernels and ranges of SylA B and SylUAU Z BZ are simply related

ker SylUAU Z BZ U ker SylA B Z

range SylUAU Z BZ U range SylA B Z

In particular SylA B isonto i SylUAU Z BZ isonto

Pro of of Prop osition

With A M and B M b oth Jordan blo cks corresp onding to F rst observe

k

T T k

that SylA B and N SylN N are the same op erator since for every X F we

k k

have

T T T

AX XB I N X X I N N X XN

k k k

To nd the range of N we use the fundamental relationship range N ker N here

k k k

k

with resp ect to the standard inner pro duct on F dened N denotes the adjointofN

k k

H

by hX Y i traceXY The computation

H H H

hLX Y i traceLX Y traceXY LhX L Y i

H

shows that the adjoint of the leftmultiplication op erator L X LX is L X L X

Similarly one sees that the adjoint of the rightmultiplication op erator R X XR is

H T

R X XR Together these imply that N SylN N

k k

Nowker SylN N iswellknown it is just the set of all To eplitz matrices of

k

dd

T

the form T when k or when k where T F with d mink is upp er

T

triangular From this known result we can obtain ker SylN N bya change of co ordinates

k

as in the discussion of ab ove Letting E e e e e denote the k k exchange

k k k

T

matrix wehave N E N E so the following diagram commutes

k k k k

SylN N

k

k k

F F

E X Y E Y X

k y y k

T

SylN N

k

k k

F F

T

Thus ker N ker SylN N E ker SylN N is the set of all k Hankel ie

k k k k

constant along antidiagonals matrices which are zero everywhere except p ossibly along the

last d antidiagonals

th k

has ones along the k i Finally consider the Hankel matrices H F where H

i i

antidiagonal and zero es everywhere else Clearly fH j i dg is a basis for ker N For

i

k

th

a matrix to b e orthogonal to H the sum of its entries along the k i antidiagonal

i

must b e zero Thus

k

fY F j sum of entries along each of the last d antidiagonals is zerog ker N

k

Pro of of Prop osition

T

For Sylvester op erators T X AX XA the appropriate co ordinate changes are given

A

nn

b

as follows Let U F be anyinvertible matrix and A UAU Then the following

diagram commutes

T

A

FSymn FSkewn

T T

X UXU Y UY U

y y

T

b

A

FSkewn FSymn

is onto Without loss of generalitythenwemay assume that A is in Thus T is onto i T

A b

A

nn

any convenient normal form in F

supp ose that A is in Jordan canonical form Beginning with the complex case F C

writing A A A A where each A is an n n Jordan blo ck M

mm ii i i n i

i

Partition X C Symninto blo cks X conformally with A The blo ckdiagonal nature of

ij

A means that the op erator T may b e treated blo ckwise

A

T T

T X AX XA A X X A

A

ij ij ii ij ij jj

th th

so the ij blo ckof T X dep ends only on the ij blo ckof X Consequently wemaydene

A

T

the blo ck op erators T X A X X A for i j m and observethat T is onto

A

ij ij ii ij ij jj

i every T is onto

ij

It is imp ortant to note a subtle dierence b etw een T with i j and T Because

ij ii

Symn X is symmetric the diagonalblo ck op erators T must b e regarded as maps C

i ii

n n n n

i j i j

C Skewn whereas the odiagonalblo ck op erators T i j are maps C C

i ij

These dierences in domain and co domain are crucial to correctly judging whether each T

ij

and hence T isonto or not

A

T

By Prop osition the odiagonalblo ck op erators T SylA A areonto i

i j

ij ii jj

By contrast the diagonalblo ck op erators T are always onto To see this rst observe that

ii

the sum of the entries along anyantidiagonal of a skewsymmetric matrix is zero Then by

T

Skew n range SylA A that is for any Y C Skew n Prop osition wehaveC

i i

ii ii

n n

i i

there is some Z C suchthat

T T

SylA A Z A Z ZA Y

ii ii ii ii

This Z however may not b e in the domain of T But taking transp ose of b oth sides of

ii

T T T T

U Y Z Z wehaveSylA A Z Y sowithU shows that SylA A

ii ii ii ii

Since U C Symn wehave T U Y showing that T is onto

i ii ii

Thus we conclude that T is onto T is onto for all i j for all i j

A ij i j

and the F Ccase is proved

The real case F IR follows almost immediately from the complex case For a real

C

C matrix A there are two op erators T IRSymn IRSkewnand T Symn

A A

T C

C Skewn dened by the same formula X AX XA The op erator T is the com

A

C C

plexication of T so dim range T dim range T Hence T is onto T is

R C

A A A A A

onto

Pro of of Prop osition

T

The strategy here is to apply a change of co ordinates to the real op erator SylA B so

that A and B are broughtinto complex Jordan form then use Prop ositions and to nd

the range of the resulting complex op erator and nally translate backinto completely real

terms using a relation analogous to We b egin by recalling the similaritiesthatconvert

real Jordan blo cks into complex Jordan form

p

is Consider the blo ckdiagonal diag where

k

ii

k k

as in Lemma and the p ermutation P e e e e j e e e It is now

k k k

straightforward to checkthat

M

k

T H

b

P A P A

k k k k

M

k

T H

where M M isthek k Jordan blo ckfor aib Similarly wehave P B P

k k

b

B diag M M so that

T

M

T H T H

b

P B P B

T

M

b

It is imp ortant to note that a ib app ears in the blo ckinA but in the

H

b

blo ckin B

H T

More generally let us consider the action of the similarities X and P YP

k k

on arbitrary k matrices X and Y WithX partitioned into blo cks X i

ij

th H H

k j it is easy to see that the ij blockof X is just X Thus

ij

k

uv

an y real blo ck X will b e transformed into a complex matrix U see Lemma

ij

v u

u v

ij ij

T

ThenP YP is the blo ck Partition Y in the same wayinto blocks Y

w z

ij

ij ij

k

UV

matrix where the k blo cks U V W and Z are assembled entrywise from the Y

ij

WZ

blo cks via

U Y u V Y v

ij ij ij ij ij ij

w Z Y z W Y

ij ij ij ij ij ij

k

Now put these two maps together dening P and letting U C denote

k k k k

k

UV

whereU V C are arbitraryWe see the set of all complex matrices of the form

V U

that the map

k

U IR

k

H

X X

k

T

is a real linear isomorphism Hence the change in co ordinates in SylA B whichtakes

T H

b b

A to A and B to B gives us the commutative diagram

T

SylA B

k k

IR IR

H H

X Y X Y

y y

k  k 

H

b b

SylA B

U U

k k



Note that the inverse of P is the p erfect shue p ermutation known to American magicians as the

k

faro shue and to English magicians as the weaveshue

Thus

T H H

b b

range SylA B range SylA B

k

H H

b b b b

Next we compute the range of SylA B The blo ckdiagonal nature of A and B means

H

b b

that SylA B may b e treated blo ckwise just as T was in the pro of of Prop osition We

A

have

T

M U V U V U V

M

k

H

b b

SylA B

T

V U M V U V U

M

k

T

T

M U U M M V VM

k k

T

T

M V V M M U UM

k k

T U T V

T V T U

Observe that since T U T U and T V T V it suces to nd the ranges of

T and T By Prop osition T is onto Prop osition gives us the range of T Thus we

UV

H

b b

U such that the sum of the entries see that range SylA B is the set of all

k

V U

k

along each of the last d antidiagonals of V C is zero

Transforming this result using and Lemma gives us the desired characterization of

T

range SylA B

Pro of of Prop osition

C

Let T denote the complexication of T that is the map

A

A

C

Symn C Skewn T C

A

T

X AX XA

C

We show rst that ker T contains at least one invertible matrix and from this deduce that

A

ker T must also have at least one invertible element Then the desired conclusion will follow

A

from basic prop erties of algebraic varieties

C

b egin as in the pro of of Prop osition Change To nd an invertible elementofker T

A

C

b

co ordinates from T to T C Symn C Skewn where A is the Jordan form of

b

A

A

T

A and then treat T blo ckwise The diagonalblo ck op erators T are just SylN N

b ii

n n

A

i i

restricted to C Symn so an argumentlike the one used to compute ker N in the pro of

i

k

of Prop osition see diagram shows that ker T ker SylN N E a certain set of

ii n n n

i i i

Hankel matrices For our purp oses it suces to observe that the invertible matrix E is in

n

i

ker T thus E diag E E E isaninvertible elementofker T Transforming

ii n n n b

m



A

C C

via yields an invertible elementofker T E backinto ker T

A A

and consider the p olynomial Now let fM M g be any xed basis for ker T

k A

px x x detx M x M x M

k k k

k

With x x IR this p olynomial p distinguishes the singular and invertible elements

k

C

of ker T ButT is the complexication of T so the real matrices fM M g also form a

A A k

A

k

C

basis for ker T thus with x x C the same p olynomial p distinguishes the singular

k

A

C

from the invertible elements of ker T Now the co ecients of p are real since each M is

i

A

real and for any p olynomial p with real co ecients the following are equivalent

i p as a formal p olynomial ie all the co ecients of p are zero

k

ii p as a function IR IR

k

C iii p as a function C

C

The existence of an invertible elementinker T means that p as a complex function

A

k k

C C Therefore p as a real function IR IR either so there must b e some

k

invertible elementinker T Consequently the zero set of p in IR equivalently the set of

A

singular matrices in ker T isaproper algebraic subset and hence a closed nowhere dense

A

set of measure zero Thus Invker T the complement of the singular matrices in ker T

A A

is op en and dense in ker T

A



References

G Alefeld and N Schneider On square ro ots of M matrices Linear Algebra Appl

R Bhatia and P Rosenthal Howandwhytosolve the op erator equation AX XB Y

Bul l London Math Soc

A Bjorck and S Hammarling A Schur metho d for the square ro ot of a matrix Linear

Algebra Appl

AJ Bosch Note on the factorization of a square matrix into two Hermitian or symmetric

matrices SIAM Review

A BunseGerstner R Byers and V Mehrmann Achart of numerical metho ds for

structured eigenvalue problems SIAM J Matrix Anal Appl

A BunseGerstner and H Fab ender A Jacobilike metho d for solving algebraic Riccati

equations on parallel computers IEEE Trans Automat Contr

GW Cross and P Lancaster Square ro ots of complex matrices Linear and Multilinear

Algebra

H Fab ender DS Mackey and N Mackey Structurepreserving Jacobi metho ds for

doublystructured eigenproblems In preparation

erlag New W Fulton and J Harris Representation Theory A First Course SpringerV

York

FR Gantmacher The Theory of Matrices vol Chelsea New York

V Guillemin and A Pollack Dierential Topology PrenticeHall Englewo o d Clis

PR Halmos FiniteDimensional Vector Spaces SpringerVerlag New York

NJ Higham Computing real square ro ots of a real matrix Linear Algebra Appl

NJ Higham Stable iterations for the matrix square ro ot Numerical Algorithms

M Hirsch and S Smale Dierential Equations Dynamical Systems and Linear Algebra

Academic Press San Diego

RA Horn and CR Johnson Matrix AnalysisCambridge University Press Cambridge

RA Horn and CR Johnson Topics in Matrix Analysis Cambridge University Press

Cambridge

T LaeyPersonal communication June

V Mehrmann The Autonomous Linear Quadratic Control Problem Theory and Nu

merical Solution Lecture Notes in Control and Information Sciences Vol Springer

Verlag BerlinHeidelb erg

V Mehrmann and H Xu Canonical forms for Hamiltonian and symplectic matrices

and p encils Technical rep ort SFB Technische Universitat Chemnitz Germany

March

KR Meyer and GR Hall Introduction to Hamiltonian Dynamical Systems and the

Nbody Problem SpringerVerlag New York

VV Prasolov Problems and Theorems in Linear Algebra American Math So c Prov

idence

GW Stewart and JiGuang Sun Matrix Perturbation Theory Academic Press San

Diego

O Taussky The role of symmetric matrices in the study of general matrices Linear

Algebra Appl

O Taussky and H Zassenhaus On the similarity transformation b etween a matrix and

its transp ose Pacic J Math

CF Van Loan A symplectic metho d for approximating all the eigenvalues of a Hamil

tonian matrix Linear Algebra Appl

VS Varadara jan Lie Groups Lie Algebras and Their Representations Springer

Verlag New York