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1 Introduction 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
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