c 2002 Society for Industrial and Applied Mathematics

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1 SIAM J MATRIX ANAL APPL Vol 3 No 4 pp c Society for Industrial and Applied Mathematics EXISTENCE UNIQUENESS AND PARAMETRIZATION OF LAGRANGIAN INVARIANT SUBSPACES GERARD FREILING VOLKER MERMANN AND ONGGUO XU Abstract The existence uniqueness and parametrization of Lagrangian invariant subspaces for amiltonian matrices is studied Necessary and sufficient conditions and a complete parametrization are given Some necessary and sufficient conditions for the existence of ermitian solutions of algebraic Riccati equations follow as simple corollaries Key words eigenvalue problem amiltonian matrix symplectic matrix Lagrangian invariant subspace algebraic Riccati equation AMS subject classifications 65F15 93B4 93B36 93C6 PII S Introduction The computation of invariant subspaces of amiltonian matrices is an important task in many applications in linear quadratic optimal and control Kalman filtering or spectral factorization; see [ and the references therein Definition 11 A matrix C nn is called amiltonian if J n =(J n ) is ermitian where J n = [ I n I n In is the n n identity matrix and the superscript denotes the conjugate transpose Every amiltonian matrix has the block form [ A M = G A with M = M G = G amiltonian matrices are closely related to algebraic Riccati equations of the form (11) A X + XA XMX + G = It is well known [15 that if X = X solves (11) then [ [ [ In In (A MX) M (1) = X I n X I n (A MX) This implies that the columns of [ I n X span an invariant subspace of associated with the eigenvalues of A MX Invariant subspaces of this form are called graph subspaces [15 The graph subspaces of amiltonian matrices are special Lagrangian subspaces Received by the editors August 9 ; accepted for publication by ACM Ran May 3 1; published electronically April 1 The research of the second and third authors was supported by Deutsche Forschungsgemeinschaft research grant Me 79/7- Fachbereich Mathematik Universität Duisburg D-4748 Duisburg Germany (freiling@math uni-duisburgde) Institut für Mathematik MA 4-5 TU Berlin Str des 17 Juni 136 D-163 Berlin Germany (mehrmann@mathtu-berlinde) Department of Mathematics University of Kansas Lawrence KS 6645 (xu@mathukansedu) 145

2 146 GERARD FREILING VOLKER MERMANN AND ONGGUO XU Definition 1 A subspace L of C n is called a Lagrangian subspace if it has dimension n and x J n y = x y L Clearly a subspace L is Lagrangian if and only if every matrix L whose columns span L satisfies rank L = n and L J n L = Despite the fact that amiltonian matrices algebraic Riccati equations and their properties have been a very active area of research for the last 4 years there are still many open problems These problems are mainly concerned with amiltonian matrices that have eigenvalues with zero real part and in particular with numerical methods for such problems In this paper we summarize and extend the known conditions for existence of Lagrangian invariant subspaces of a amiltonian matrix Based on these results we then give a complete parametrization of all possible Lagrangian invariant subspaces and also discuss necessary and sufficient conditions for the uniqueness of Lagrangian invariant subspaces Most of the literature on this topic is stated in terms of ermitian solutions for algebraic Riccati equations; see [15 For several reasons we will however be mainly concerned with the characterization of Lagrangian invariant subspaces First of all the concept of Lagrangian invariant subspaces is a more general concept than that of ermitian solutions of the Riccati equation since only graph subspaces are associated with Riccati solutions A second and more important reason is that in most applications the solution of the Riccati equation is not the primary goal but rather a dangerous detour; see [1 Finally even most numerical solution methods for the solution of the algebraic Riccati equations (with the exception of Newton s method) proceed via the computation of Lagrangian invariant subspaces to determine the solution of the Riccati equation; see [ These methods employ transformations with symplectic matrices Definition 13 A matrix S C nn is called symplectic if S J n S = J n If S is symplectic then by definition its first n columns span a Lagrangian subspace Conversely if the columns of S 1 span a Lagrangian subspace then it generates a symplectic matrix given for example by S =[S 1 J n S 1 (S1 S 1 ) 1 ence the relation between Lagrangian subspaces and symplectic matrices can be summarized as follows Proposition 14 If S C nn is symplectic then the columns of S [ I n span a Lagrangian subspace If the columns of S 1 C nn span a Lagrangian subspace then there exists a symplectic S such that range S [ I n = range S1 Considering Lagrangian invariant subspaces L of a amiltonian matrix we immediately have the following important equivalence Proposition 15 Let C nn be a amiltonian matrix There exists a Lagrangian invariant subspace L of if and only if there exists a symplectic matrix S such that range S [ I n = L and (13) S 1 S = [ R D R The form (13) is called amiltonian block triangular form and if furthermore R is upper triangular (or quasi-upper triangular in the real case) it is called amiltonian triangular form or amiltonian Schur form Note that for the existence of Lagrangian

3 LAGRANGIAN INVARIANT SUBSPACES 147 invariant subspaces it is not necessary that R in (13) is triangular if one is not interested in displaying actual eigenvalues Most numerical methods however will return a amiltonian triangular or quasi-triangular form Necessary and sufficient conditions for the existence of such transformations were given in [18 and in full generality in [19 and we will briefly summarize these conditions in the next section Numerically backward stable methods to compute such forms have been developed in [1 3 4 The contents of this paper are summarized as follows In section after recalling some of the results on amiltonian triangular forms we discuss the existence of Lagrangian invariant subspaces corresponding to all possible eigenvalue selections In section 3 we give complete parametrizations of all possible Lagrangian subspaces of a amiltonian matrix associated with a particular set of eigenvalues Based on these results we summarize necessary and sufficient conditions for the existence and uniqueness of Lagrangian invariant subspaces in section 4 Finally we apply these results to give some simple proofs of (mostly known) theorems on existence and uniqueness of ermitian solutions to algebraic Riccati equations in section 5 amiltonian block triangular forms and existence of Lagrangian invariant subspaces To study an invariant subspace problem we first need to discuss the possible selection of associated eigenvalues We denote by Λ(A) the spectrum of a square matrix A counting multiplicities For a amiltonian matrix if λ Λ() and Re λ then it is easy to see that also λ Λ(); see [15 Furthermore if has the block triangular form (13) and if iα is a purely imaginary eigenvalue (including zero) then it must have even algebraic multiplicity It follows that the spectrum of a amiltonian matrix in the form (13) can be partitioned into two disjoint subsets (1) Λ 1 () ={λ 1 λ }{{} 1 λ 1 λ 1 λ }{{} µ λ µ λ µ λ µ } }{{}}{{} n 1 n 1 n µ n µ Λ () ={iα 1 iα }{{} 1 iα ν iα ν } }{{} m 1 m ν where λ 1 λ µ are pairwise disjoint eigenvalues with positive real part and iα 1 iα ν are pairwise disjoint purely imaginary eigenvalues (including zero) If a matrix is transformed as in (13) then the spectrum associated with the Lagrangian invariant subspace spanned by the first n columns of S is Λ(R) Since Λ() =Λ(R) Λ( R ) it follows that Λ(R) must be associated to a characteristic polynomial µ (λ λ j ) tj (λ + λ ν j ) nj tj (λ iα j ) mj j=1 where t j are integers with t j n j for j =1µ We denote the set of all possible such selections of eigenvalues by Ω() Note that Ω() contains µ j=1 (n j +1) different selections In most applications it is desirable to determine Lagrangian invariant subspaces associated with eigenvalue selections for which only one of the eigenvalues of the pair λ j λ j (which are not purely imaginary) can be chosen in Λ(R) In another words t j must be either or n j Such subspaces all called unmixed and the associated Riccati solution if it exists is called the unmixed solution of the Riccati equation; see j=1

4 148 GERARD FREILING VOLKER MERMANN AND ONGGUO XU [6 We denote the subset of all possible such selections by Ω() Obviously Ω() contains µ different elements Note that all selections in Ω() contain the same purely imaginary eigenvalues Note further that if cannot be transformed to the amiltonian block triangular form (13) then the set Ω() may be empty A simple example for this is the matrix J 1 We now recall some results on the existence of amiltonian triangular forms In the following we denote a single Jordan block associated with an eigenvalue λ by N r (λ) =λi r + N r where N r is a nilpotent Jordan block of size r We also frequently use the antidiagonal matrices () P r = ( 1) r ( 1) and denote by e j the jth unit vector of appropriate size Lemma 1 (see [19) Suppose that iα is a purely imaginary eigenvalue of a amiltonian matrix and that the Jordan block structure associated with this eigenvalue is N(iα) :=iαi + N where N = diag(n r1 N rs ) Then there exists a full column rank matrix U such that U = UN(iα) and 1 U J n U = diag(π 1 P r1 π s P rs ) where π k {1 1} if r k is even and π k {i i} if r k is odd Using the indices and matrices introduced in Lemma 1 the structure inertia index associated with the eigenvalue iα is defined as Ind S (iα) ={β 1 β s } where β k =( 1) r k π k if r k is even and β k =( 1) r k 1 iπ k if r k is odd Note that the β i are all ±1and there is one index associated with every Jordan block The structure inertia index is closely related to the well-known sign characteristic for ermitian pencils (see [15) since every amiltonian matrix can be associated with the ermitian pencil λij J Although the sign characteristic is a more general concept since it also applies to general ermitian pencils we prefer to use the structure inertia index because it is better suited for the analysis of amiltonian triangular forms; see [19 For the following analysis the tuple Ind S (iα) is partitioned into three parts Ind e S(iα) Ind c S(iα) Ind d S(iα) where Ind e S(iα) contains all the structure inertia indices corresponding to even r k Ind c S(iα) contains the maximal number of structure inertia indices corresponding to odd r k in ±1pairs and Ind d S(iα) contains the remaining indices; ie all indices in Ind d S(iα) have the same sign; see [19 Necessary and sufficient conditions for the existence of a symplectic similarity transformation to a amiltonian triangular Jordan-like form (13) are given in the following theorem Theorem Let be a amiltonian matrix let iα 1 iα ν be its pairwise distinct purely imaginary eigenvalues and let the columns of U k k =1νspan the associated invariant subspaces of dimension m k Then the following are equivalent:

5 LAGRANGIAN INVARIANT SUBSPACES 149 (i) There exists a symplectic matrix S such that S 1 S is amiltonian block triangular (ii) There exists a unitary symplectic matrix U such that U U is amiltonian block triangular (iii) Uk JU k is congruent to J mk for all k =1ν (iv) Ind d S(iα k ) is void for all k =1ν Moreover if any of the equivalent conditions holds then the symplectic matrix S can be chosen such that S 1 S is in amiltonian triangular Jordan form (3) R r R e D e R c D c Rr Re Rc where the blocks with subscript r are associated with eigenvalues of nonzero real part and have the substructure R r = diag(r r 1R r µ) R r k = diag(n dk1 (λ k )N dkpk (λ k )) k =1µ The blocks with subscript e are associated with the structure inertia indices of even r k for all purely imaginary eigenvalues and have the substructure R e = diag(r1r e ν) e Rk e = diag(n lk1 (iα k )N lkqk (iα k )) D e = diag(d1d e ν) e Dk e = diag(βk1e e lk1 e l k1 βkq e k e lkqk e l kqk ) The blocks with subscript c are associated with pairs of blocks of inertia indices associated with odd-sized blocks for purely imaginary eigenvalues and have the substructure R c = diag(r1r c ν) c Rk c = diag(b k1 B krk ) D c = diag(d1d c ν) c Dk c = diag(c k1 C krk ) where B kj = C kj = N mkj (iα k ) e m kj iβc kj N nkj (iα k ) e n kj iα k e mkj e nkj e m kj e n kj Proof The proof of equivalence for (i) and (iv) is given in Theorem 13 in [5 The equivalence of the other conditions and the structured amiltonian triangular Jordan form (3) was derived in [19 Remark 1 For real amiltonian matrices a real quasi-triangular Jordan form analogous to (3) and a similar set of equivalent conditions as in Theorem can be given We refer the reader to [5 and Theorem 4 in [19 for details The necessary and sufficient conditions in Theorem guarantee the existence of only one Lagrangian invariant subspace associated to one selection in Ω() But the

6 15 GERARD FREILING VOLKER MERMANN AND ONGGUO XU following theorem shows they also guarantee the existence of a Lagrangian invariant subspace associated to every selection in Ω() Theorem 3 Let be a amiltonian matrix If any of the conditions in Theorem holds then for every eigenvalue selection ω Ω() there exists at least one corresponding Lagrangian invariant subspace Proof A proof for this result based on condition (iv) was given in [3 5 but a simple proof follows directly from (3) Note that any ω contains half the number of eigenvalues for every purely imaginary eigenvalue So a basis for a corresponding invariant subspace is easily determined from (3) For an eigenvalue pair λ k λ k we need to consider only the small amiltonian block [ R r k (R Note that R r r k ) k is upper triangular Suppose that the selection ω contains t k copies of λ k and s k copies of λ k A corresponding basis of the invariant subspace can then be chosen based on a symplectic permutation which exchanges trailing s k s k blocks in Rk r and (Rk r) In this section we have reviewed some results on the existence of (unitary) symplectic transformations to amiltonian block triangular form and the existence of Lagrangian invariant subspaces In the next section we use these results to give a full parametrization of all possible Lagrangian subspaces and therefore also a parametrization of all symplectic similarity transformations to amiltonian block triangular form 3 Parametrization of all Lagrangian invariant subspaces In the previous section we have shown that if has a amiltonian block triangular form then for every eigenvalue selection ω Ω() there exists at least one corresponding invariant subspace In this section we will parametrize all possible Lagrangian invariant subspaces associated to a given selection ω For this we will need some technical lemmas Lemma 31 Consider pairs of matrices (π k P rk N rk ) k =1 where r 1 r are either both even or both odd Let π 1 π {1 1} if both r k are even and π 1 π {i i} if both r k are odd; let ([ π1 P (P c N c ):= r1 π P r [ Nr1 N r ) ; and let d := r1 r Ifπ 1 =( 1) d+1 π ie β 1 = β for the corresponding β 1 and β then we have the following transformations to amiltonian triangular form: 1 If r 1 r then with I d Z 1 := I r 1 π Pr 1 π 1 P 1 d I r 1 π Pr 1 we obtain Z 1 P c Z 1 = J r 1 +r and Z 1 1 N cz 1 = [ N r1 +r D N r 1 +r where D = τe d e r 1 +r + τer 1 +r e d τ = 1 π

7 If r 1 <r then with LAGRANGIAN INVARIANT SUBSPACES 151 Z = π 1 P r1 1 I r 1 π P d π 1 P r1 1 I r 1 I d we obtain that Z P c Z = J r 1 +r and Z 1 N cz = [ N r1 +r D N r 1 +r where D = τe 1 e r τ 1 e r1+1e 1 τ = 1 π 1 Proof The proof is a simple modification of the proof of Lemma 18 in [19 Lemma 3 Consider a nilpotent matrix in Jordan form N = diag(n r1 N rp ) (i) If the columns of the full column rank matrix X form an invariant subspace of N ie NX = XA for some matrix A then X = UZ where Z is nonsingular and (31) U = I t1 V 1 V 1p 1 V 1p I t V p 1 V p I tp 1 V p 1p I tp ere for k =1p t k r k and for i =1p 1 and j = i+1p we have V ij C sitj with s i = r i t i Moreover if M s = diag(n s1 N sp ) M t = diag(n t1 N tp ) and E = diag(e t1 e 1 e tp e 1 ) then V = V 1 V 1p Vp 1p satisfies the algebraic Riccati equation M s V VM t VEV = (ii) If the columns of the full column rank matrix X form an invariant subspace of N ie N X = XA for some matrix A then X = ÛZ where Z is

8 15 GERARD FREILING VOLKER MERMANN AND ONGGUO XU nonsingular and (3) Û = Iˆt 1 ˆV 1 Iˆt ˆV p 11 ˆVp 1 Iˆt p 1 ˆV p1 ˆVp ˆVpp 1 Iˆt p ere for k =1p ˆt k r k and for i =p and j =1i 1 we have ˆV ij Cŝiˆt j with ŝ i = r i ˆt i Moreover if Mŝ = diag(nŝ1 Nŝp ) Mˆt = diag(nˆt 1 Nˆt p ) and E = diag(e 1 e ŝ 1 e 1 e ŝ p ) then ˆV = satisfies the algebraic Riccati equation ˆV 1 ˆV p1 ˆVpp 1 M ŝ ˆV ˆVM ˆt ˆVEˆV = Proof We will derive the structure of X by multiplying nonsingular matrices to X from the right Let us first prove part (i) Partition X = [ X 1 X so that X has r p rows Then using the QR or singular value decomposition [1 there exists a nonsingular (actually unitary) matrix Y 1 such that X =[X Y 1 where X C rptp and rank X = t p (This implies that t p r p ) Then we have the partition ˆX = XY1 1 = [ X 11 X 1 X Since range X is an invariant subspace of N so is range ˆX ence there exists a matrix  such that (33) N ˆX = ˆX If we partition  = [ A 11 A 1 A 1 A conformally with ˆX then (33) implies that A1 = and N rp X = X A Because X has full column rank and N rp is a single Jordan block it is clear that A is similar to N tp ie there exists a nonsingular matrix Y such that Y 1 A Y = N tp and hence N rp (X Y )=(X Y )N tp By Lemma 4411 in [14 X Y = [ T where T is an upper triangular Toeplitz matrix and T must be nonsingular since X has full column rank Therefore by setting X = ˆXY with Y = diag(iy )T 1 it follows that X 1 X X = I tp and (33) becomes N X = X [ à 11 à 1 N tp Setting Ñ = diag(n r1 N rp 1 ) it follows that Ñ X 1 = X 1 à 11 and since X has full column rank X1 also has full column rank

9 LAGRANGIAN INVARIANT SUBSPACES 153 By inductively applying the construction that leads from X to X we determine a nonsingular matrix Z 1 such that XZ1 1 = X where X has the block structure X = I t1 W 1 W 1p 1 W 1p V 1 V 1p 1 V 1p I t W p 1 W p V p 1 V p I tp 1 W p 1p V p 1p I tp with t i r i The blocks W ij in X can be eliminated by performing a sequence of block Gaussian type eliminations from the right ence there exists a nonsingular 1 matrix Z such that XZ = U where U is in (31) Therefore by setting Z := Z Z 1 we have X = UZ From the block form of U we can determine a block permutation matrix Q such that QU = [ I V and QNQ 1 = [ M t [ E M s Since I V is invariant to QNQ 1 we have M s V VM t VEV = Part (ii) is proved analogously by beginning the reduction from the top and compressing in each step to the left Using these lemmas we are able to parametrize the set of all Lagrangian invariant subspaces of a amiltonian matrix associated with a fixed eigenvalue selection in ω Ω() Let be in amiltonian block triangular form (13) and let the spectrum of be as in (1) Then (see [19) there exists a symplectic matrix S such that S 1 S = [ R D R where R = diag(r1 R µ+ν ) and D = diag(d 1 D µ+ν ) Furthermore the blocks are reordered such that k := [ R k D k R is amiltonian k block triangular and associated with an eigenvalue pair λ k λ k with nonzero real part for k =1µ and purely imaginary eigenvalues iα k for k = µ +1µ+ ν Furthermore Λ(R) =ω and range S [ I = L For this block diagonal form there exists a block permutation matrix P such that (34) P JP = diag(j n1 J nµ ; J m1 J mν )=: J P 1 S 1 SP = diag( 1 µ ; µ+1 µ+ν ) Suppose that there exists another Lagrangian invariant subspace L corresponding to ω Using the same argument there exists a symplectic matrix S such that for the same block permutation matrix P we have P 1 S 1 SP = diag( 1 µ ; µ+1 µ+ν ) where again all k are amiltonian block triangular and Λ( k ) = Λ( k ) for all k =1µ+ ν Therefore we have SP = SPE for some block diagonal matrix E = diag(e 1 E µ+ν ) satisfying k E k = E k k Since P JP = J and since S and S are symplectic it follows that E = P 1 S 1 SP satisfies E JE = J which implies that all blocks E k are symplectic Since S = SPEP 1 the difference between S and S (and therefore L and L) is completely described by the first half of the columns of the symplectic matrices E k ie the Lagrangian invariant subspaces of the small

10 154 GERARD FREILING VOLKER MERMANN AND ONGGUO XU amiltonian matrices k (note that all are amiltonian block triangular) Following this argument it is sufficient to parametrize all possible Lagrangian invariant subspaces of a amiltonian matrix with either a single purely imaginary eigenvalue iα or a single eigenvalue pair λ λ with Re λ Consider first the case of a single purely imaginary eigenvalue In this case Ω() has only one element So all Lagrangian invariant subspaces are associated to the same eigenvalue To simplify our analysis we need the following amiltonian Jordan form Lemma 33 Let be a amiltonian matrix that has only one eigenvalue iα Then there exists a symplectic matrix S such that (35) R := S 1 S = [ N(iα) D N(iα) where N = diag(n r1 N rp ) D = diag(d 1 D p ) ere either D j = βj ee r j e r j so that has a Jordan block N rj with structure inertia index βj e {1 1} ord j = τ j e dj e r j + τ j e rj e d j with τ j = 1 ( 1) r j +d j +1 iβ j if r j +d j is odd and τ j = 1 ( 1) r j +d j β j if r j + d j is even for some β j { 1 1} so that has two Jordan blocks N rj+d j N rj d j with structure inertia indices β j β j respectively Proof Since iαi is amiltonian we may without loss of generality (wlog) consider the problem with α = ie that has only the eigenvalue zero Since has only one multiple eigenvalue the columns of every nonsingular matrix span a corresponding invariant subspace so that condition (iii) of Theorem holds The canonical form (35) then is obtained in a similar way as for (3); see [19 The only difference is that here we match all possible pairs of Jordan block with opposite structure inertia indices in such a way that even blocks are matched with even blocks and odd blocks with odd blocks and furthermore the blocks are ordered in decreasing size Finally we use the technique given in Lemma 31 The complete parametrization is then as follows Theorem 34 Let be a amiltonian matrix that has only one purely imaginary eigenvalue Let S be symplectic such that S 1 S is in amiltonian canonical form (35) Then all possible Lagrangian subspaces can be parametrized by range SU where (36) U = I t1 V 1 V 1p 1 V 1p W 11 W 1p 1 W 1p I t V p 1 V p W1 W p 1 W p I tp 1 V p 1p W1p 1 W p 1p 1 W p 1p I tp W1p Wp 1p W pp I s1 V1 V1p 1 I sp 1 V1p Vp 1p I sp

11 LAGRANGIAN INVARIANT SUBSPACES 155 with block sizes s j t j r j and s j + t k = r j Then setting partitioning the ermitian blocks and setting it follows that the block matrices M t = diag(n t1 N tp ) M s = diag(n s1 N sp ) E = diag(e t1 e 1 e tp e 1 ) D j = [ Gj F j Fj K j K = diag(k 1 K s ) F = diag(f 1 F s ) G = diag(g 1 G s ) V := V 1 V 1p Vp 1p W 11 W 1p W := = W W1p W pp satisfy (37) [ [ Ms F W V Mt V [ [ W V E V E G [ W V + V [ W V V [ M s F M t [ K = or equivalently V W satisfy (38) (39) =M s V VM t VEV =(M s VE)W + W (M s VE) +(VF) + VF VGV K Every Lagrangian invariant subspace is uniquely determined by a set of parameters t 1 t p with t j r j and a set of matrices V ij i =1p 1 j = i +1p and W ij i =1p j = ip satisfying (38) and (39) Moreover all symplectic matrices that transform to amiltonian block triangular form can be parametrized as SUY where Y is a symplectic block triangular matrix

12 156 GERARD FREILING VOLKER MERMANN AND ONGGUO XU U =[U Ũ with U as in (36) and I t1 I t I tp 1 I Ũ = tp (31) I s1 I s I sp 1 I sp Proof As in Lemma 33 we assume that the only eigenvalue of is zero Considering the form (35) it is sufficient to prove that every basis X of a Lagrangian invariant subspace of R can be expressed as X = UY To prove this we first compress the X 1 X bottom square block of X ie we determine a matrix Y 1 such that XY 1 = [ X 11 where X has full column rank Obviously X 11 also has full column rank Then since XY 1 is still a basis of an invariant subspace of R the block triangular form of R implies that the columns of X 11 and X form bases of the invariant subspace of N and N respectively Applying Lemma 3 there exist matrices Z 1 and Z such that U 11 := X 11 Z 1 and U := X Z have structures as the matrices in (31) and (3) associated with the integer parameters t 1 t p and ˆt 1 ˆt p respectively Now let [ U := XY Z1 1 Z Y = [ U 11 U 1 U where Y is used to eliminate the blocks in X 1 Z using the identity blocks in U 11 Since X and hence also U is Lagrangian we have that U11U = Thus we have ˆt j = m j t j =: s j for all j =1p and ˆV ij = Vji for all i =p j =1p 1 Since U1U is ermitian it follows that U 1 has the desired form To prove (37) as in the proof of Lemma 3 there exists a block permutation matrix P such that P [U 11 U 1 = [ I V symplectic Then I PU = V W V P 1 R P = I W Let P = diag(p P) which is M t E G F M s F K Nt E Ns Since the columns of PU form an invariant subspace for P 1 R P it follows that the matrices VW satisfy (37) Conditions (38) and (39) follow directly from (37) To show the uniqueness of a particular Lagrangian invariant subspace suppose that there are two matrices U 1 U of the same form as U such that range SU 1 = range SU Then U JU 1 = and from this it follows first that the associated integer parameters t 1 t p must be the same and thus all the blocks V ij W ij must be the same To prove the second part let X be a symplectic matrix which triangularizes Since the first n columns of X form a Lagrangian invariant subspace there exists a matrix U of the form (36) such that range X [ I = range SU Then the matrix

13 LAGRANGIAN INVARIANT SUBSPACES 157 U =[U Ũ with Ũ as in (31) is symplectic Since both X and SU are symplectic and their first n columns span the same subspace there exists a symplectic block triangular matrix Y such that X = SUY These results show that the parameters that characterize a Lagrangian invariant subspace are integers t j with t j m j and the matrices V ij W ij satisfying the Riccati equations (37) or equivalently (38) and (39) Note that the equation for W is a singular Lyapunov equation The equation for V is quadratic But if we consider it blockwise it is equivalent to a sequence of singular Sylvester equations (311) N si V ij V ij N tj j 1 k=i+1 V ik E k V kj = for i = p 11 j = i +1p For results on nonsymmetric Riccati equations see [1 In general not much more can be said about this parametrization In the special case of a amiltonian matrix that has only two Jordan blocks we have the following result Corollary 35 Consider a amiltonian matrix that has exactly two Jordan blocks N r1 (iα)n r (iα) with < r r 1 and the corresponding structure inertia indices β 1 = β Then there exists a symplectic matrix S such that S 1 S = [ Nm (iα) D N m (iα) where m =(r 1 + r )/ d =(r 1 r )/ and D = τe d e m + τe m e d and τ = ±i/ if r 1 is odd and τ = ±1/ if r is even All Lagrangian invariant subspaces of can be parametrized by I t range S W I s and all symplectic matrices that transform to amiltonian block triangular form can be parametrized as I p S W I q I p Y I q where Y is symplectic block triangular d t m t + s = m W = W satisfying (31) N s W + WN s = which has infinitely many solutions for every s> Proof Note that r 1 + r is the size of the amiltonian matrix which must be even So r 1 and r must be both even or odd The canonical form and the form of the parametrization follow directly from Theorem 34 by setting there p = 1 So we need to prove only that d t m and that (31) holds For p = 1 (38) reduces to N s W + WN s = K

14 158 GERARD FREILING VOLKER MERMANN AND ONGGUO XU where K is the trailing s s block of D Then K = if t d (s r ) and K = τe d t e s + τe s e d t if t<d(s >r ) If t d then the singular Lyapunov equation has infinitely many ermitian solutions W ; see [11 14 If t<dand r 1 r are both even then it follows that τ is real By comparing the elements it follows that the Lyapunov equation has no solution The same conclusion follows for the case that r 1 r are both odd Consequently W exists if and only if d t m In this simple case the parameters are completely given But more importantly this result also gives a sufficient condition that a amiltonian matrix has infinitely many Lagrangian invariant subspaces Corollary 36 If a amiltonian matrix has exactly one eigenvalue iα and has at least two even-sized or two odd-sized Jordan blocks with opposite structure inertia indices then has infinitely many Lagrangian invariant subspaces Proof We may assume wlog that the two (even or odd) Jordan blocks are arranged in trailing position of R in the canonical form (35) Choosing t j = r j for all j =1p 1implies that all V ij are void W = W pp and I I tp U = W pp I sp By Corollary 35 there are infinitely many Lagrangian invariant subspaces (that are parametrized by W pp ) for the small amiltonian matrix p := [ Nmp (iα) D p N mp (iα) and hence there are also infinitely many Lagrangian invariant subspaces for This corollary shows that to obtain a unique Lagrangian invariant subspace all structure inertia indices of must have the same sign Moreover by Theorem this also implies that has only even-sized Jordan blocks In the next section we will prove that this is also sufficient In order to complete the analysis we need to study amiltonian matrices that have only two eigenvalues λ λ that are not purely imaginary If C nn then the algebraic multiplicities of λ λ are both n and hence Ω() consists of n +1 selections ω(m) m =n where ω(m) contains m copies of λ and n m copies of λ It follows from Theorem that in this case there exists a symplectic matrix S such that (313) R := S 1 S = [ N(λ) N(λ) where N(λ) =λi + N N = diag(n r1 N rp ) For every ω(m) m n the parametrization of all possible Lagrangian invariant subspaces can be derived in a similar way as in the case of purely imaginary eigenvalues Theorem 37 Let C n n be a amiltonian matrix that has only eigenvalues λ λ which are not purely imaginary Let S be a symplectic matrix that transforms to the form (313) For every selection ω(m) Ω() all the corresponding

15 LAGRANGIAN INVARIANT SUBSPACES 159 invariant subspaces can be parametrized by range SU where U has the form (314) I t1 V 1 V 1p 1 V 1p I t V p 1 V p I tp 1 V p 1p I tp I s1 V1 I s V1p 1 Vp 1 I sp 1 V 1p V p V p 1p I sp with s j t j r j s j + t j = r j and p j=1 t j = m Ifweset M t = diag(n t1 N tp ) M s = diag(n s1 N sp ) E = diag(e t1 e 1 e tp e 1 ) then the matrix V := V 1 V 1p Vp 1p must satisfy the Riccati equation (315) =M s V VM t VEV Every Lagrangian invariant subspace associated with ω(m) is uniquely determined by a set of parameters {t 1 t p } with t j r j and p j=1 t j = m and a set of matrices V ij i =1p 1 j = i +1p satisfying (315) Moreover all symplectic matrices that transform to amiltonian block triangular form can be parametrized by SUY where Y is symplectic block triangular

16 16 GERARD FREILING VOLKER MERMANN AND ONGGUO XU U =[U Ũ with U as in (31) and I t1 I t I tp 1 Ũ = I tp I s1 I s I sp 1 I sp Proof It is sufficient to consider the Lagrangian invariant subspaces of R in (313) Let the columns of X span a Lagrangian invariant subspace of R associated with ω(m) Then RX = XA and Λ(A) =ω(m) Since λ λ there exists a matrix Y such that Y 1 AY = [ A 1 A where A1 is m m and has only the eigenvalue λ and A is (n m) (n m) and has only the eigenvalue λ If we partition XY = [ X 11 X 1 X 1 X conformally with Y 1 AY then from the block diagonal form of R we obtain X 1 =X 1 = and N(λ)X 11 = X 11 A 1 N(λ) X = X A since X 11 X must have full column rank We apply Lemma 3 and then the result follows as in the case of purely imaginary eigenvalues The parametrization in this case is essentially the same as in the case of purely imaginary eigenvalues except that here W is void and p j=1 t j is fixed for a given ω(m) In both cases the blocks V ij still satisfy a sequence of Sylvester equations (311) Again we have a corollary Corollary 38 Let C n n be a amiltonian matrix that has only the eigenvalues λ λ which are not purely imaginary If has exactly two Jordan blocks with respect to λ then for every fixed ω(m) Ω() the corresponding Lagrangian invariant subspaces can be parametrized as S I t1 V I t I s1 V I s where t 1 + t = m t j + s j = r j and t j s j r j for j =1

17 LAGRANGIAN INVARIANT SUBSPACES 161 Furthermore V = [T if s 1 < t and V = [ T if s1 t where T is an arbitrary square upper triangular Toeplitz matrix So for every ω(m) with <m<n there are infinitely many Lagrangian invariant subspaces Proof Applying Theorem 37 for p = we obtain the parametrization and the restrictions for t 1 t The expression for V follows from the fact that V satisfies the Sylvester equation N s1 V VN t = In this special case we have the following uniqueness result Corollary 39 Let C n n be a amiltonian matrix that has only the eigenvalues λ λ which are not purely imaginary Then we have the following: (i) For ω() or ω(n) the corresponding Lagrangian subspace is unique (ii) If has only a single Jordan block with respect to λ then for every fixed ω(m) Ω() with m n the corresponding Lagrangian invariant subspace is unique In this case there exists a symplectic matrix Ŝ such that [ Ŝ 1 R D (316) Ŝ = R with R = diag(n m (λ) N n m (λ) ) D = e m e m+1 + e m+1 e m (iii) If has at least two Jordan blocks with respect to λ then for every fixed ω(m) Ω() with <m<nthere are infinitely many corresponding Lagrangian invariant subspaces Proof (i) For ω() all t j must be zero so U = [ I n is unique Analogously for ω(n) the unique Lagrangian invariant subspace is U = [ I n (ii) By assumption p = 1 so for a fixed ω(m) U is unique as I m I n m Then (316) follows from (313) and the special form U for p =1 (iii) In this case we can choose the integers t j such that t 1 <r 1 and t p > We set V ij = except for V 1p which is chosen to satisfy N s1 V 1p V 1p N tp = Since s 1 t p > there are infinitely many solutions V 1p and hence infinitely many U In the next section we will use the parametrizations to characterize the existence and uniqueness of Lagrangian invariant subspaces 4 Existence and uniqueness of Lagrangian invariant subspaces In this section we summarize all results given in the previous sections and give a complete characterization of the existence and the uniqueness of Lagrangian invariant subspaces for a amiltonian matrix This complete result includes previous results based on the structure inertia indices of [3 5 Theorem 41 (existence) Let C n n be a amiltonian matrix let iα 1 iα ν be its pairwise distinct purely imaginary eigenvalues and let λ 1 λ 1 λ µ λ µ be its pairwise distinct nonimaginary eigenvalues The following are equivalent: (i) has a Lagrangian invariant subspace for one ω Ω() (ii) has a Lagrangian invariant subspace for all ω Ω() (iii) There exists a symplectic matrix S such that S 1 S is amiltonian block triangular (iv) There exists a unitary symplectic matrix U such that U U is amiltonian block triangular

18 16 GERARD FREILING VOLKER MERMANN AND ONGGUO XU (v) For all k =1νifU k span the invariant subspace associated with iα k then Uk JU k is congruent to J mk (vi) Ind d S(iα k ) is void for all k =1ν Proof This result in different notation is known; see [ Theorem 4 (uniqueness for Ω()) Let C n n be a amiltonian matrix Let iα 1 iα ν be its pairwise distinct purely imaginary eigenvalues and let λ 1 λ 1 λ µ λ µ be its pairwise distinct nonimaginary eigenvalues Suppose that any of the equivalent conditions of Theorem 41 for the existence of Lagrangian invariant subspaces holds Then the following are equivalent: (i) For every ω Ω() there exists a unique associated Lagrangian invariant subspace (ii) If ω Ω() and if S 1 and S are symplectic matrices such that S1 1 S 1 = S 1 S = [ R D R [ R1 D 1 R and Λ(R1 )=Λ(R )=ω then S1 1 S is 1 symplectic block triangular (iii) There exists an ω Ω() such that has a unique associated Lagrangian invariant subspace (iv) There exists an ω Ω() such that if S 1 and S are symplectic matrices satisfying S1 1 S 1 = [ R 1 D 1 R 1 then S 1 S 1 S = [ R D R and Λ(R1 )=Λ(R )=ω 1 S is symplectic block triangular (v) Let [ A B A be an arbitrary amiltonian block triangular form of If for a purely imaginary eigenvalue iα k the columns of Φ k form a basis of the left eigenvector subspace of A ie Φ k A = iα kφ k then Φ k BΦ k is positive definite or negative definite (vi) For every purely imaginary eigenvalue iα k there are only even-sized Jordan blocks which furthermore have all structure inertia indices of the same sign If the uniqueness conditions do not hold then for every ω Ω() there are infinitely many Lagrangian invariant subspaces They can be parametrized by applying Theorem 34 for every iα k Proof The proof of the equivalence of (i) and (vi) has been given (in different notation) in Theorem 13 of [5 For completeness we give the whole proof in our terminology By the argument in section 3 it suffices to consider a amiltonian matrix that has either a single purely imaginary eigenvalue iα or an eigenvalue pair λ and λ In the first case we again take iα = Since by Corollary 39 for nonimaginary eigenvalues the corresponding invariant subspaces are unique we need to consider only the case of a purely imaginary eigenvalue The proofs of (i) (ii) and (iii) (iv) are obvious Corollary 36 implies that (ii) (vi) If (vi) holds then by Theorem there exists a symplectic matrix S such that (41) R := S 1 S = [ R D R where R = diag(n l1 N lq ) and D = β diag(e l1 e l 1 e lq e l q ) (Recall that iα = ) We need to prove only that for every symplectic Z satisfying Z 1 RZ = [ R is block triangular Partitioning Z = [ Z 11 Z 1 Z 1 Z it follows that D R Z (4) RZ 11 + DZ 1 = Z 11 R

19 LAGRANGIAN INVARIANT SUBSPACES 163 and (43) R Z 1 = Z 1 R Suppose that Z 1 ; then by (43) it follows that range Z 1 is an invariant subspace of R ence there exists a vector x such that Z 1 x and (44) R Z 1 x = ie Z 1 x is a left eigenvector of R Multiplying (Z 1 x) and x on both sides of (4) and using (44) we get (Z 1 x) D(Z 1 x)= x Z 1Z 11 Rx Since Z is symplectic we have Z 1Z 11 = Z 11Z 1 Combining (43) and (44) we get and therefore x Z 1Z 11 Rx = x Z 11Z 1 Ãx = x Z 11R Z 1 x = (Z 1 x) D(Z 1 x)= On the other hand since Z 1 x is a left eigenvector of R by the structure of R there must exist a nonzero vector y such that Z 1 x = Ey where (45) E := [e p1 e pq with p k = k j=1 l j for k =1q But E DE = βi q and hence =(Z 1 x) D(Z 1 x)=y E DEy = βy y which is a contradiction (i) (iii) is obvious and (iii) (i) follows from (iii) (vi) by Corollary 36 and (vi) (i) To prove (vi) (v) let ˆR = [ A B A be an arbitrary amiltonian triangular form of and let R be as in (41) Since (vi) holds and (vi) (ii) there exists a symplectic block triangular matrix S = [ S 1 S S (see [5) such that ˆR = S 1 RS 1 ence S1 1 RS 1 = A and B = S1 1 RS + S1 1 DS 1 + S R S1 Since A is similar to R a left eigenvector subspace of A can be chosen as Φ = S1 E where E is as in (45) Then a simple calculation yields Φ BΦ=βI q For (v) (vi) suppose that ˆR = [ A B A satisfies (v) Using the same argument as for (vi) (ii) and replacing R by ˆR we obtain that (v) (ii) Since (ii) (vi) it follows that (v) also implies (vi) Theorem 43 (uniqueness for Ω()) Let C n n be a amiltonian matrix Let iα 1 iα ν be its pairwise distinct purely imaginary eigenvalues and let λ 1 λ 1 λ µ λ µ be its pairwise distinct nonimaginary eigenvalues Suppose that any of the equivalent conditions of Theorem 41 for the existence of Lagrangian invariant subspaces holds Then the following are equivalent: (i) For every ω Ω() there exists a unique associated Lagrangian invariant subspace

20 164 GERARD FREILING VOLKER MERMANN AND ONGGUO XU (ii) Let ω Ω() If S 1 and S are symplectic matrices such that S1 1 S 1 = [ R1 D 1 R S 1 S = [ R D 1 R and Λ(R1 )=Λ(R )=ω then S1 1 S is symplectic block triangular (iii) There exists an ω Ω() but ω Ω() such that has a unique associated Lagrangian invariant subspace (iv) There exists an ω Ω() but ω Ω() such that if S 1 and S are symplectic matrices satisfying S1 1 S 1 = [ R 1 D 1 R S 1 S = [ R D 1 R and Λ(R 1 )=Λ(R )=ω then S1 1 S is symplectic block triangular (v) Let [ A B A be an arbitrary amiltonian block triangular form of Then either A has one of λ k λ k as its eigenvalue and has a unique corresponding left eigenvector or A has both λ k λ k as eigenvalues and has unique corresponding left eigenvectors x k and y k such that x k By k Furthermore for every iα k if the columns of Φ k form a basis of the left eigenvector subspace of A ie Φ k A = iα kφ k then Φ k BΦ k is positive definite or negative definite (vi) For every nonimaginary eigenvalue has only one corresponding Jordan block and for every purely imaginary eigenvalue iα k has only even-sized Jordan blocks with all structure inertia indices of the same sign If the uniqueness conditions do not hold then for every ω Ω() there are infinitely many Lagrangian invariant subspaces They can be parametrized by applying Theorem 34 for every iα k and Theorem 37 for every pair λ k λ k Proof The proof of the equivalence of (i) and (vi) has again been given (in different notation) in Theorem 13 of [5 For completeness we again give the whole proof in our terminology By the argument in section 3 it suffices to consider that the amiltonian matrix has only either a single purely imaginary eigenvalue iα or an eigenvalue pair λ and λ and in the first case we will assume iα = For the purely imaginary eigenvalue the proof is as that of Theorem 4 ence consider with an eigenvalue pair λ λ The parts (i) (ii) and (iii) (iv) are obvious (i) (vi) follows from Corollary 39 (i) (iii) follows since (iii) (vi) and (vi) (i) and since ω Ω() implies that both λ and λ have been chosen in ω It remains to prove (v) (vi) We may assume that both λ λ are in Λ(A) since otherwise ω Ω() For (vi) (v) let ˆR = [ A B A be an arbitrary amiltonian triangular form of Since (vi) holds by (316) in Corollary 39 the amiltonian canonical form is R = [ R D R where R = diag(nt (λ) N s (λ) ) D = e t e t+1 + e t+1 e t and t is the multiplicity of λ in Λ(A) By (ii) there exists a symplectic matrix S = [ S 1 S S such 1 that ˆR = S 1 RS ence S1 1 RS 1 = A and B = S1 1 RS + S1 1 DS 1 + S R S1 If only one of λ λ is in Λ(A) then since A is similar to R it is also in Λ(R) ence either t =ors = and R (and thus also A) has only one corresponding left eigenvector If both λ λ are in Λ(A) then s t > In this case R has only left eigenvectors e t and e t+1 with respect to λ and λ respectively Therefore A also has only left eigenvectors S 1 e t and S 1 e t+1 for λ and λ respectively Then it is easy to see that e t S 1 BS 1 e t+1 = e t De t+1 =1 For (v) (vi) if A has only one of λ λ as its eigenvalue and has a unique left eigenvector then A also has only one right eigenvector Since Λ(A) Λ( A )= in this case ˆR has also a unique corresponding right eigenvector Therefore there is only one corresponding Jordan block By the canonical form (313) for the other eigenvalue there is also only one Jordan block If A has both λ λ then we first show that for left eigenvectors x y of A such that x By condition (ii) holds Then we

21 LAGRANGIAN INVARIANT SUBSPACES 165 show that x y must be unique As in the proof of Theorem 4 we need only prove [ that for every symplectic matrix Z satisfying Z 1 ˆRZ = R D R Λ( R) =Λ(A) it follows that Z is block triangular Partitioning Z = [ Z 11 Z 1 Z 1 Z it follows that (46) AZ 11 + BZ 1 = Z 11 R and A Z 1 = Z 1 R Suppose that Z 1 ; then range Z 1 is an invariant subspace of A ence there exists z 1 with either Rz 1 = λz 1 or Rz 1 = λz 1 such that Z 1 z 1 which implies that Z 1 z 1 is the left eigenvectors of A corresponding to λ or λ Wlog assume that z 1 satisfies Rz 1 = λz 1 Let z satisfy z A = λz Multiplying z and z 1 on both sides of (46) a simple calculation yields z1 B(Z 1 z ) = which is a contradiction Suppose that x y are not unique Then let X form the left eigenvector subspace of A with respect to λ Since X By has more than one row there always exists a vector z such that z X By = which is a contradiction So x and y must be unique Remark For real amiltonian matrices it is reasonable to consider real Lagrangian invariant subspaces For this problem we have to give a natural additional restriction on the eigenvalue selections Note that in this case if λ is a nonreal eigenvalue of then λ λ and λ are also eigenvalues of To obtain real invariant subspaces it is necessary to keep the associated eigenvalues in conjugate pairs So if we choose a nonreal λ we must choose λ with same multiplicity But essentially we can use the same construction as for the complex case to solve this problem (see [19) since if is real then for real eigenvalues the corresponding invariant subspaces can be chosen real So for these eigenvalues we can still use Theorems 37 and 34 by choosing V and W real In this section we have given necessary conditions for the existence and uniqueness of Lagrangian invariant subspaces In the following section we obtain as corollaries several results on the existence and uniqueness of ermitian solutions of the algebraic Riccati equation 5 ermitian solutions of Riccati equations In this section we apply the existence and uniqueness results for Lagrangian invariant subspaces to analyze the existence and uniqueness of ermitian solutions of the algebraic Riccati equation (51) A X + XA XMX + G = with M = M and G = G The related amiltonian matrix is = [ A M G A The following result is well known; see eg [15 Proposition 51 The algebraic Riccati equation (51) has a ermitian solution if and only if there exists a n n matrix L = [ L 1 L with L1 L C n n and L 1 invertible such that the columns of L span a Lagrangian invariant subspace of the related amiltonian matrix associated to ω Ω() In this case X = L L 1 1 is ermitian and solves (51) and Λ(A MX)=ω It follows that we can study the existence and uniqueness of solutions of algebraic Riccati equations via the analysis of Lagrangian invariant subspaces of the associated amiltonian matrices

22 166 GERARD FREILING VOLKER MERMANN AND ONGGUO XU Unlike the Lagrangian invariant subspace problem which only depends on the Jordan structure ermitian solutions of the Riccati equation depend further on the top block of the basis of the Lagrangian invariant subspace and the choice of the associated eigenvalues In other words for a given amiltonian block triangular form R all amiltonian matrices which are similar to R have Lagrangian invariant subspaces while for Riccati equation solutions these amiltonian matrices may be partitioned into three groups which (i) have ermitian solutions for all selections Ω(R) (ii) have ermitian solutions for some ω Ω(R) (iii) have no ermitian solution for any ω Ω(R) Example 1 Consider three Riccati equations with matrices [ [ [ i 1 1 i 1+i (a) A = M = G = 1 [ [ [ i 1 1 i (b) A = M = G = 1 1+i [ [ [ i 1 1+i (c) A = M = G = 1 1 i In all three cases the related amiltonian matrices have the same amiltonian Jordan canonical form i 1 1 i 1 and Ω() ={ω 1 ω } with ω 1 = {i 1} ω = {i 1} Certainly for both ω 1 ω all amiltonian matrices have a unique Lagrangian invariant subspace But the ermitian solutions of the Riccati equation are different In case (a) for ω 1 the solution is X = and for ω there is no solution In case (b) for ω 1 the solution is and for ω the solution is X = [ 1 In case (c) for both ω1 and ω there is no solution at all It is also possible that the Riccati equation has no ermitian solution while the related amiltonian matrix has infinitely many Lagrangian invariant subspaces Example For A = M = [ [ 1 G = 1 the Riccati equation (51) has no solution But for the associated amiltonian matrix the bases of the Lagrangian invariant subspace can be parametrized as iβ 1 1 α iβ γ where α β γ are real By using the parametrizations in section 3 we can give a necessary and sufficient condition for the existence of the ermitian solutions of the Riccati equation (51) Note that for the solvability of the Riccati equation it is necessary that the amiltonian matrix associated to (51) has a amiltonian block triangular form So there

23 LAGRANGIAN INVARIANT SUBSPACES 167 exists a symplectic matrix S such that (5) S 1 S = [ R D R with R = diag(r 1 R µ ; R µ+1 R µ+ν ) D = diag(; D µ+1 D µ+ν ) A submatrix k := [ R k R has the Jordan form (313) with respect to the eigenvalues k λ k λ k for k =1µ and a submatrix k := [ R k D k R has the Jordan form (35) k with respect to iα k µ for k = µ +1µ+ ν Theorem 5 Let be the amiltonian matrix associated with the algebraic Riccati equation (51) and assume that has a amiltonian block triangular form Let S be a symplectic matrix satisfying (5) and let P be a permutation matrix such that (53) P 1 S 1 SP = diag( 1 µ ; µ+1 µ+ν ) P JP = diag(j n1 J nµ ; J m1 J mν ) where k = [ R k R Then for an eigenvalue selection ω Ω() the Riccati k equation (51) has a ermitian solution X with Λ(A MX)=ω if and only if there exist matrices U 1 U µ and Q 1 Q ν with the following properties The matrices U k are n k n k and have the block form (31) with blocks satisfying (315) and the matrices Q k are m k m k and have the block form (36) with blocks satisfying (38) and (39) such that (54) L 1 := [I n SP diag(u 1 U µ ; Q 1 Q ν ) is nonsingular Moreover X = [I n SP diag(u 1 U µ ; Q 1 Q ν )L 1 1 Proof Since has a amiltonian block triangular form we have (5) and P can easily be determined to obtain (53) A given ω specifies the number elements λ k λ k and hence by Theorems 37 and 34 we obtain the parametrizations for the bases of the associated Lagrangian invariant subspaces of Thus by Proposition 51 we have the conclusion Remark 3 If in the amiltonian matrix = [ A M G A the matrix M is positive or negative semidefinite then the invertibility of L 1 in (54) is ensured by a controllability assumption; see Theorem 31 and Remark 3 in [9 or [15 for details If (51) has a ermitian solution with respect to a selection ω then the uniqueness follows directly from the uniqueness results for Lagrangian invariant subspaces Theorem 53 Let X = X be a ermitian solution of (51) with Λ(A MX)= ω Then X associated to ω is unique if and only if the related amiltonian matrix has a unique Lagrangian invariant subspace associated to ω Moreover in this case if ω Ω() then for every selection in Ω() for which the associated ermitian solutions exists it is unique If the uniqueness condition for the Lagrangian invariant subspaces of does not hold and if (51) has at least one ermitian solution associated with a selection ω then (51) has infinitely many ermitian solutions associated to ω Proof The uniqueness conditions for the ermitian solutions follows from the equivalence of (i) and (iii) in Theorems 4 and 43