Linear Operators on the Real Symmetric Matrices whose Exponential Preserves the Inertia Robert Stelzer∗ 13th December 2006

Abstract The linear operators B on the real symmetric matrices whose exponential exp(Bt) leaves the inertia invariant and maps the positive (semi-)definite matrices onto themselves for all t ∈ R are fully characterized.

AMS Subject Classification 2000: 15A04, 15A48, 15A57 Keywords: exponential operator, inertia, linear preserver, positive semi-definite matrices, real symmetric matrices

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Introduction

Let Sd be the space of real symmetric d × d matrices and denote for a matrix A ∈ Sd its inertia by In(A) = (r, s, t) where the triplet of natural numbers r, s, t means that A has r strictly positive, s strictly negative and t zero eigenvalues (counted with their respective algebraic multiplicities). In this paper we show that the exponential group generated by a linear operator B : Sd → Sd preserves the inertia, i.e. In(eBt A) = In(A) for all t ∈ R and A ∈ Sd , and that eBt maps the positive (semi)-definite matrices onto themselves for all t ∈ R, if and only if there is a d×d matrix B such that B is representable as X → BX +XB T . That the exponential of linear operators of this form preserves the inertia is not hard to be seen (cf. the upcoming Theorem 2.2) and well-known. Thus the main contribution of this paper is to establish the converse. The proof given is based on results on the linear preservers of the inertia and the positive (semi-)definite matrices, respectively. For an overview over linear preserver problems and references to the original literature see Pierce, Lim, Loewy, Li, Tsing, McDonald & Beasley (1992), in particular its third chapter Loewy (1992), or Li & Pierce (2001). The complete characterization of linear operators having the above mentioned properties is of particular interest in applications, since one often wants to construct ∗

Graduate Programme Applied Algorithmic Mathematics, Centre for Mathematical Sciences, Munich University of Technology, Boltzmannstraße 3, D-85747 Garching, Germany. Email: [email protected], www.ma.tum.de/stat

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models for the time evolution of a positive definite matrix based on (stochastic) differential equations whose solution involves the exponential of a given linear operator. As regards stochastic processes, Ornstein-Uhlenbeck type processes taking values in the positive (semi-)definite matrices were recently introduced in Barndorff-Nielsen & Stelzer (2006). These processes can be used as a flexible and very tractable model for the stochastic evolution of some covariance matrix in continuous time, as can be seen in Pigorsch & Stelzer (2006) who present a detailed analysis of the second-order moment structure and an application to the modelling of financial data. BarndorffNielsen & Stelzer (2006) used linear operators B on the symmetric d × d matrices of the form X → BX + XB T for some d × d matrix B in order to successfully construct Ornstein-Uhlenbeck type processes in the positive (semi-)definite matrices. The results of the present paper show that these are indeed the only linear operators possible when one demands that exp(Bt) maps the positive semi-definite cone onto itself at all times. The latter is a very realistic demand, as it ensures that the OrnsteinUhlenbeck type process has a non-degenerate distribution whenever the driving L´evy process has one. Furthermore linear operators of the type X → BX + XB T have also been used in ordinary differential equations in order to ensure positive (semi-)definiteness of the solution (see Dragan, Freiling, Hochhaus & Morozan (2004), for instance). For example, the elementary differential equation x (t) = Bx(t) in the symmetric d × d matrices has the solution x(t) = eBt x0 . In this set-up demanding that eBt maps the positive semi-definite matrices onto themselves thus means that x(t) varies over all of the positive semi-definite cone when x0 does so.

Notation Throughout this paper we write R+ for the positive real numbers including zero and we denote the set of real d × d matrices by Md (R) and the group of invertible d × d matrices by GLd (R), the linear subspace of symmetric matrices by Sd (R), the positive ++ semidefinite cone by S+ d (R) and the open positive definite cone by Sd (R). Id stands T for the d × d identity matrix. Finally, A is the transposed of a matrix A ∈ Md (R) and the exponential of a matrix or linear operator A is denoted by exp(A) or eA . Moreover, for a matrix A we denote by Aij the element in the i-th row and j-th column and this notation is extended to matrix-valued functions in a natural way. The standard basis matrices (i.e. the matrices which have only zero entries except for a one in the i-th row and j-th column) of Md (R) are denoted by E (ij) for i, j = 1, 2, . . . , d.

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Exponential inertia preservers

Definition 2.1 (Exponential inertia preserver). A linear operator B : Sd → Sd is said to be an exponential inertia preserver, if In(eBt A) = In(A) for all t ∈ R and A ∈ Sd . We start by recalling a well-known result showing that a special class of linear operators are exponential inertia preservers. 2

Theorem 2.2. Assume that a linear operator B : Sd → Sd can be represented as X → BX + XB T for some C ∈ Md (R). Then B is an exponential inertia preserver. T

Proof: It is easy to see that eBt A = eBt AeB t using e.g. Horn & Johnson (1991, pp. 255 and 440). Thus the result follows immediately from the general results on inertia preservers (see Loewy (1992) or Li & Pierce (2001) and the original articles cited therein), as the exponential of a matrix is necessarily invertible. It is important to note that the linear operator B above is uniquely characterized by the matrix B. Proposition 2.3. Let B, C ∈ Md (R). Then the linear operators B : Sd → Sd , X → BX + XB T and C : Sd → Sd , X → CX + XC T are the same, if and only if B = C. Moreover, for any operator B : Sd → Sd , X → BX +XB T the matrix B is already uniquely identified by the values {BE (ii) }i=1,...,d . Proof: It suffices to show the second claim that {BE (ii) }i=1,...,d already uniquely characterizes B. It is easy to see that ⎧ for k = l = i 2Bii ⎪ ⎪ ⎨ Bil for k = i and k = l (BE (ii) + E (ii) B T )kl = for l = i and k = l Bik ⎪ ⎪ ⎩ 0 otherwise Thus BE (ii) uniquely characterizes the i − th column of B and, hence, {BE (ii) }i=1,...,d uniquely characterizes B. In the following we establish the converse result that all exponential inertia preservers are of the form given in Theorem 2.2. Lemma 2.4. Let B : Sd → Sd be a linear operator and assume that there exists an  > 0 and a function D :] − , [→ Md (R) such that eBt A = D(t)AD(t)T for all t ∈  > 0, a continuously differentiable function ] − , [ and A ∈ Sd . Then there exists an   :] − , [→ Md (R) and a unique matrix B ∈ Md (R) such that eBt A = D(t)A   T D D(t) for all t ∈] −  ,  [ and A ∈ Sd and such that BA = BA + AB T for all A ∈ Sd .  Proof: We first show the existence of a matrix D(t) stated properties. Ob  with the(11) = 0 the operator serve that we obtain as a side result that provided exp(Bt)E 11 (11) and exp(Bt)(E (1,j) + exp(Bt) is already fully identified by the values exp(Bt)E (j,1) E ) with j = 2, 3, . . . , d and the fact that it can be represented as X → D(t)XD(t)T . Note that D(t)XD(t)T = (−D(t))X(−D(t))T for all X ∈ Sd , so the matrix D(t) can only be unique up to a multiplication by minus one. Elementary calculations give 2

D11 D11 D21 D11 D31 · · · D11 Dd1 (11) exp(Bt)E = (2.1) ∗ ∗ ∗ ··· ∗ and   exp(Bt) E (1j) + E (j1) = (2.2)

D11 D1j + D11 D1j D1j D21 + D11 D2j · · · D1j Dd1 + D11 Ddj ∗ ∗ ··· ∗ 3

for j = 2, 3, . . . , d. Here Dkl denotes Dkl (t) for notational convenience and ∗ represents entries which are of no interest in the following. It is easy to see that the above characterize the matrix D(t) up to the sign of D11 (t), as long as  equations uniquely (11) exp(Bt)E = D11 (t)2 = 0. 11   Since exp(B · 0) is the identity on Sd and thus exp(B · 0)E (11) 11 = 1, the con→ exp(Bt) ensures that there exists an   > 0 with   ≤  such that tinuity of t(11) exp(Bt)E > 0 for all t ∈] −  ,  [. Thus it follows from (2.1) and (2.2) that 11  ∈ Md (R) defined by D(t)   11 (t) = (exp(Bt)E (11) ) D (2.3) 11   exp(Bt)E (11) i1  Di1 (t) = for i = 2, 3, . . . , d (2.4)  11 (t) D   exp(Bt)(E (1j) + E (j1) ) 11  1j (t) = D for j = 2, 3, . . . , d (2.5)  11 (t) 2D       exp(Bt)(E (1j) + E (j1) ) 1i exp(Bt)(E (1j) + E (j1) ) 11 exp(Bt)E (11) i1  ij (t) = D −  11 (t)  11 (t)3 D 2D for i, j = 2, 3, . . . , d (2.6) is well-defined for all t ∈] −  ,  [ and satisfies   T ∀ t ∈] − , [ and A ∈ Sd . exp(Bt)A = D(t)A D(t) The continuous differentiability of t → exp(Bt) and the one of the square root function on R+ \{0} imply together with (2.3) to (2.6) and the strict positivity of    is continuously differenexp(B · t)E (11) 11 that the map ] − , [→ Md (R), t → D(t) tiable. Using the notion of Fr´echet derivatives (see Rudin (1976) or Bhatia (1997, Section X.4) for a review in connection with matrix analysis) and denoting the linear operators on Sd by L(Sd ), it follows immediately from (X + H)A(X + H)T = XAX T + XAH T + HAX T + HAH T ∀ A ∈ Sd for X, H ∈ Md (R) that the map f : Md (R) → L(Sd ), X → f (X) with f (X)A = XAX T for all A ∈ Sd is continuously differentiable and the derivative Df (X) is given by the linear map Md (R) → L(Sd ), H → Df (X)(H) with Df (X)(H)A = XAH T + HAX T for A ∈ Sd . Thus





d d  d   exp(Bt) A = f D(t) A = Df (D(t)) D(t) A dt dt dt

T

d d    T  D(t) D(t) AD(t) + = D(t)A dt dt for all t ∈] − , [ and A ∈ Sd . Since dtd exp(Bt) = exp(Bt)B, it follows that B = exp(−Bt) dtd exp(Bt). Moreover,  ∈ GLd (R) and exp(Bt)−1 A = exp(−Bt)A = D(t)  −1 AD(t)  −T it is easy to see that D(t) 4

for A ∈ Sd and hence





T

d d   −T    T D(t)  −1 D(t)A D(t) D(t) AD(t) BA = D(t) + dt dt

T

d d −1 −1     = A D(t) D(t) D(t) A + D(t) dt dt

(2.7) (2.8)

    for A ∈ Sd and all t ∈] − , [. As by construction D(0) = Id , setting B = dtd D(t) t=0 concludes the proof now noting that Proposition 2.3 ensures the uniqueness of B ∈ Md (R). For d ≥ 3 we can use the above Lemma to fully characterize exponential inertia preservers. Theorem 2.5. Let d ∈ N with d ≥ 3. Then the following holds (i) A linear mapping B : Sd → Sd is an exponential inertia preserver, if and only if there exists a matrix B ∈ Md (R) such that BX = BX + XB T for all X ∈ Sd . (i) A linear mapping B : Sd → Sd is an exponential inertia preserver, if there exists an  > 0 such that In(eBt A) = In(A) for all A ∈ Sd and t ∈] − , [. That we need to restrict ourselves to d ≥ 3 is clear from the proof below and Johnson & Pierce (1985, Remark 1.3). Proof: (i): The ’if’ part is given in Theorem 2.2. Regarding the ’only if’ part we have that for d ≥ 3 all linear preservers B : Sd → Sd on Sd of the inertia class (d − 1, 1, 0) are of the form X → CXC T for some C ∈ Md (R) (cf. Loewy (1992) and references therein). Hence, there is a function D(t) : R → Md (R) such that eBt A = D(t)AD(t)T for all t ∈ R and A ∈ Sd . Lemma 2.4 immediately concludes now. (ii): This follows from Theorem 2.2 and Lemma 2.4 in a straightforward manner. Moreover, we can now also characterize the linear operators whose exponential group maps the positive semi-definite matrices onto themselves. Corollary 2.6. Let d ∈ N. Then the following holds + (i) A linear mapping B : Sd → Sd satisfies eBt (S+ d ) = Sd for all t ∈ R, if and only if there exists a matrix B ∈ Md (R) such that BX = BX + XB T for all X ∈ Sd . + (i) A linear mapping B : Sd → Sd satisfies eBt (S+ d ) = Sd for all t ∈ R, if there + exists an  > 0 such that eBt (S+ d ) = Sd for all t ∈] − , [. T Proof: Since all linear maps that map S+ d onto itself are of the form X → CXC for some C ∈ Md (R) (cf. Loewy (1992) or the original article Schneider (1965)), the proof is analogous to the one of the last theorem. ++ S+ in the above corollary. Yet we cannot extend the result d can be replaced by Sd + + + Bt to the case e (Sd ) ⊂ Sd , as there are linear operators C such that C(S+ d ) ⊂ Sd which are not representable by X → CXC T for some C ∈ Md (R) (cf. Choi (1975)). Furthermore note that from the characterization of linear preservers of various other fixed inertia classes (see Loewy (1992) for an overview) many results analogous to the above ones follow immediately, since we obviously have the following general result:

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Theorem 2.7. Let P be some property of linear maps on Sd and assume that a linear map C : Sd → Sd has the property P, if and only if there is a matrix C ∈ Md (R) such that CA = CAC T for all A ∈ Sd . Then the following holds: (i) The exponential eBt of a linear map B : Sd → Sd has the property P for all t ∈ R, if and only if there exists a matrix B ∈ Md (R) such that BX = BX + XB T for all X ∈ Sd . (i) The exponential eBt of a linear map B : Sd → Sd has the property P for all t ∈ R, if there exists an  > 0 such that eBt has the property P for all t ∈ R t ∈]−, [.

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