A matrix function useful in the estimation of linear continuous-time models

Statistics & Operations Research Transactions SORT 30 (1) January-June 2006, 85-90 Statistics & Operations Research Transactions A matrix function useful in the estimation of linear continuous-time models c Institut d’Estad´ ıstica de Catalunya [email protected] ISSN: 1696-2281 www.idescat.net/sort Heinz Neudecker Cesaro Abstract In a recent publication Chen & Zadrozny (2001) derive some equations for efficiently computing e A and ∇ e A , its derivative. They employ an expression due to Bellman (1960), Snider (1964) and Wilcox (1967) for the differential de A and a method due to Van Loan (1978) to find the derivative ∇eA.The present note gives a ) a short derivation of ∇ e A by way of the Bellman-Snider-Wilcox result, b ) a shorter derivation without using it. In both approaches there is no need for Van Loan’s method. MSC: 15A69 Keywords: matrix derivatives, vectorization, matrix exponential 1 Introduction In a recent publication Chen & Zadrozny (2001) consider the matrix exponential eA=In+A+1 2!A2+···+1 k!Ak+···. Their aim is to find ∇eAwhich is implicitly defined by dvec eA=∇eAdvec A Address for correspondence: Oosterstraat, 13. 1741 GH Schagen. The Netherlands. E-mail: [email protected] Received: February 2005 Accepted: June 2006 86 A matrix function useful in the estimation of linear continuous-time models or explicitly as ∇eA=∂vec eA ∂(vec A). They introduce a matrix function K(t)=t τ=0 e(t−τ)A⊗eτAdτ and recall that de A=1 τ=0 eτA(dA )e(1−τ)Adτ. See Bellman (1960, p. 171, (7)), Snider (1960) and Wilcox (1967). It then follows by vectorization that dvec eA=K(1) dvec A. The authors compute K(1) by employing a method due to Van Loan (1978). It turns out that K(1) is the submatrix in the northeast corner of eC, where C=⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ A⊗InIn2 0In⊗A⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠. The submatrix is denoted by G1(1). In this note we shall use two methods to find ∇eA: a) one by way of the Bellman-Snider-Wilcox result, b) another more direct one without using that earlier result. 2 The derivation through K (1) Starting from the expression K(1) =1 τ=0e(1−τ)A⊗InIn⊗eτAdτ, which is also to be found in Chen & Zadrozny, we further develop K(1) =1 τ=0 e(1−τ)A⊗IneτIn⊗Adτ= =1 τ=0 e(1−τ)A⊗In+τIn⊗Adτ= Heinz Neudecker 87 =1 τ=0 eA⊗In+τ(In⊗A−A⊗In)dτ= =eA⊗In1 τ=0 eτ(In⊗A−A⊗In)dτ= =eA⊗InIn2+1 2! (In⊗A−A⊗In)+···+1 (k+1)! (In⊗A−A⊗In)k+···= =In2+1 2! (In⊗A+A⊗In)+···+1 (k+1)! (In⊗A+A⊗In)(k)+··· where for commuting A and B: (A+B)(i)= i−1  j=1 AjBi−j+Ai+Bi,(A+B)(1) =A+B. Clearly In⊗Aand A⊗Incommute. We used Properties 3, 4 and 5 of the Appendix. It is clear that K(1) =G1(1), given the following computation. Consider C=⎛ ⎜ ⎜ ⎜ ⎜ ⎝ PI OQ ⎞ ⎟ ⎟ ⎟ ⎟ ⎠and eC=⎛ ⎜ ⎜ ⎜ ⎜ ⎝ RS TU ⎞ ⎟ ⎟ ⎟ ⎟ ⎠. Then T=O,R=I+ ∞  i=1 1 i!Pi=eP,U=I+ ∞  i=1 1 i!Qi=eQand S=In2+ ∞  i=1 1 (i+1)!(P+Q)(i). Hence G1(1) =In2+ ∞  i=1 1 (i+1)! A⊗I+I⊗A(i)=K(1). We can also define C=⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ In⊗AI n2 0A⊗In⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ to get the same G1(1). 88 A matrix function useful in the estimation of linear continuous-time models 3 A direct derivation of ∇ ∇ ∇ e A Still simpler is to proceed as follows. Differentiation of eAyields de A=dA +1 2! {(dA)A+AdA}+1 3! (dA)A2+A(dA)A+A2dA+··· and from this by vectorization dvec eA=dvec A+1 2! (In⊗A+A⊗In)dvec A+1 3! In⊗A2+A⊗A+(A)2⊗I ×dvec A+··· = =In2+1 2! (In⊗A+A⊗In)+1 3! (In⊗A+A⊗In)(2) +···dvec A= =In2+ ∞  i=1 1 (i+1)! In⊗A+A⊗In(i)dvec A. Hence ∇eA=∂vec eA ∂(vec A)=In2+ ∞  i=1 1 (i+1)! In⊗A+A⊗I(i). 4 Appendix Some algebraic properties: 1. vec ABC =(C⊗A)vec B. 2. (A⊗B)( C⊗D)=AC ⊗BD. 3. eI⊗A=I⊗eA,eA⊗I=eA⊗I. 4. eA·eB=eA+Bfor commuting Aand B. 5. eBI+1 2!(A−B)+1 3!(A−B)2+···=I+1 2!(A+B)+1 3!(A+B)(2) +··· where (A+B)(i)= i−1  j=1 AjBi−j+Ai+Bifor commuting Aand B. Heinz Neudecker 89 References Bellman, R. (1960). Introduction to Matrix Analysis. McGraw-Hill, New York. Chen, B. and Zadrozny, P. A. (2001). Analytic derivatives of the matrix exponential for estimation of linear continuous-time models. Journal of Economic Dynamics &Control, 25, 1867-1879. Neudecker, H. (1971). On a theorem of Snider and Wilcox. METU Journal of Pure and Applied Sciences, 4, 217-220. Snider, R. F. (1964). Perturbation variation methods for a quantum Boltzmann equation. Journal of Mathematical Physics, 5, 1580-1587. Van Loan, C. F. (1978). Computing integrals involving the matrix exponential. IEEE Transactions on Automatic Control, 23, 395-404. Wilcox, R. M. (1967). Exponential operators and parameter differentiation in quantum physics. Journal of Mathematical Physics, 8, 962-982.