EXPANSIONS FOR FUNCTIONS OF DETERMINANTS OF POWER SERIES

Transcription

1 CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 18, Number 1, Spring 010 EXPANSIONS FOR FUNCTIONS OF DETERMINANTS OF POWER SERIES CHRISTOPHER S. WITHERS AND SARALEES NADARAJAH ABSTRACT. Let Aɛ) be an analytic n n matrix function of a scalar ɛ. We obtain expansions in powers of ɛ for det Aɛ) and smooth functions of it including its powers and its log. An application is given to multivariate statistics. 1 Introduction Suppose that Aɛ) is an n n complex matrix function of a complex scalar ɛ expandable in the form 1.1) Aɛ) = a k where {a k } are n n complex matrices. Suppose also that det a 0 0. We show how to obtain power series expansions for functions of det Aɛ). As examples we give the coefficients in the expansions 1.) 1.3) 1.4) log det Aɛ) = det Aɛ) = { } α det Aɛ) = 1 + det A0) g k c k c k k!. Functions of the form 1.) 1.4) arise often in multivariate analysis see Srivastava [10], Anderson [1], Kollo and von Rosen [8], Eaton [4], Härdle and Simar [6], Johnson and Wichern [7]). Keywords: Determinants, expansions, power series. Copyright c Applied Mathematics Institute, University of Alberta. 107

2 108 C. S. WITHERS AND S. NADARAJAH Main results We first show how to obtain power series expansions for functions of gɛ) = log det Aɛ), say.1) fgɛ)) = b k where f : C C is any function which is analytic about.) g 0 = g0) = log det a 0 and C is the set of complex numbers. By Taylor s series,.1) holds with b k = d/dɛ) k fgɛ)) ɛ=0, so that b 0 = fg 0 ) for g 0 of.). By Faa di Bruno s formula [, page 137],.3) b k = f j B kj g) for k 1, where f j = f j) g 0 ), g = g 1, g,...), g j = g j) 0), and B kj g) is the partial exponential Bell polynomial tabled on pages of [] to k = 1, and defined by g k /k! ) j j! = k=j B kj g) ɛk k! for j 1. So, in terms of f = f 0, f 1,...) and g, the first seven coefficients in.3) are.4).5) b 0 = f 0 = fg 0 ), b 1 = f 1 g 1,.6) b = f 1 g + f g1,.7) b3 = f1g3 + f3g1g) + f3g1 3, b4 = f1g4 + f 4g1 g 3 + 3g ) ) + f3 6g.8) 1 g + f4 g1, 4 b 5 = f 1 g 5 + f 5g 1 g g g 3 ) + f 3 10g 1 g g 1 g ).9) + f 4 10g 3 1 g ) + f 5 g 5 1,

3 FUNCTIONS OF DETERMINANTS OF POWER SERIES ) b 6 = f 1 g 6 + f 6g1 g g g g3 ) + f 3 15g 1 g g 1 g g g 3 ) + f 4 0g 3 1 g g 1 g ) + f5 15g 4 1 g ) + f6 g 6 1. We now give expressions for {g j } of 1.). Set.11) A k = ) k d Aɛ), A k = A 1 0 dɛ A k. Since d/dɛ)a 1 0 = A 1 0 A 1A 0, direct calculation gives g 1) ɛ) = tr A 1 = tr A 1 say, g ) ɛ) = tr A A 1), g 3) ɛ) = tr A 3 3A 1 A + A 3 1), and so on. Set a k = a 1 0 a k. Since A k ɛ=0 = a k, we have.1) g 1 = tr a 1,.13) g = tr a a 1 ),.14) g3 = tr a3 3a1a + a 3 1),.15).16).17) g 4 = tr a 4 4a 1 a 3 a + 1a 1a 6a 4 1), g 5 = tr a 5 5a 1 a 4 10a a 3 + 0a 1 a a 1 a 60a3 1 a + 4a 5 1 ), g 6 = tr a 6 6a 1 a 5 15a a 4 10a a 1 a a1 a a a 3 10a3 1 a 3 6 ) a 1 a + 360a4 1 a 10a 6 1, and so on, where.18) a1 a a 3 = a 1 a a 3 + a 3 a ),

4 110 C. S. WITHERS AND S. NADARAJAH.19) 6 a 1 a = 4a 1 a + a 1a ). To obtain the general term, consider first the univariate case n = 1: by.1) and.3) with fg) = log g, f j = 1) j 1 j 1)!g j 0, and for k 1, g k = f j B kj a) = 1) j 1 j 1)! B kj a), where a = a 1, a,...) and a = a 1, a,...). Comparison with.1).17) shows that for general n for k 1,.0) g k = 1) j 1 j 1)! B kj a), where B kj a) is tr B kj a) with B 1 B r replaced by B 1 B r = r!) 1 r B π1 B πr, where r sums over all r! permutations π 1,..., π r ) of 1,..., r). Here, B 1,..., B r is any sequence of matrices from {a j }. One then simplifies using tr B 1 B = tr B B 1. For example, tr B r 1 B = tr B r 1 B, tr B 1 B B 3 = 1 tr B 1 B B 3 + B 3 B ), tr B 1 B = 6 1 tr 4B 1 B + B 1B ) ), tr B 1 B B r = tr B 1 B B r. The second and third of these four equations give.18) and.19). Example.1. Suppose A = I n + ɛa for I n the identity matrix and a an n n matrix. Then for k 1, g k = 1) k 1 k 1)! tr a k, so that log det I n + ɛa) = tr log I n + ɛa), where logi n +ɛa) is defined as the matrix power series 1) k 1 ɛa) k /k. This is a special case of log det A = tr log A,

5 FUNCTIONS OF DETERMINANTS OF POWER SERIES 111 a result that is given in Dunford and Schwarz [3, p. 109] and Glazman and Ljubic [5, p. 140] or in an equivalent form by Perko [9, p. 16]. A proof for any nonsingular square matrix A is easily obtained by defining log A in terms of the Jordan canonical form of A. Example.. Take fg) = expg). So, f j det a 0 and det Aɛ) = c k where c 0 = det a 0, and by.3), for k 1, c k = c 0 c k, where c k = B kj g) = B k g) say with g given by.1).0). That is, det Aɛ) det a 0 = 1 + c k k! with c k = b k of.3).10) with f j 1. Comtet [] calls B k g) the complete exponential Bell polynomial. Example.3. Take fg) = expαg) for some scalar α. So, f j = α j and { } α det Aɛ) = 1 + c k det a 0 where c k = b k of.3).10) with f j = α j. By the above result, for F g) = log g and a 0 = I n, F detaɛ)) = F 1) + G k where G k = F j) 1)B kj a).

6 11 C. S. WITHERS AND S. NADARAJAH By Faa di Bruno s formula the same holds with B kj a) replaced by B kj c) for c of 1.3). However, B kj c) B kj a) for a 0 = I n, k 1. So, ) does not hold for general smooth F. For general smooth F we do have in terms of c 0 = det a 0 and c of 1.3), even if a 0 I n, where F detaɛ)) = F c 0 ) + G k = F j) c 0 )B kj c). G k However, unless n is small and Aɛ) is a finite series, that is a polynomial in ɛ, these coefficients are better obtained using 1.) and.) since the coefficients g of 1.) each being the trace of a matrix, are more fundamental than the coefficients c of 1.3) which are nonlinear functions of traces derived from g. Our results have been presented in terms of power series. Alternatively, they can be written as derivatives of functions. So,.3) can be written as d/dɛ) k f log det Aɛ)) = f j) log det Aɛ)) B kj g) for k 1, where g k is given by.0) with a k replaced by A k of.11). For many purposes a first approximation will suffice: F det Aɛ)) = F det A0))+ɛF 1) det A0)) tr A0) 1 A 1) 0)+O ɛ ). 3 Application Here, we present an example of an application to the theory of multivariate statistics. Suppose that X 1 and X are independent p-variate normal random variables with means µ 1 and µ, respectively, and covariances V 1 and V, respectively. For example, X 1 has density p X1 x) = det V) 1/ π) p/ exp { x µ 1 ) V 1 x µ 1 ) / }. Then Y = X 1 + ɛx is p-variate normal with mean µ 1 + ɛµ and covariance V 1 + ɛ V. From 1.) it follows that the density of Y is given in terms of that of X 1 by log p Y x) = log p X1 x) 1 d k,

7 FUNCTIONS OF DETERMINANTS OF POWER SERIES 113 where d j = 1) j j 1 tr V 1 1 V ) j + x µ1 ) b j xµ 1 ) + µ b j 1 µ, d j+1 = µ b j x µ 1 ), b 1 = 0, b j = 1) j V1 1 V ) j V 1 1 for j 1. REFERENCES 1. T. W. Anderson, An Introduction to Multivariate Statistical Analysis, 3rd ed., John Wiley and Sons, Hoboken, New Jersey, L. Comtet, Advanced Combinatorics, Reidel Publishing Company, Dordrecht, Holland, N. Dunford and J. T. Schwarz, Linear Operators, Part II. John Wiley and Sons, New York, M. L. Eaton, M. L. 007). Multivariate Statistics: A Vector Space Approach, Institute of Mathematical Statistics Lecture Notes Monograph Series 53, Institute of Mathematical Statistics, Beachwood, Ohio, I. M. Glazman and J. I. Ljubic, Finite Dimensional Linear Analysis: A Systematic Presentation in Problem Form, MIT Press, Cambridge, Massachusetts, W. Härdle and L. Simar, Applied Multivariate Statistical Analysis, nd ed., Springer, Berlin, R. A. Johnson and D. W. Wichern, Applied Multivariate Statistical Analysis, sixth edition, Pearson Prentice Hall, Upper Saddle River, New Jersey, T. Kollo and D. von Rosen, Advanced Multivariate Statistics with Matrices: Mathematics and Its Applications, Springer, Dordrecht, L. Perko, Differential Equations and Dynamical Systems, Springer Verlag, New York, M. S. Srivastava, Methods of Multivariate Statistics, John Wiley and Sons, New York, 00. Applied Mathematics Group, Industrial Research Limited, Lower Hutt, New Zealand School of Mathematics, University of Manchester, Manchester M13 9PL, UK

8