THE nth POWER OF A MATRIX AND APPROXIMATIONS FOR LARGE n. Christopher S. Withers and Saralees Nadarajah (Received July 2008)

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1 NEW ZEALAND JOURNAL OF MATHEMATICS Volume 38 (2008), THE nth POWER OF A MATRIX AND APPROXIMATIONS FOR LARGE n Christopher S. Withers and Saralees Nadarajah (Received July 2008) Abstract. When a square matrix A, is diagonalizable, (for example, when A is Hermitian or has distinct eigenvalues), then A n can be written as a sum of the nth powers of its eigenvalues with matrix weights. However, if a 1 occurs in its Jordan form, then the form is more complicated: A n can be written as a sum of polynomials of degree n in its eigenvalues with coefficients depending on n. In this case to a first approximation for large n, A n is proportional to n m 1 λ n with a constant matrix multiplier, λ is the eigenvalue of maximum modulus and m is the maximum multiplicity of λ. 1. Introduction and Summary Let A be an s s complex matrix with eigenvalues λ 1,, λ s ordered so that r 1 def = λ 1 = = λ N > r 0 def = λ N+1 λ s. (1.1) If all eigenvalues have the same modulus then (1.1) should be interpreted as r 1 def = λ 1 = = λ s with N = s. If A is Hermitian, or more generally if it has diagonal Jordan form, then for n 1, A n = s λn j W j, for certain matrices {W j }. Let us write r j and θ j for the amplitude and phase of λ j, that is, for 1 j s. Then this implies the approximations λ j = r j e iθj (1.2) A n = r n 1 C n + r n 0 E n, A n = λ n 1 W 1 + r n 0 F n (1.3) (with the latter holding if N = 1), N C n = e inθj W j with E n, F n and C n bounded as n increases. These cases are covered in Sections 2 and 3. Otherwise, A n can be written as a sum of polynomials in each eigenvalue: A n = f j (λ j ), 2000 Mathematics Subject Classification Primary 15A21 Canonical forms, reductions, classification Secondary 15A24 Matrix equations and identities. Key words and phrases: powers of matrices, approximations, Jordan form, Singular value decomposition.

2 172 C.S. WITHERS AND S. NADARAJAH f j (λ) = m j 1 k=0 λ n k W jk k for n 1 and certain matrices {W jk }. Here m j is the dimension of the jth Jordan block of A and r is the number of blocks. So, λ j has multiplicity at least m j. Suppose the blocks for λ 1 are ordered so that with (1.4) interpreted as m = m 1 = = m M > m j, M < j N (1.4) m = m 1 = = m N when all Jordan blocks have dimension m. Then for n 1, A n = r1 n [C n + n 1 E m 1 n], C n = M e i(n m+1)θj W j,m 1 with En and C n bounded as n increases. For example, if there is only one Jordan block with eigenvalue λ 1 then A n = λ n 1 [W 1,m 1 + n 1 F m 1 n], F n is bounded as n increases. Details are given in Section 4. Section 5 gives results similar to (1.3) for (AA T ) n and related forms for large n based on the singular-value decomposition (SVD). In this section A need not be square. Throughout this note, a n = O(b n ) means that an integer m and a positive constant K exist such that for all n m, a n K b n. 2. A n for A Hermitian Consider the case A is Hermitian, that is A T = A, A T is the transpose of the complex conjugate of A. Then the eigenvalues are real, s A = HΛH T = λ j p j p T j, (2.1) Λ = λ 1,, λ s, H = (p 1,, p s ), and p j is the eigenvector of λ j, that is, a solution of Ap j = λ j p j. The eigenvectors can be taken as orthogonal and scaled to have unit norm, so that p T j p k = δ jk and s p jp T j = I s, δ jj = 1 and δ jk = 0 for j k. If A is real symmetric then H can be taken as real. So, by (1.1) we have the well known expression s A n = HΛ n H T = λ n j p j p T j. (2.2) It follows that A n = r1 n C n + r0 n E n, n = 0, 1,, C n = N sn j p jp T j, s j = sign(λ j ) and E n is bounded as n increases. So, for n 1, if N = 1 then

3 THE nth POWER OF A MATRIX AND APPROXIMATIONS FOR LARGE n 173 A n = λ n 1 D + r0 n F n, D = p 1 p T 1 and F n is bounded as n increases. On the other hand, if N 2 then the multiplier C n alternates between C +, its form for n even, and C, its form for n odd, C + = s p jp T j, C = s p jp T j s p j= 1 jp T j. Also if det(a) 0 then (2.2) extends to n = 1, 2,, and in fact to complex n. For example, A 1 = HΛ 1 H T = s λ 1 j p j p T j. Section 5 gives results similar to those of Section 3 but for (AA T ) n, (AA T ) n A. These are based on the SVD of A which need not be square. 3. Diagonal Jordan Form Now suppose that A has diagonal Jordan form, that is A = P ΛP 1 for some matrix P and Λ of (2.1). Such a matrix is said to be diagonalizable. If A and its eigenvalues are real, then P can be taken as real. Writing P = (p 1,, p s ), Q T = P 1 = (q 1,, q s ) T, we have qj T p k = δ jk, s p jqj T = I s. By (1.1) and (1.2), for n 1, A n = P Λ n P 1 = s λ n j p j qj T = r1 n C n + r0 n E n, (3.1) C n = N p einθj j qj T and E n is bounded as n increases. For example, if N = 1 then A n = λ n 1 D + r0 n F n, D = p 1 q1 T and F n is bounded as n increases. On the other hand, if N = 2 then A n = λ n 1 D n + r0 n F n, D n = p 1 q1 T +e in(θ2 θ1) p 2 q2 T and F n is bounded as n increases. In particular if θ 2 θ 1 = 2π/K for some K = 1, 2,, then D n takes K distinct forms: D n = D m when n modulo(k) = m, m = 0, 1,, K 1. If det(a) 0 then (3.1) extends to n = 1, 2,, and to complex n. For example, A 1 = P Λ 1 P 1 = s λ 1 j p j qj T. 4. General Jordan Form Now suppose that A has non-diagonal Jordan form, that is, it has one or more multiple eigenvalues, and A = P JP 1, J = J 1,, J r, J j = J mj (λ j ), J m (λ) = λi m + U m = λ λ λ for some matrix P, and U m is the m m matrix with 1s on the superdiagonal and 0s else: (U m ) jk = δ j,k 1. Again if A and its eigenvalues are real, then P can be taken as real. Methods for obtaining P are known, see, for example, Dunford and Schwartz (1958), Finkbeiner II (1978), Horn and Johnson (1985) and Golub and van Loan

4 174 C.S. WITHERS AND S. NADARAJAH (1996). The m m matrix Um n is defined as J m (0) n, so that Um 0 = I, Um =, Um =, Um =, Um m 1 = with U k m = 0 for k m. Also by the binomial expansion, for λ 0 and n 1, J m (λ) n = min(n,m 1) j=0 λ n j Um j (4.1) j = λ n e n,m 1 (λ) [U m 1 m + n 1 Ω n ] = d n,m 1 (λ) + n m 2 λ n Ω n = n m 1 λ n Ω e nk (λ) = ( ) n k λ k, d nk (λ) = λ n e nk (λ)um m 1 and Um m 1 is the m m matrix of 0s except for a 1 in the upper right corner. Further, Ω n, Ω n and Ω n are bounded as n increases. Another way to write (4.1) is 1 e n1 e n2 e n,m 1 J m (λ) n = λ n 0 1 e n1 e n,m n,

5 THE nth POWER OF A MATRIX AND APPROXIMATIONS FOR LARGE n 175 at e nk = e nk (λ). So, we obtain the exact formula A n = P J n P 1 = P J n 1,, J n r P 1, Jj n is given by (4.1). Suppose now that N of the Jordan blocks {J j } have eigenvalues with the maximum amplitude r 1, and that these are ordered so that r 1 = λ 1 = = λ N > r 0 = max N<j s λ j. Because of multiplicities, this N is not same as N of (1.1). Suppose also that the first M N blocks have the maximum multiplicity m, that is, (1.4) holds, and that m 0 is the maximum multiplicity of eigenvalues of amplitude r 0. Then for n 1, A n = P J n 1,, J n N, 0,, 0 P 1 + n m0 1 r n 0 Ψ n = r n 1 e n,m 1 (r 1 )P d n,m 1 (e iθ1 ),, d n,m 1 (e iθ M ),, 0,, 0 P 1 + n m 2 r n 1 Ψ n for {θ j } of (1.2), Ψ n and Ψ n are bounded as n increases. Now partition P and its inverse into r r blocks to match J, say P = (P jk ), P 1 = (P jk ). So, A n is the r r block matrix with (j, k) element giving (A n ) jk = A n = = P jc Jc n P ck (4.2) N P jc Jc n P ck + O(n m0 1 r0 n ) = r n 1 e n,m 1 (r 1 ) M P jc d n,m 1 (e iθc ) P ck + O(n m 1 r1 n ), r1 n m+1 [C n + n 1 Ψ n ] = n m 1 r1 n Ψ n, m 1 (C n ) jk = M e i(n m+1)θc P jc U m 1 m P ck = O(1) and Ψ n and Ψ n are bounded as n increases. The last multiplier P jc Um m 1 is an m m matrix with (p, q) element (P jc ) p1 (P ck ) mq. For example, if λ 1 > λ j for j > 1, then M = 1 and A n = λ n m+1 1 [D + n 1 Ψ n ], m 1 [D jk ] pq = (P j1 ) p1 (P 1k ) mq and Ψ n is bounded as n increases. If det(a) 0 then (4.2) and (4.1) extend to n = 1, 2,, and to complex n. For example, (A 1 ) jk = P jc J 1 c P ck, P ck

6 176 C.S. WITHERS AND S. NADARAJAH Example 4.1. Take m 1 J m ( λ) 1 = λ 1 j Um. j A = j= Then r = 3, J 1 = 1, J 2 = 2, J 3 = J 2 (4), P = P 1 = J n = n n n4 n n A n = 4 n (Bn/4 + C) + 2 n D + E = n4 n [B/4 + n 1 Φ n ], B = C = D = E = and Φ n is bounded as n increases.

7 THE nth POWER OF A MATRIX AND APPROXIMATIONS FOR LARGE n 177 Example 4.2. Take A = Then r = 2, J 1 = J 2 (2), J 2 = J 1 (1) = 1, P = P 1 = J n = 2n n2 n n So, 18A n = (Bn/2 + C)2 n + D = n2 n [B/2 + n 1 Φ n], B = C = D = and Φ n is bounded as n increases. 5. Behaviour of (AA T ) n and Related Forms for Large n This section gives results similar to those of Section 3 but for (AA T ) n and (AA T ) n A. These are based on the SVD of A. The SVD of an m n matrix A can be written A = θ j l j rj T, r = min(m, n), lj T l k = rj T r k = δ jk and 0 < θ 1 θ r. So, for 1 j r, Ar j = θ j l j and A T l j = θ j r j. So, AA T = θj 2 l j lj T, A T A = AA T A = θj 2 r j rj T, θj 3 l j rj T.

8 178 C.S. WITHERS AND S. NADARAJAH That is, {θj 2} are the non-zero eigenvalues of AAT and A T A, and the corresponding eigenvectors are {l j } and {r j }. Similarly, we have for n 1, (AA T ) n = θj 2n l j lj T, (A T A) n = (AA T ) n A = (A T A) n A T = θ 2n j r j r T j, θ 2n+1 j l j r T j, θ 2n+1 j r j l T j. So, if max 1 j s θ j < θ s+1 = = θ r then as n, (AA T ) n = θr 2n L sr + θs 2n n = θr 2n Ξ n, (A T A) n = θr 2n R sr + θs 2n n = θr 2n Ξ n, (AA T ) n A = θr 2n+1 B sr + θs 2n n = θr 2n (A T A) n A T = θ 2n+1 r C sr + θ 2n s n = θr 2n = r Ξ n, Ξ n, L sr = r j=s+1 l jlj T, R sr = r j=s+1 r jrj T, B sr j=s+1 l jrj T and C sr = r j=s+1 r jlj T with n, n, n, n, Ξ n, Ξ n, Ξ n, Ξ n, L sr, R sr, B sr and C sr bounded as n increases. Acknowledgments The authors would like to thank the Managing Editor and the referee for carefully reading the paper and for their comments which greatly improved the paper. References [1] N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory, Interscience, [2] D. T. Finkbeiner II, Introduction to Matrices and Linear Transformations, Third Edition, Freeman, [3] G. H. Golub and C. F. van Loan, Matrix Computations, Third Edition, Johns Hopkins University Press, Baltimore, [4] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, C.S. Withers Applied Mathematics Group Industrial Research Limited Lower Hutt NEW ZEALAND c.withers@irl.cri.nz S. Nadarajah School of Mathematics University of Manchester Manchester M13 9PL UNITED KINGDOM mbbsssn2@manchester.ac.uk