MATRIX ROOTS OF NONNEGATIVE AND EVENTUALLY NONNEGATIVE MATRICES PIETRO PAPARELLA

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1 MATRIX ROOTS OF NONNEGATIVE AND EVENTUALLY NONNEGATIVE MATRICES By PIETRO PAPARELLA A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY WASHINGTON STATE UNIVERSITY Department of Mathematics JULY 2013 Copyright by PIETRO PAPARELLA, 2013 All Rights Reserved

2 Copyright by PIETRO PAPARELLA, 2013 All Rights Reserved

3 ii To the Faculty of Washington State University: The members of the Committee appointed to examine the dissertation of PIETRO PAPARELLA find it satisfactory and recommend that it be accepted. Michael J. Tsatsomeros, Ph.D., Chair Judith J. McDonald, Ph.D., Chair Bala Krishnamoorthy, Ph.D.

4 iii MATRIX ROOTS OF NONNEGATIVE AND EVENTUALLY NONNEGATIVE MATRICES Abstract by Pietro Paparella, Ph.D. Washington State University July 2013 Chair: Michael J. Tsatsomeros and Judith J. McDonald Eventually nonnegative matrices are real matrices whose powers become and remain nonnegative. As such, eventually nonnegative matrices are, a fortiori, matrix roots of nonnegative matrices, which motivates us to study the matrix roots of nonnegative matrices. Using classical matrix function theory and Perron-Frobenius theory, we characterize, classify, and describe in terms of the complex and real Jordan canonical form the pth-roots of nonnegative and eventually nonnegative matrices.

5 iv Acknowledgements This dissertation would not have been possible without the unwavering encouragement, patience, and love of my wife Allison L. Paparella. I wish to thank my children Quintin L. Paparella and Alessia R. Paparella for providing me with inspiration to help me finish this endeavor. I also would like to acknowledge my mother Maria Aquino for always supporting me and my siblings and for her tremendous efforts and sacrifice in raising three children on her own. I owe a deep debt of gratitude to my advisors Michael J. Tsatsomeros and Judtih J. McDonald. Their vision, guidance, wisdom, and expertise were paramount in the completion of this project. I wish to extend a debt of gratitude to Bala Krishnamoorthy, who went above and beyond the duties and responsibilities of a committee member: thank you for responding to my pesky s, the sage advice, and the encouraging words. I would also like to thank David S. Watkins for serving on my committee and K. A. Ari Ariyawansa for generously providing me a research assistantship from , so that I was able to return to Washington State to complete my Ph.D in mathematics. I am also grateful for wonderful friends and family for their encouraging words and support: Travis W. Arnold; Aaron L. Carr-Callen; Kirk J. Griffith; Jasmin L. and Jeffrey A. Harshman; Jameus C. Hutchens; Francesco Paparella; Stefano Paparella; Christine S. and V. Joshua Phillips; Lena Y. and V. Jefferson Phillips; Daniela Paparella and Joseph W. Thornton; and Frank and Chiara D. Trejos. Finally, I would like to acknowledge several graduate students from my time at Washington State and Louisiana State for the wonderful memories, collaboration, and continued

6 v inspiration: Baha M. Alzalg, Jeremy J. Becnel, Timothy D. Breitzman, Mihály Kovács, Suat Namli, and Rokas Varaneckas.

7 To my wife Allison, my children Quintin and Alessia, and my mother Maria. vi

8 If I have seen further it is by standing on ye sholders of Giants Sir Isaac Newton. vii

9 viii Contents Abstract iii Acknowledgements iv Dedication vi Inspiration vii 1 Introduction Background and Motivation Notation and Definitions Preliminaries Combinatorial Structure of a Matrix Upper-triangular Toeplitz Matrices Theoretical Background Perron-Frobenius Theory of Nonnegative Matrices Theory of Functions of Matrices Matrix Roots of Nonnegative Matrices Matrices > λ > 0 > λ λ =

10 ix 3.2 Primitive Matrices Imprimitive Irreducible Matrices Complete Residue Systems Jordan Chains of h-cyclic matrices Main Results Reducible Matrices Preliminary Results Bibliography 73

11 1 Chapter 1 Introduction 1.1 Background and Motivation A real matrix A is said to be eventually nonnegative (respectively, positive) if there exists a nonnegative integer p such that A k is entrywise nonnegative (respectively, positive) for all k p. If p is the smallest such integer, then p is called the power index of A and is denoted by p(a). Eventual nonnegativity was introduced and studied in [5] and was the central subject of study in various publications (see, e.g., [3, 4, 10, 14, 19, 20, 21, 24, 25]). It is well-known that the notions of eventual positivity and nonnegativity are associated with properties of the eigenspace corresponding to the spectral radius. A real matrix A has the Perron-Frobenius property if its spectral radius is a positive eigenvalue corresponding to an entrywise nonnegative eigenvector. The strong Perron-Frobenius property further requires that the spectral radius is simple; it dominates in modulus every other eigenvalue of A; and it has an entrywise positive eigenvector. Several challenges regarding the theory and applications of eventually nonnegative matrices remain unresolved. For example, eventual positivity of A is equivalent to A and A T having the strong Perron-Frobenius property, however, the Perron-Frobenius property for A and A T is a necessary but not sufficient condition for eventual nonnegativity of A. The motivation of this thesis is the following observation: an eventually nonnegative

12 2 (respectively, positive) matrix with power index p = p(a) is, a fortiori, a matrix pth-root of the nonnegative matrix A p (i.e., it is a solution to the matrix equation X p A p = 0). As a consequence, in order to gain insight into eventual nonnegativity, it is natural to examine the roots of matrices that possess the (strong) Perron-Frobenius property. 1.2 Notation and Definitions The following notational conventions are adopted throughout the thesis; when convenient, additional notation and definitions are introduced and adopted throughout. (1) Denote by C the set of complex numbers, R the set of real numbers, Z the set of integers, and N the set of natural numbers. Denote by i the imaginary unit, i.e., i := 1. (2) When convenient, an indexed set of the form {x i, x i+1,..., x i+j } is abbreviated to {x k } i+j k=i. (3) Denote by M n (C) (M n (R)) the algebra of complex (respectively, real) n n matrices. (4) Given A M n (C), [a ij ] = [a ij ] i,j=1 n denotes the entries of A, where a ij C is the (i, j)-entry of A. When convenient, the (i, j)-entry of A may also be denoted by [A] ij. (5) Denote by C n (R n ) the n-dimensional complex (respectively, real) vector space. Given x C n (x R n ), [x] i denotes the ith-component of x. (6) Denote by I n the n n identity matrix. Whenever the context is clear, the subscript may be suppressed.

13 3 (7) For λ C, J n (λ) M n (C) denotes the n n Jordan block, i.e, λ, i = j; [J n (λ)] ij := 1, j = i + 1; and 0, otherwise. (8) Given A M n (C): (i) the spectrum of A, denoted by σ (A), is the multiset σ (A) := {λ C : det(λi A) = 0}, i.e., σ (A) contains the eigenvalues of A; (ii) the spectral radius of A, denoted by ρ = ρ (A), is the quantity ρ (A) := max λ i σ(a) { λ i }; and (iii) the peripheral spectrum, denoted by π (A), is the (multi-)set given by {λ σ (A) : λ = ρ}. (9) The direct sum of the matrices A 1, A 2,..., A k, where A i M ni (C), denoted by A 1 A 2 A k, k i=1 A n i, or diag (A 1, A 2,..., A n ), is the matrix where n = k i=1 n i. A 1 A 2... A k M n (C)

14 4 (10) For A M n (C), let J = Z 1 AZ = t i=1 J n i (λ i ) = t i=1 J n i, where n i = n, denote a Jordan canonical form. (11) The order of the largest Jordan block of A corresponding to the eigenvalue λ is called the index of λ i and is denoted by m = m i = index λi A. (12) For c = (c 1, c 2,..., c n ), where c i C, i, the circulant matrix (or simply circulant) of c, denoted by circ (c) is the n n matrix given by c 1 c 2 c n c n c 1 c n 1. M n (C) c 2 c 3 c 1 (13) For A, B M n (C), the hadamard product of A and B, denoted by A B, is the n n matrix whose (i, j)-entry is given by [A B] ij := a ij b ij. (14) For A M n (C) and X M n (C), X is said to be a matrix pth-root, pth-root, matrix root, or simply root of A if X satisfies the matrix equation X p A = 0. (15) For k N, let R(k) denote the canonical complete residue system (mod k), i.e., R(k) = {0, 1,..., k 1}. 1.3 Preliminaries Because the combinatorial stucture of a matrix (i.e., the location of its zero-nonzero entries) will be central to the analysis of the matrix roots of nonnegative, irreducible, imprimitive

15 5 matrices, we devote Subsection to notation, concepts, and background concerning the combinatorial structure of a matrix. In addition, in Subsection 1.3.2, we establish ancillary results pertaining to upper-triangular Toeplitz matrices Combinatorial Structure of a Matrix For notation and definitions concerning the combinatorial stucture of a matrix, i.e., the location of the zero-nonzero entries of a matrix, we follow [2] and [10]. For further results concerning combinatorial matrix theory, see [2] and references therein. A directed graph (or simply digraph) Γ = (V, E) consists of a finite, nonempty set V of vertices, together with a set E V V of arcs. Digraphs may contain both arcs (u, v) and (v, u) as well as loops, i.e., arcs of the form (v, v). In this thesis, we do not consider directed general graphs (or simply general digraphs), i.e., digraphs that contain multiple copies of the same arc. For A M n (C), the directed graph (or simply digraph) of A, denoted by Γ = Γ (A), has vertex set V = {1,..., n} and arc set E = {(i, j) V V : a ij 0}. If R, C {1,..., n}, then A[R C] denotes the submatrix of A whose rows and columns are indexed by R and C, respectively. If R = C, then A[R R] is abbreviated as A[R]. For a digraph Γ = (V, E) and W V, the induced subdigraph Γ[W ] is the digraph with vertex set W and arc set {(u, v) E : u, v W }. For a square matrix A, Γ(A[W ]) is identified with Γ (A) [W ] by a slight abuse of notation. A digraph Γ is strongly connected if for any two distinct vertices u and v of Γ, there is a walk in Γ from u to v (as a consequence, there must also be a walk from v to u;

16 6 thus, both conditions could be employed as the definition). Following [2], we consider every vertex of V as strongly connected to itself. For a strongly connected digraph Γ, the index of imprimitivity is the greatest common divisor of the the lengths of the closed walks in Γ. A strong digraph is primitive if its index of imprimitivity is one, otherwise it is imprimitive. The strong components of Γ are the maximal strongly connected subdigraphs of Γ. For n 2, a matrix A M n (C), is reducible if there exists a permutation matrix P such that P T AP = A 11 A 12 0 A 22 where A 11 and A 22 are nonempty square matrices and 0 is a (possibly rectangular) block consisting entirely of zero entries. If A is not reducible, then A is called irreducible. The connection between reducibility and the digraph of A is as follows: A is irreducible if and only if Γ (A) is strongly connected 1 (see, e.g., [2, Theorem 3.2.1] or [11, Theorem ]). If A M n (C) is reducible, then there exists a permutation matrix P such that A 11 A 12 A 1k A P T AP = A 2k.... A kk (1.3.1) where each A ii is irreducible (see, e.g., [2, Theorem 3.2.4] or [11, Exercise 8, 8.3]. The matrix on the right-hand side of (1.3.1) is called the Frobenius (or irreducible) normal form of A. For h 2, a digraph Γ = (V, E) is cyclically h-partite if there exists an ordered partition Π = (π 1,..., π h ) of V into h nonempty subsets such that for each arc (i, j) E, there 1 Following [2], vertices are strongly connected to themselves so we take this result to hold for all n N and not just n N, n 2.

17 7 exists l {1,..., h} with i π l and j π l+1 (where, for convenience, we take V h+1 := V 1 ). For h 2, a strong digraph Γ is cyclically h-partite if and only if h divides the index of imprimitivity (see, e.g., [2, p. 70]). For h 2, a matrix A M n (C) is called h-cyclic if Γ (A) is cyclically h-partite. If Γ (A) is cyclically h-partite with ordered partition Π, then A is said to be h-cyclic with partition Π or that Π describes the h-cyclic structure of A. The ordered partition Π = (π 1,..., π h ) is consecutive if π 1 = {1,..., i 1 }, π 2 = {i 1 + 1,..., i 2 },..., π h = {i h 1 + 1,..., n}. If A is h-cyclic with consecutive ordered partition Π, then A has the block form 0 A A A (h 1)h A h (1.3.2) where A i,i+1 = A[π i π i+1 ] ([2, p. 71]). For any h-cyclic matrix A, there exists a permutation matrix P such that P AP T is h-cyclic with consecutive ordered partition. The cyclic index of A is the largest h for which A is h-cyclic. An irreducible nonnegative matrix A is primitive if Γ (A) is primitive, and the index of imprimitivity of A is the index of imprimitivity of Γ (A). If A is irreducible and imprimitive with index of imprimitivity h 2, then h is the cyclic index of A, Γ (A) is cyclically h-partite with ordered partition Π = (π 1,..., π h ), and the sets π i are uniquely determined (up to cyclic permutation of the π i ) (see, for example, [2, p. 70]). Furthermore, Γ ( A h) is the disjoint union of h primitive digraphs on the sets of vertices π i, i = 1,..., h (see, e.g., [2, 3.4]). Following [10], for an ordered partition Π = (π 1,..., π h ) of {1,..., n} into h nonnempty subsets, the cyclic characteristic matrix, denoted by χ Π, is the n n matrix whose (i, j)-entry

18 8 is 1 if there exists l {1,..., h} such that i π l and j π l+1 and 0 otherwise. For an ordered partition Π = (π 1,..., π h ) of {1,..., n} into h nonnempty subsets, note that (1) χ Π is h-cyclic and Γ (χ Π ) contains every arc (i, j) for i π l and j π l+1 ; and (2) A M n (C) is h-cyclic with ordered partition Π if and only if Γ (A) Γ (χ Π ) Upper-triangular Toeplitz Matrices In this subsection we establish the following lemmas concerning upper-triangular Toeplitz matrices. Lemma For n 2, let λ, c, y 1,..., y n 1 C, and y n 1 0. If y := (y 1, y 2,..., y n 1 ) T C n 1, ȳ := (c, y 1,..., y n 2 ) T C n 1, Ĵ := J n 1(λ) y M n (C), z T λ and ˆQ := I n 1 z T ȳ y n 1 M n (C) then ˆQJ ˆQ 1 = J n (λ). Proof. Note that ˆQ 1 = I n 1 ȳ y n 1 z T 1 y n 1 ˆQĴ ˆQ 1 = I n 1 z T, hence ȳ y n 1 J n 1(λ) z T y I n 1 λ ȳ y n 1 z T 1 y n 1

19 9 J n 1 (λ) = z T y + λȳ λy n 1 I n 1 ȳ y n 1 z T 1 y n 1 1 J n 1 (λ) y n 1 (y + λȳ J n 1 (λ)ȳ) =. z T λ If N n = [n ij ] C n n is the nilpotent matrix defined by 1, j = i + 1 n ij = 0, otherwise then it is well known that J n (λ) = λi n + N n. Furthermore, N n 1 ȳ = (y 1, y 2,..., y n 2, 0) T and y + λȳ J n 1 (λ)ȳ y n 1 = y + λȳ (λi n 1 + N n 1 )ȳ y n 1 = y + λȳ λȳ + N n 1ȳ y n 1 = (y 1, y 2,..., y n 2, y n 1 ) T (y 1, y 2,..., y n 2, 0) T y n 1 = (0, 0,..., 0, 1) T, which completes the proof.

20 10 Lemma For n 2, consider the upper-triangular Toeplitz matrix x 1... x k... x n F := x 1... x k M n (C).... where x 1,..., x n C and x i 0 for all i = 2,..., n. Then there exists an invertible matrix Q such that QF Q 1 = J n (x 1 ). Proof. Proceed by induction on n: for the base-case, i.e., n = 2, note that 1 0 x 1 x = x 1 x = x x 2 0 x 1 0 x 1 x 2 0 x 1 so that the base-case holds. 0 1 x 2 x x 2 Assume the assertion holds when n = k 1. Let F be the submatrix of F obtained from deleting the n th -row and column of F, i.e., x 1... x n 1 F =.... M n 1(C), x 1 and let x = (x n, x n 1,..., x 2 ) T C n 1 F. Then F = x and, by the inductive hypoth- z T x 1 esis, there exists an invertible matrix Q such that Q F Q 1 = J n 1 (x 1 ). Note that the matrix Q z Q := is invertible, Q 1 Q 1 z =, and z T 1 z T 1 QF Q 1 Q z F x Q 1 z = z T 1 z T x 1 z T 1,

21 11 Q F Qx Q 1 z = z T x 1 z T 1 1 Q F Q Qx = z T x 1 = J n 1(x 1 ) Qx. (1.3.3) z T x 1 If y := Qx C n 1, ȳ := (0, y 1,..., y n 2 ) T C n 1, then, following Lemma 1.3.1, a similaritytransform by the matrix ˆQ := I n 1 ȳ brings the matrix in (1.3.3) into the desired z T y n 1 form. Thus, the desired similarity matrix transformation is given by Q = ˆQ Q. In Table 1, we present the transformation Q and its inverse for several dimensions. Table 1: The transformation matrix Q and Q 1. n F Q Q 1 2 x 1 x x 1 0 x 2 0 x 2 3 x 1 x 2 x x 1 x 2 0 x 2 x 3 0 x 2 x x x x x 1 x 2 x 3 x x 1 x 2 x 3 0 x 2 x 3 x 4 0 x 2 x 3 x x 1 x x 2 2 2x 2 x x x x x 3 2 2x 2 3 x 2x 4 x 5 2 2x 3 x 4 2

22 12 Chapter 2 Theoretical Background 2.1 Perron-Frobenius Theory of Nonnegative Matrices We review some well-known results, concepts, and definitions from the theory of nonnegative matrices. For further results, see [11, Chapter 8] or [1] and references therein. Recall that a matrix A = [a ij ] M n (R) is said to be (entrywise) nonnegative (respectively, positive), denoted A 0 (respectively, A > 0), if a ij 0 (respectively, a ij > 0) for all 1 i, j n. Recall that a matrix A M n (R) is eventually positive (nonnegative) if there exists a nonnegative integer p such that A k is entrywise positive (nonnegative) for all k p. If p is the smallest such integer, then p is called the power index of A and is denoted by p(a). We recall the Perron-Frobenius theorem for positive matrices (see [11, Theorem ]). Theorem If A M n (R) is positive, then (a) ρ > 0; (b) ρ σ (A); (c) there exists a positive vector x such that Ax = ρx; (d) ρ is an algebraically (and hence geometrically) simple eigenvalue of A; and (e) λ < ρ for every λ σ (A) such that λ ρ.

23 13 There are nonnegative matrices with zero-entries that satisfy Theorem Recall that a nonnegative matrix A M n (R) is said to be primitive if it is irreducible and has only one eigenvalue of maximum modulus. The conclusions to Theorem apply to primitive matrices (see [11, Theorem 8.5.1]), and the following theorem is a useful characterization of primitivity (see [11, Theorem 8.5.2]). Theorem If A M n (R) is nonnegative, then A is primitive if and only if A k > 0 for some k 1. Perron-Frobenius type results also hold when A is an irreducible imprimitive matrix (see [11, 8.4]). Theorem Let A M n (R) and suppose that A is irreducible, nonnegative, and h is the cyclic index of A. Then (a) ρ > 0; (b) ρ σ (A); (c) there exists a positive vector x such that Ax = ρx; (d) ρ is an algebraically (and hence geometrically) simple eigenvalue of A; and (e) π (A) = {ρ exp (i2πk/h) : k R(h)}. The following result is the widest-reaching forrm of the Perron-Frobenius Theorem. Theorem If A M n (R) and A 0, then ρ (A) σ (A) and there exists a nonnegative vector x 0, x 0, such that Ax = ρ (A) x.

24 14 One can verify that the matrix possesses properties (a) through (e) of Theorem 2.1.1, is irreducible, but obviously contains a negative entry. This motivates the following concept. Definition A matrix A M n (R) is said to possess the strong Perron-Frobenius property if A possesses properties (a) through (e) of Theorem The following theorem characterizes the strong Perron-Frobenius property (see [6, Lemma 2.1], [14, Theorem 1], or [20, Theorem 2.2]). Theorem A real matrix A is eventually positive if and only if A and A T possess the strong Perron-Frobenius property. 2.2 Theory of Functions of Matrices We review some basic notions and results from the theory of matrix functions (for further results on the theory of matrix functions, see [8], [12, Chapter 9], or [15, Chapter 6]). Definition Let f : C C be a function and f (k) denote the kth derivative of f. The function f is said to be defined on the spectrum of A if the values f (k) (λ i ), k = 0,..., m i 1, i = 1,..., s, called the values of the function f on the spectrum of A, exist. Definition (Matrix function via Jordan canonical form). If f is defined on the spectrum of A M n (C), then ( k ) f(a) := Zf(J)Z 1 = Z f(j ni ) Z 1, i=1

25 15 where f(λ i ) f (λ i )... f(λ f(j ni ) := i ) f (λ i ) f(λ i ) f (n i 1) (λ i ) (n i 1)!. (2.2.1) The following equivalent definition (see [8, Theorem 1.12]) serves to illuminate the notion of a primary root, and will be used to examine the 2 2 case in Section 3.1. Definition (Matrix function via Hermite interpolation). Let f be defined on the spectrum of A M n (C) and let ϕ be the minimal polynomial of A. Then f(a) := p(a), where p is the polynomial of degree less than s n i = deg ϕ i=1 that satisfies the interpolation conditions p (j) (λ i ) = f (j) (λ i ), j = 0,..., n i 1, i = 1,..., s (2.2.2) There is a unique such p and it is known as the Hermite interpolating polynomial. Remark The Hermite interpolating polynomial p is given explicitly by the Lagrange- Hermite formula p(t) = s i=1 [( ni 1 j=0 1 j! φ(j) i (λ i )(t λ i ) j ) j i (t λ j ) n j ], (2.2.3) where φ i (t) = f(t)/ j i (t λ j) n j. For a matrix with distinct eigenvalues, i.e., n i = 1, s = n, (2.2.3) reduces to p(t) = n i=1 f(λ i )l i (t), l i (t) = j=1 j i ( t λj λ i λ j ). (2.2.4)

26 16 Formulae (2.2.3) and (2.2.4) will be used to classify the primary matrix roots 2 2 matrices in Section 3.1. Theorem Let f be defined on the spectrum of a nonsingular matrix A M n (C). If J = t i=1 J n i (λ i ) = Z 1 AZ is a Jordan canonical form of A, then is a Jordan canonical form of f(a). Proof. Because f(a) = Z (f(j)) Z 1 J f := t J ni [f(λ i )] i=1 = Z ( k ) i=1 f(j n i ) Z 1, note that, for each i = 1,..., t, f(j ni ) is an upper-triangular, Toeplitz matrix. Following Lemma 1.3.1, for each i = 1,..., k, there exists an invertible matrix Q i M ni (C) such that f(j ni ) = Q i J ni (f(λ i ))Q 1 i, and hence f(a) = Z = Z = and the result is established. ( t f(j ni )Z 1 i=1 [ t Z i=1 t Q i J ni [f(λ i )]Q 1 i t Q i) i=1 i=1 ( t ) = Z J ni [f(λ i )] i=1 ] J ni [f(λ i )] Z 1 Z 1 ( t i=1 Q 1 i Z 1 ) Definition For z = r exp (iθ) C, where r > 0, and an integer p > 1, let z 1/p := r 1/p exp (iθ/p), and, for j {0, 1,..., p 1}, define f j (z) = z 1/p exp (i2πj/p) = r 1/p exp (i [θ + 2πj] /p), (2.2.5)

27 17 i.e., f j is the (j + 1)st-branch of the pth-root function. Note that f (k) j [ (z) = 1 k 1 ] ( i[2πj + θ(1 kp)] (1 ip) r (1 kp)/p exp p k p i=0 ), (2.2.6) where k is a nonnegative integer and the product k 1 i=0 (1 ip) is empty when k = 0. Next we present several technical, but easily establishable lemmas on the branches of the pth-root function. Lemma If z = r exp (iθ), z = r exp (iθ ) C, and r < r, then. Proof. Note that f j (z) = r 1/p exp (i [θ + 2πj] /p) = r 1/p exp (i [θ + 2πj] /p) = r 1/p < (r ) 1/p = (r ) 1/p exp (i [θ + 2πj ] /p) = (r ) 1/p exp (i [θ + 2πj ] /p) = f j (z ). Lemma For z C, I(z) 0, j, j R(p), and f (k) j f (k) j ( z) if and only if j + j 0 (modp). as in (2.2.6), we have f (k) j (z) = Proof. Note that f (k) j (z) = f (k) ( z) j

28 18 exp (i[2πj + θ(1 kp)]/p) = exp (i[ 2πj + θ(1 kp)]/p) exp (i[2πj + θ(1 kp)]/p) = exp (i[ 2πj + θ(1 kp)]/p) exp (i2πlp/p) exp (i[2πj + θ(1 kp)]/p) = exp (i[2π(pl j ) + θ(1 kp)]/p) i[2πj + θ(1 kp)]/p = i[2π(pl j ) + θ(1 kp)]/p j + j = lp j + j 0 (modp), where l Z. Finally, note that j + j 0 (modp) j = j = 0 or j = p j. Lemma Let z = r exp (iπ), r > 0. For j, j {0, 1,..., p 1} and f (k) j as in (2.2.6), we have f (k) j (z) = f (k) j (z) if and only if j + j = p 1. Proof. Note that f (k) j (z) = f (k) (z) exp (i[2πj + π(1 kp)]/p) = exp (i[ 2πj π(1 kp)]/p) j exp (i[2πj + π(1 kp)]/p) = exp (i[ 2πj π(1 kp)]/p) exp (i2π(p kp)) exp (i[2πj + π(1 kp)]/p) = exp (i[ 2πj π(1 kp) + 2π(p kp)]/p) i[2πj + π(1 kp)]/p = i[ 2πj π(1 kp) + 2π(p kp)]/p 2πj + π(1 kp) = 2πj π(1 kp) + 2π(p kp) 2πj + 2π(1 kp) = 2πj + 2π(p kp) j + j = p 1. The following theorem classifies all pth-roots of a general nonsingular matrix [23, Theorems 2.1 and 2.2].

29 19 Theorem (Classification of pth-roots of nonsingular matrices). If A M n (C) is nonsingular, then A has precisely p s pth-roots that are expressible as polynomials in A, given by ( t ) X j = Z f ji (J ni ) Z 1, (2.2.7) i=1 where j = (j 1,..., j t ), j i {0, 1,..., p 1}, and j i = j k whenever λ i = λ k. If s < t, then A has additional pth-roots that form parameterized families ( t ) X j (U) = ZU f ji (J ni ) U 1 Z 1, (2.2.8) i=1 where U is an arbitrary nonsingular matrix that commutes with J and, for each j, there exist i and k, depending on j, such that λ i = λ k, while j i j k. In the theory of matrix functions, the roots given by (2.2.7) are called primary functions (or roots) of A and are expressible as polynomials in A, and the roots given by (2.2.8), which exist only if A is derogatory (i.e., if some eigenvalue appears in more than one Jordan block), are called the nonprimary functions or roots of A and are not expressible as polynomials in A [8, Chapter 1]. The next result provides a necessary and sufficient condition for the existence of a root for a general matrix (see [22]). Theorem (Existence of pth-root). A matrix A M n (C) has a pth-root if and only if the ascent sequence of integers d 1, d 2,... defined by d i = dim (null (A i )) dim (null (A i 1 )) has the property that for every integer ν 0 no more than one element of the sequence lies strictly between pν and p(ν + 1).

30 20 Before we state results concerning the matrix roots of a real matrix, we state some wellknown results concerning the real Jordan canonical form for real matrices (see [11, Section 3.4], [15, Section 6.7]). Theorem (Real Jordan canonical form). If A M n (R) has r real eigenvalues (including multiplicities) and c complex conjugate pairs of eigenvalues (including multiplicities), then there exists a real, invertible matrix R M n (R) such that r R 1 k=1 AR = J R = J n k (λ k ), r+c k=r+1 C n k (λ k ) where: 1. C(λ) I 2 C(λ) C k (λ) :=.... M 2k (R); (2.2.9).. I2 C(λ) 2. C(λ) := R(λ) I(λ) I(λ) M 2 (R); (2.2.10) R(λ) 3. λ 1,..., λ r are the real eigenvalues (including multiplicities) of A; and 4. λ r+1, λ r+1,..., λ r+c, λ r+c are the complex eigenvalues (including multiplicities) of A.

31 21 Lemma Let λ C and suppose C k (λ) and C(λ) are defined as in (2.2.9) and ( k ) (2.2.10), respectively. If S k := i=1 S M 2k (R), where S := i i, then 1 1 S 1 k C k(λ)s k = D k (λ) := where D(λ) := λ 0 0 λ. D(λ) I 2 D(λ).... M 2k (C), (2.2.11).. I 2 D(λ) Proof. Proceed by induction on k, the number of 2 2-blocks: when k = 1 one readily obtains S 1 C(λ)S = 1 i 1 R(λ) 2 i 1 I(λ) I(λ) i i = D(λ). R(λ) 1 1 Now assume the assertion holds for all matrices of the form (2.2.9) of dimension 2(k 1). Note that the matrices in the product S 1 k C k(λ)s k can be partitioned as S 1 k 1 Z C k 1(λ) Y S k 1 Z, Z T S 1 Z T C(λ) Z S. where Z M 2(k 1),2 (R) is a rectangular zero matrix, Y = M 2(k 1),2 (R), and Z 2 Z 2 is the 2 2 zero matrix. With the above partition in mind, and following the induction Z 2 I 2

32 22 hypothesis, we obtain S 1 C k (λ)s = D k 1(λ) Z T S 1 k 1 Y S, S 1 C(λ)S but note that S 1 k 1 Y S = Y and S 1 C(λ)S = D(λ). Lemma Let λ C and suppose D k (λ) and D(λ) are defined as in Lemma If P k is the permutation matrix given by ] P k = [e 1 e 3... e 2k 1 e 2 e 4... e 2k M 2k (R), (2.2.12) where e i denotes the canonical basis vector in R n of appropriate dimension, then P T k D k (λ)p k = J k (λ) J k ( λ). Proof. Proceed by induction on k: the base-case when k = 1 is trivial, so we assume the assertion holds for matrices of the form (2.2.11) of dimension 2(k 1). If z n denotes the n 1 zero vector, then λ e T 2 0 D(λ) = z 2(k 1) D 2(k 1) ( λ) e 2(k 1), 0 z2(k 1) T λ and if ˆP is the permutation matrix defined by 1 z2(k 1) T 0 ˆP := z 2(k 1) P 2(k 1) z 2(k 1) M 2k(R), 0 z2(k 1) T 1

33 23 then, following the induction-hypothesis, λ zk 1 T e T 1 0 ˆP T D(λ) ˆP z = k 1 J k 1 ( λ) Z k 1 z k 1. (2.2.13) z k 1 Z k 1 J k 1 (λ) e k 1 0 zk 1 T zk 1 T λ A permutation-similarity by the matrix P defined by 1 I P := k 1 M 2k (R) I k 1 1 brings the matrix in the right-hand side of (2.2.13) to the desired form. The proof is completed by noting that 1 ] ˆP P = [e 1 e 2... e 2(k 1) e 3... e 2k 1 e 2k I k 1 I k 1 = P k, 1 since right-hand multiplication by P permutes columns 2 through k with columns k + 1 through 2k 1. Corollary Let λ C, λ 0, and let f be a function defined on the spectrum of J k (λ) J k ( λ). For j a nonnegative integer, let f (j) λ denote f (j) (λ). If C k (λ) and C(λ) are

34 24 defined as in (2.2.9) and (2.2.10), respectively, then ( ) C(f λ ) C(f λ )... C f (k 1) λ (k 1)! f(c k (λ)) = C(f λ )..... M 2k (R)... C(f λ ) C(f λ ) if and only if f (j) λ = f (j) λ. Proof. Following Lemmas Lemma and Lemma , P T k S 1 k C k(λ)s k P k = J k (λ) J k ( λ). (2.2.14) Since f(a) = f(x 1 AX) ([8, Theorem 1.13(c)]), f(a B) = f(a) f(b) [8, Theorem 1.13(g)], and f (j) λ = f (j) λ for all j, applying f to (2.2.14) yields Pk T S 1 k f(c k(λ))s k P k = f(j k (λ)) f(j k ( λ) ) = f(jk (λ)) f(j k (λ)). Hence, and ] k f(c k(λ))s k = P k [f(j k (λ)) f(j k (λ)) Pk T ( ) D(f λ ) D(f λ )... D f (k 1) λ (k 1)!. = D(f λ ) D(f λ ) D(f λ ) S 1 ( ) D(f λ ) D(f λ )... D f (k 1) λ (k 1)! f(c k (λ)) = S k D(f λ ) D(f λ ) D(f λ ) S 1 k

35 25 ( ) C(f λ ) C(f λ )... C f (k 1) λ (k 1)!. = C(f λ ) C(f λ ) C(f λ ) The converse follows by noting that, for µ, ν C, the product S µ 0 S 1 = i i µ 0 1 i ν ν i 1 = 1 iµ iν i 1 2 µ ν i 1 = 1 µ + ν i(ν µ) 2 i(µ ν) µ + ν is real if and only if ν = µ. Remark In general, our analysis reveals that f(c λ ) f f (C λ )... (k 1) (C λ ) (k 1)! f(c f(c k (λ)) = λ )..... M 2k (C),.. f (C λ ) f(c λ ) which bears a striking resemblance to (2.2.1). Corollary If λ C, I(λ) 0 and F j (C k (λ)) := S k P k f j 1 (J k (λ)) 0 Pk 0 f j2 (J k ( λ) T S 1 k M 2k (C), (2.2.15) )

36 26 where j = (j 1, j 2 ) and j 1, j 2 {0, 1,..., p 1}, then ( ) C(f j1 (λ)) C(f f (k 1) j (λ) j 1 (λ))... C 1 (k 1)!. F j (C k (λ)) = C(f j1 (λ))... M 2k (R)... C(f j1 (λ)) C(f j1 (λ)) if and only if j 1 = j 2 = 0 or j 1 = j 2 p. Proof. Follows from Lemma and Corollary Corollary If λ = r exp (iπ) C, where r > 0, and F j is defined as in (2.2.15), then ( ) C(f j1 (λ)) C(f f (k 1) j (λ) j 1 (λ))... C 1 (k 1)!. F j (C k (λ)) = C(f j1 (λ)).... M 2k (R).. C(f j1 (λ)) C(f j1 (λ)) if and only if j 1 + j 2 = p 1. Proof. Follows from Lemma and Corollary The next theorem provides a necessary and sufficient condition for the existence of a real pth-root of a real A (see [9, Theorem 2.3]) and our proof utilizes the real Jordan canonical form. Theorem (Existence of real pth-root). A matrix A M n (R) has a real pth-root if and only if it satisfies the ascent sequence condition specified in Theorem and, if p is even, A has an even number of Jordan blocks of each size for every negative eigenvalue.

37 27 Proof. Case 1: p is even. Following Theorem , there exists a real, invertible matrix R such that A = R J 0 J + J C R 1 where J 0 collects the singular Jordan blocks; J + collects the Jordan blocks with positive real eigenvalues; J collects Jordan blocks with negative real eigenvalues; and C collects blocks of the form (2.2.9) corresponding to the complex conjugate pairs of eigenvalues of A. By hypothesis, if J k (λ) is a submatrix of J, it must appear an even number of times; for every such pair of blocks, it follows that P k [J k (λ) J k (λ)]p T k = C k (λ), where P k is defined as in (2.2.12). Thus, there exists a permutation matrix P such that J 0 J 0 J A = RP T P + P T P R 1 = R J p R 1, J C C where R = RP T, and C collects all the blocks of the form (2.2.9). Since the ascent sequence condition holds for A it must also hold for J 0, so J 0 has a pth-root, say W 0, and W 0 can be taken real in view of the construction given in [22, Section 3]; clearly, there exists a real matrix W + such that W p + = J + and, following Corollaries and , there exists a real matrix W c such that Wc p = C. Hence, the matrix X = R[W 0 W + W c ] R 1 is a real pth-root of A.

38 28 Conversely, if A satisfies the ascent sequence condition and has an odd number of Jordan blocks corresponding to a negative eigenvalue, then the process just described can not produce a real matrix pth-root, as one of the Jordan blocks can not be paired, so that the root of such a block is necessarily complex. Case 2: p is odd. Follows similarly to the first case since real roots can be taken for J 0, J +, J, and C. We now present an analog of Theorem for real matrices. Theorem (Classification of pth-roots of nonsingular real matrices). Let F k be defined as in (2.2.15). If A M n (R) is nonsingular, then A has precisely p s primary pth-roots, given by r k=1 X j = R f j k (J nk (λ k )) 0 R 1, (2.2.16) 0 r+c k=r+1 F j k (C nk (λ k )) where j = (j 1,..., j r, j r+1,..., j r+c ), j k = (j k1, j k2 ) for k = r + 1,..., r + c, and j i = j k whenever λ i = λ k. If s < t, then A has additional nonprimary pth-roots that form parameterized families given by r k=1 X j (U) = RU f j k (J nk (λ k )) 0 U 1 R 1, (2.2.17) 0 r+c k=r+1 F j k (C nk (λ k )) where U is an arbitrary nonsingular matrix that commutes with J R, and for each j there exist i and k, depending on j, such that λ i = λ k while j i j k. Proof. Following Theorem , there exists a real, invertible matrix R such that r R 1 k=1 AR = J R = J n k (λ k ) ; r+c k=r+1 C n k (λ k )

39 29 if T = I, then r+c k=r+1 S n k P nk r T 1 k=1 J R T = J = J n k (λ k ) r+c k=r+1 [ )], Jnk (λ k ) J nk ( λk following Lemmas and Following Theorem , J R has p s primary roots given by r k=1 T f j k (J nk (λ k )) [ ] T 1 r+c k=r+1 f jk1 (J nk (µ k )) f jk2 (J nk ( µ k ) r k=1 = f j k (J nk (λ k )) 0, 0 r+c k=r+1 F j k (C nk (µ k )) where j k = (j k1, j k2 ) for k = r + 1,..., r + c, which establishes (2.2.16). If A is derogatory, then J R has additional roots of the form r k=1 T W f j k (J nk (λ k )) [ ] W 1 T 1, r+c k=r+1 f jk1 (J nk (µ k )) f jk2 (J nk ( µ k ) where W is arbitray nonsingular matrix that commutes with J. Note that r k=1 T W f j k (J nk (λ k )) [ ] W 1 T 1 r+c k=r+1 f jk1 (J nk (µ k )) f jk2 (J nk ( µ k ) r k=1 = U f j k (J nk (λ k )) U 1, r+c k=r+1 F j k (C nk (λ k )) where U = T W T 1. Following [15, Theorem 1, 12.4], U is an arbitary, nonsingular matrix that commutes with J R, which establishes (2.2.17).

40 30 The next theorem identifies the number of real primary pth-roots of a real matrix (c.f. [9, Theorem 2.4]) and our proof utilizes the real Jordan canonical form. Corollary Let the nonsingular real matrix A have r 1 distinct positive real eigenvalues, r 2 distinct negative real eigenvalues, and c distinct complex-conjugate pairs of eigenvalues. If p is even, there are (a) 2 r 1 p c real primary pth-roots when r 2 = 0; and (b) no real primary pth-roots when r 2 > 0. If p is odd, there are p c real primary pth-roots. Proof. Following Theorem , there exists a real, invertible matrix R such that r R 1 k=1 AR = J R = J n k (λ k ). r+c k=r+1 C n k (λ k ) Case 1: p is even. If r 2 > 0, then A does not possess a real primary root, since A must have an even number of Jordan blocks of each size for every negative eigenvalue and the same branch of the pth-root function must be selected for every Jordan block containing the same negative eigenvalue. If r 2 = 0, then, following Corollary , for every complex-conjugate pair of eigenvalues, there are p choices such that F jk (C nk (λ k )) is real. For every real eigenvalue, there are two choices such that f jk (J nk (λ k )) is real, yielding 2 r 1 p c real primary roots. Case 2: p is odd. The matrix X j is real provided that the principal-branch of the pth-root function is chosen for every real eigenvalue. Similar to the first case, there are p choices such that F jk (C nk (λ k )) is real, yielding p c real primary roots. The following theorem extends Theorem to include singular matrices (see [9, Theorem 2.6]). Theorem (Classification of pth-roots). Let A M n (C) have the Jordan canonical form Z 1 AZ = J = J 0 J 1, where J 0 collects together all the Jordan blocks corresponding to

41 31 the eigenvalue zero and J 1 contains the remaining Jordan blocks. If A possesses a pth-root, then all pth-roots of A are given by A = Z (X 0 X 1 ) Z 1, where X 1 is any pth-root of J 1, characterized by Theorem , and X 0 is any pth-root of J 0.

42 32 Chapter 3 Matrix Roots of Nonnegative Matrices Matrices We analyze and categorize the primary matrix roots of a 2 2 nonnegative matrix via the interpolating-polynomial definition of a matrix function (see Definition 2.2.3, (2.2.3), and (2.2.4)): let A M 2 (R) and suppose A 0. Following Theorem and the fact that the spectrum of a real matrix must be self-conjugate (see, e.g., [11, Appendix C]), without loss of generality, we may assume that σ (A) = {1, λ}, where 1 λ 0 and λ R. Note that the Jordan form of A must be or 1 0 (3.1.1) 0 λ 1 1. (3.1.2) 0 1 As in (2.2.5), let f j denote the (j + 1)st-branch of the pth-root function. For 2 2 matrices having the Jordan form (3.1.1), with the exception when λ = 1, following (2.2.4), the required Lagrange-Hermite interpolating polynomial is given by p(t) = f j1 (1) t λ ( ) t 1 1 λ + f j j2 (λ) = ωj 1 p f j2 (λ) t + f j 2 (λ) λω j 1 p. λ 1 1 λ 1 λ Thus, if j = (j 1, j 2 ), then A has p 2 primary pth-roots given by A 1/p j = ωj 1 p f j2 (λ) 1 λ A + f j 2 (λ) λω j 1 p I 2. (3.1.3) 1 λ

43 > λ 0 Let A 1/p j be the matrix given in (3.1.3) and consider the two cases depending on the parity of p: (1) p is even. The root A 1/p j is real if and only if j 1, j 2 {0, p/2}. If j 1 = j 2 = 0, then A 1/p j = 1 p λ 1 λ A + p λ λ 1 λ I 2 0, because (1 p λ)/(1 λ) > 0 and ( p λ λ)/(1 λ) > 0. If j 1 = j 2 = p/2, then A 1/p j = If j 1 = p/2 and j 2 = 0, then A 1/p j p λ 1 1 λ A + λ p λ 1 λ I 2 0. can not be nonnegative since the off-diagonal entries of A are negative and not affected by a positive multiple of the identity. If j 1 = 0 and j 2 = p/2, then A 1/p j = 1 p λ 1 λ A + λ p λ 1 λ I 2, and A 1/p j 0 if and only if (1 p λ)a ii (λ p λ) for i = 1, 2. (2) p is odd. The root A 1/p j is real if and only if j 1 = j 2 = 0 and, if j 1 = j 2 = 0, then A 1/p j = 1 p λ 1 λ A + p λ λ 1 λ I 2 0, because (1 p λ)/(1 λ) > 0 and ( p λ λ)/(1 λ) > 0. If λ = 0, then A is a rank-one matrix and A 1/p j = ω j 1 p A, j 1 R(p)

44 > 0 > λ Let A 1/p j be the matrix given in (3.1.3) and consider the two cases depending on the parity of p: (1) p is even. If λ < 0 and p is even, then A 1/p j M 2 (R). (2) p is odd. Same as (2) when 1 > λ > λ = 1 If the Jordan form of A is (3.1.1), then (trivially) A = I 2 and { } ω j 1 p 1 p I 2 are the pth-roots j 1 =0 of A. If the Jordan form of A is (3.1.2), then (2.2.3) reduces to p(t) = f j1 (1) + f j 1 (1)(t 1) = ω j 1 p + ωj 1 p p (t 1) so that j = f j1 (1)I 2 + f j 1 (1)(A I 2 ) = ωj 1 p p A + A 1/p [ ω j 1 p ωj 1 p p ] I 2, which is real if and only if j 1 = 0 (or j 1 = p/2, when p is even). When j 1 = 0, note that A 1/p j = 1 [ p A ] I 2 = A + (p 1)I 2. p p As a special case, note that if A = 1 1, then J p := p N. p 1 is a pth-root for all

45 Primitive Matrices We present our main results for primitive matrices. Theorem Let the nonsingular primitive matrix A have r 1 distinct positive real eigenvalues, r 2 distinct negative real eigenvalues, and c distinct complex-conjugate pairs of eigenvalues. If p is even, there are (a) 2 r1 1 p c eventually positive primary pth-roots when r 2 = 0; and (b) no eventually positive primary pth-roots if r 2 > 0. If p is odd, there are p c eventually positive primary pth-roots. Proof. If r 2 > 0, then, following Theorem , the matrix A does not have a real primary pth-root, hence, a fortiori, it can not have an eventually positive primary pth-root. If r 2 = 0, then, following Theorems and 2.1.1, there exists a real, invertible matrix R such that ] ρ A = [x R r k=2 J n k (λ k ) yt, (3.2.1) (R 1 ) r+c k=r+1 C n k (λ k ) ] where R = [x R M n (R), R 1 = y M n (R), x > 0 is the right Perron-vector, (R 1 ) and y > 0 is the left Perron-vector. Following Lemma 2.2.7, f j (z) < f j (z ) for all j, j {0, 1,..., p 1}, so that any primary pth-root X of the form p ρ X j = R r k=2 f j k (J nk (λ k )) r+c k=r+1 F j k (C nk (λ k )) R 1

46 36 inherits the strong Perron-Frobenius property from A. Since f(a) T = f(a T ) ([8, Theorem 1.13(b)]), a similar argument demonstrates that X T inherits the strong Perron-Frobenius property from A T. Case 1: p is even. For all k = 2,..., r, there are two possible choices such that f jk (J nk (λ k )) is real; following Corollary , for all k = r + 1,..., r + c, there are p choices such that F jk (C nk (λ k )) is real. Thus, there are 2 r1 1 p c possible ways to select X and X T to be real. Case 2: p is odd. For all k = 2,..., r, the principal-branch of the pth-root function must be selected so that f jk (J nk (λ k )) is real and, similar to the previous case, there are p choices such that F jk (C nk (λ k )) is real. Hence, there are p c ways to select X and X T real. Regardless of the parity of p, following Theorem 2.1.6, the matrices X and X T are eventually positive. Example We demonstrate Theorem via an example. Consider the matrix A =

47 37 Note that A = RJ R R 1, where J R = , R = , and R 1 = Because σ (A) = {10, 1 + i, 1 + i, 1 i, 1 i}, following Theorem 3.2.1, A has eight primary matrix square-roots, of which the matrices X j = R R 1

48 = , where j = (0, (0, 0)), and X j = R = , where j = (0, (1, 1)), are eventually positive square-roots of A. R 1 The following question arises from Example 3.2.2: are X j and X j the only eventually positive square-roots of A? This is answered in the following result, which yields an explicit description of the eventually positive primary roots of a nonsingular primitive matrix A. Theorem Let A be a nonsingular, primitive matrix and X j be any primary pth-root

49 39 of A of the form f j1 (ρ) X j = R r k=2 f j k (J nk (λ k )) R 1, r+c k=r+1 F j k (C nk (λ k )) where j = (j 1,..., j r, j r+1,..., j r+c ) and j k = (j k1, j k2 ), for k = r + 1,..., r + c. If p is odd, then X j is eventually positive if and only if 1. j 1 = 0; 2. j k = 0 for all k = 2,..., r; and 3. j k = (0, 0) or j k = (j k1, p j k1 ) for all k = r + 1,..., r + c. If p is even, then X j is eventually positive if and only if 1. j 1 = 0; 2. j k = 0 or j k = p/2 for all k = 2,..., r; and 3. j k = (0, 0) or j k = (j k1, p j k1 ) for all k = r + 1,..., r + c. Proof. We demonstrate necessity as sufficiency is shown in the proof of Theorem To this end, we demonstrate the contrapositive. Case 1: p is odd. If the principal branch of the pth-root function is not selected for the Perron eigenvalue, then X j can not possess the strong Perron-Frobenius property; if j k 0 for some k {2,..., r}, then f jk (J nk (λ k )) is not real so that X j can not be real; similarly, X j is not real if j k (0, 0) or j k (j k1, p j k1 ) for some k {r + 1,..., r + c}. Case 2: p is even. Result is similar to the first case, but we note that, without loss of generality, we may assume that A does not have any negative eigenvalues (else it can not possess a primary root).

50 40 Theorem Let A be a primitive, nonsingular, derogatory matrix that possesses a real root, and let X j (U) be any nonprimary root of A of the form f j1 (ρ) X j (U) = RU r k=2 f j k (J nk (λ k )) U 1 R 1, r+c k=r+1 F j k (C nk (λ k )) where j = (j 1,..., j r, j r+1,..., j r+c ) and j k = (j k1, j k2 ), for k = r + 1,..., r + c, subject to the constraint that for each j, there exist i and k, depending on j, such that λ i = λ k, while j i j k. If p is even, then X j (U) is eventually positive if and only if (1) j 1 = 0; (2) j k = 0 or j k = p/2, for all k = 2,..., r; (3) j k = (0, 0) or j k = (j k1, p j k1 ) for all k = r + 1,..., r + c; (4) U is a real nonsingular matrix; and (5) Jordan blocks containing negative eigenvalues are transformed, via a permutation matrix, to blocks of the form (2.2.9) (see proof of Theorem ) and branches for these blocks are selected in accordance with Corollary If p is odd, then X j (U) is eventually positive if and only if (1) j 1 = 0; (2) j k = 0 for all k = 2,..., r; (3) j k = (0, 0) or j k = (j k1, p j k1 ) for all k = r + 1,..., r + c; and

51 41 (4) U is a real nonsingular matrix. Example We demonstrate Theorem via an example. Consider the matrix A = Note that A = RJ R R 1, where J R =

52 42 R = , and R 1 = If j = (0, (0, 0), (1, 1)), then, following Theorem 3.2.4, any matrix of the form X j (U) = RU 20 F (0,0) (C 2 (1 + i)) F (1,1) (C 2 (1 + i)) R 1 U 1,

53 43 where U = u u 2 u u 4 u u 3 u u 5 u u 2 u u 4 u u 3 u u 5 u 4 0 u 6 u u 8 u u 7 u u 9 u u 6 u u 8 u 9, u 7 u u 9 u 8 u 1 0, and u i R, for i = 2,..., 9, is an eventually positive square-root of A. Next, we present an analog of Theorem for real matrices. Theorem Let the primitive matrix A M n (R) have the real Jordan canonical form R 1 AR = J R = J 0 J 1, where J 0 collects all the singular Jordan blocks and J 1 collects the remaining blocks. If A possesses a real root, then all eventually positive pth-roots of A are given by A = R (X 0 X 1 ) R 1, where X 1 is any pth-root of J 1, characterized by Theorem or Theorem 3.2.4, and X 0 is a real pth-root of J 0. Remark It should be clear that Theorems 3.2.1, 3.2.3, 3.2.4, and remain true if the assumption of primitivity is replaced with eventually positivity. Recall that for A M n (C) with no eigenvalues on R, the principal pth-root, denoted by A 1/p, is the unique pth-root of A all of whose eigenvalues lie in the segment {z : π/p <

54 44 arg(z) < π/p} [8, Theorem 7.2]. In addition, recall that a nonnegative matrix A is said to be stochastic if n j=1 a ij = 1, for all i = 1,..., n. In [9], being motivated by discrete-time Markov-chain applications, two classes of stochastic matrices were identified that possess stochastic principal pth-roots for all p. As a consequence, and of particular interest, for these classes of matrices the twelfth-root of an annual transition matrix is itself a transition matrix. A (monthly) transition matrix that contains a negative entry or is complex is meaningless in the context of a model, however the following remark demonstrates that, under suitable conditions, a matrix-root of a primitive stochastic matrix will be eventually stochastic (i.e., eventually positive with row sums equal to one). Eventual stochasticity may be useful in the following manner: consider, for example, the application of discrete-time Markov chains in credit risk: let P = [p ij ] M n (R) be a primitive transition matrix where p ij [0, 1] is the probability that a firm with rating i transitions to rating j. Such matrices are derived via annual data, and the twelfth-root of such a matrix would correspond to a monthly transition matrix if the entries are nonnegative. However, in application, the twelfth-root would be used for forecasting purposes (e.g., to estimate the likelihood that a firm with credit-rating i, transitions to rating j, m months into the future). Thus, if R is the twelfth-root of P and R m 0, then r (m) ij is a candidate for the aforementioned probability. Moreover, it is well-known that n 2 2n + 2 is a sharp upper-bound for the primitivity index of a primitive matrix (see, e.g., [11, Corollary 8.5.9]) so that eventual stochasticity is feasible in application. Remark Theorems 3.2.1, 3.2.3, 3.2.4, and remain true if stochasticity is added as an assumption on the matrix A and the conclusion of eventually postivity is replaced with eventually stochasticity.

55 45 We conclude with the following remark. Remark If A is an eventually positive matrix matrix, then B := A q is eventually positive for all q N: thus, all the previous results hold for fractional powers of A. 3.3 Imprimitive Irreducible Matrices Consider the matrices A = 0 0 1, B = and C = One can verify that B 2 = C 2 = A, yet the matrix C is not eventually nonnegative despite possessing left and right positive eigenvectors. In this section, we investigate why it is guaranteed that B is an eventually nonnegative and C is not eventually nonnegative Complete Residue Systems Our discussion of complete residue systems begins with the following definition. Definition If R = {a 0, a 1,..., a h 1 } is a set of integers, then R is said to be a complete residue system (modh), denoted CRS (h), if the map f : R R(h) defined by a i q i := a i (modh) is injective or, equivalently, surjective. With the aforementioned definition, we readily glean the following equivalent properties if R is a CRS(h):