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NTH ROOTS OF MATRICES by Eric Jarman An Abstract presented in partial fulllment of the requirements for the degree of Master of Science in the Department of Mathematics and Computer Science University of Central Missouri June, 2012 ABSTRACT by Eric Jarman This paper investigates the feasibility of nding any power, for an arbitrary m×m m × m square matrix. matrices and complex m×m nth root, and by extension any rational Root nding methods are investigated for real matrices. NTH ROOTS OF MATRICES by Eric Jarman A Thesis presented in partial fulllment of the requirements for the degree of Master of Science in the Department of Mathematics and Computer Science University of Central Missouri June, 2012 iv NTH ROOTS OF MATRICES by Eric Jarman ACCEPTED: C~o:::hematics and Computer Science Contents 1 Introduction 1 2 Iterative Methods 3 2.1 Babylonian Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Netwon's Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Denman-Beavers Iteration 2.5 Weaknesses 3 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Complex Number Equivalents 13 3.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3 Cayley-Dickson construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Matrix Forms and Properties 22 4.1 Powers of a Matrix and Diagonal Matrices . . . . . . . . . . . . . . . . . . . 22 4.2 Matrix Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.3 Examples: Diagonalizing Matrix Forms of Complex Numbers . . . . . . . . . 29 4.3.1 Example 1: 2×2 Matrix of a Complex Number . . . . . . . . . . . . 29 4.3.2 Example 2: 2×2 Matrix of a Quaternion . . . . . . . . . . . . . . . . 34 v vi Contents 4.3.3 5 4×4 Matrix of a Quaternion . . . . . . . . . . . . . . . . 38 4.4 Roots of Complex Numbers vs. Roots of Diagonal Matrices . . . . . . . . . . 42 4.5 Jordan Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.6 Nilpotent Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.7 Existence of Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Hypercomplex Numbers 55 5.1 Multicomplex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.1.1 Bicomplex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.1.2 Tessarines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.2 Split-Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.3 Dual Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.4 Split-Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.4.1 Roots of Split-Quaternions . . . . . . . . . . . . . . . . . . . . . . . . 62 Biquaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.5.1 68 5.5 6 Example 3: Conclusion Bibliography Roots of Biquaternions . . . . . . . . . . . . . . . . . . . . . . . . . . 72 73 Chapter 1 Introduction The operations for nding any integer power of a matrix are well-dened. The same, however, cannot be said for non-integer powers of a matrix. If we can nd a method for calculating some root of a matrix integer n > 1, A, that is to say, nd some matrix then we can dene this matrix A0 to be an It is important to note that for an arbitrary matrix of A. A0 nth A, where root of (A0 )n = A for some A. there may be multiple nth roots It is easy to construct a simple example which shows this: 2 2 4 0 −2 0 2 0 . = = 0 4 0 −2 0 2 When there are multiple solutions, we can choose one of them as the principal root. If we do so, we can denote the principal root by 1 An . We should note that it is possible to construct examples of complex matrix roots as well as real matrix roots: 2 2 i 0 −i 0 −1 0 = = . 0 i 0 −i 0 −1 So, we would ideally like to have root nding methods which will work for real matrices and 1 CHAPTER 1. 2 INTRODUCTION complex matrices. This denition can also be used to nd any rational power the principal nth root to the pth p of a matrix by simply taking n power A 1 n p p = An . We note that this construction only has meaning when matrices. There is no need to examine m×n construct matrices whose product is a given matrices where m×n A and m 6= n, A0 are square m×m as we might of course matrix, but such matrices would not be equal in values or dimensions, and so cannot be called roots. Chapter 2 Iterative Methods We can begin by searching for methods of directly calculating a root. The simplest place to start is to look at methods of calculating square roots of real numbers, and expand them to be used on matrices. 2.1 Babylonian Method The Babylonian method, also known as Heron's method, is based on a historical method of calculating square roots. The earliest reference to this method is listed in a numerical approximation for the square root of 2 on the Babylonian clay tablet YBC 7289 [20]. The rst explicit denition of the method was given by Hero of Alexandria in the rst century [1]. This method for nding the square root of s ∈ R starts with root, and can be fully described as follows: √ s, 1 s = xn + 2 xn x0 ≈ xn+1 3 for n > 0. some approximation for the CHAPTER 2. 4 ITERATIVE METHODS This sequence is guaranteed to converge quadratically to the principal root if both s and x0 are positive. This method can be extended to matrices by starting with the identity matrix I, and proceeding as follows: X0 = I Xk+1 = 1 Xk + AXk−1 . 2 This method is numerically unstable, i.e., small perturbations of the error matrix in the k th iteration may induce perturbations with increasing norm in successive iterations, causing the sequence to diverge from the root [24]. When it does converge, this sequence will converge quadratically to the root. The computational expense of this method is low, requiring the calculation of a single matrix inverse per iteration. Note that the norm in the preceding paragraph is the norm of a square matrix. norm of A, denoted N (A) or kAk, is a nonnegative number associated with A The having the properties [50] 1. kAk > 0 2. kkAk = |k|kAk 3. kA + Bk ≤ kAk + kBk, 4. kABk ≤ kAkkBk. when A 6= 0, and kAk = 0 ⇔ A = 0, for any scalar k, There are multiple methods which may be used to compute the norm of a matrix of the simplest to compute norms [46] are NI (B) = sX X i j b2ij B. Two CHAPTER 2. 5 ITERATIVE METHODS and NII (B) = max X i 2.2 |bij | . j Binomial Theorem The binomial theorem can be used to nd the coecients of any power of the quantity For x, y ∈ R, n ∈ Z, n ≥ 0, (x+y). we state: n n X n n−k k X n k n−k (x + y) = x y = x y . k k k=0 k=0 n The well-known generalization of the binomial theorem by Sir Isaac Newton allows for any real exponent, rather than being limited to positive integers, by replacing the nite sum with an innite series. ∞ X r r−k k (x + y) = x y . k k=0 r In this case, it is necessary to dene what is meant by a binomial coecient with any real upper index. If n is an integer we have n n! = k k!(n − k)! For any real number r, we factor out (n − k)! the usual binomial coecient, and substitute r from the top and bottom of the formula for for n to yield [21] r r(r − 1) · · · (r − k + 1) (r)k = = , k k! k! where (r)k denotes the falling factorial, dened as To nd an arbitrary root of any real a, (r)k = r(r − 1) · · · (r − k + 1). we substitute a = 1+x into the generalized CHAPTER 2. 6 ITERATIVE METHODS binomial equation above to yield a formula involving only a single variable. (1 + x) r = ∞ X r k k=0 xk = 1 + rx + r(r − 1) 2 r(r − 1)(r − 2) 3 x + x + ··· . 2! 3! Note that we can use the rst few terms of this series as an approximation for a root of This method can be extended for nding a root of a real matrix Rm×m where A=I +B (I + B) r = ∞ X r k r(r − 1) 2 r(r − 1)(r − 2) 3 B + B + ··· . 2! 3! This series will converge if it can be shown that of B as A, B ∈ Bk = I + rB + B k → [0] [46] by taking and rewriting the power series as k=0 show that A a. k B k → [0] as k increases. It is possible to increases by either showing that the dominant characteristic root is less than unity, or by showing that the norm of the matrix If the norm of the matrix B is close to 1, N (B) < 1. the series may converge slowly, in which case the norm of the matrix in the series can be reduced by introducing a scalar coecient. For A, C ∈ Rm×m , and a scalar k ∈ R, we take A = k(I + C), where C= 1 k A−I to make the series become r A =k where k r r(r − 1) 2 r(r − 1)(r − 2) 3 I + rC + C + C + ··· , 2! 3! is any scalar value selected that minimizes the norm of the matrix C. The norm CHAPTER 2. NI (C) 7 ITERATIVE METHODS is minimized [46] by using P P 2 i j aij k= P . i aii Other norms may be more dicult to minimize. The calculation of this series is even cheaper than the Babylonian square root method shown earlier, since only one matrix multiplication is required for each term, rather than one matrix inversion. If the series can be shown to converge, or modied appropriately to ensure faster convergence, only the rst few terms of the series are necessary to nd an approximation of a root. Also, since we may choose any real value for r, we are not limited to nding only square roots. 2.3 Netwon's Method Newton's method, or the Newton-Raphson method, is a method for nding roots (or zeroes) of real valued functions. The method starts with an initial guess using the function f and its derivative f (xn ) f 0 (xn ) This method can be used to nd a square root of a number and its derivative f0 and proceeds to iterate f 0: xn+1 = xn − f x0 , as: f (x) = x2 − a f 0 (x) = 2x a by simply dening the function CHAPTER 2. 8 ITERATIVE METHODS We should note that given this denition for the real function f (x), we see that the Baby- lonian method is just a special case of Newton's Method for the square root. f (xn ) f 0 (xn ) x2 − a xn − n 2xn 2 2xn − x2n − a 2x 2 n 1 xn − a 2 xn 1 a xn − . 2 xn xn+1 = xn − = = = = The proof of quadratic convergence [51] shows us the properties of successive error terms of the sequence {xn }, n+1 ≈ f 00 (α) 2 2f 0 (α) k from which it is possible to determine conditions on x0 that guarantee convergence of the sequence: Let I be the interval 1. f 0 (x) 6= 0; ∀x ∈ I 2. f 00 (x) 3. x0 Then [α − r, α + r] r ≥ |α − x0 | where , is not unbounded, is suciently close to {xn } for some ∀x ∈ I α , . will converge to a square root α of a. Since the function used for nding a square root of a number see that the rst derivative f 00 (x) = 2 means that f 0 (x) = 2x will only equal is constant, and thus not unbounded. x0 0 at x = 0, a is f (x) = x2 − a, we can and the second derivative For the initial guess, suciently close is in a neighborhood of a root within which the sequence can be guaranteed to converge. One set of conditions for nding such a neighborhood is found in Kantorovich's Theorem [25]: CHAPTER 2. 9 ITERATIVE METHODS Let a0 be a point in Rn , U an open neighbor- Theorem 2.3.1 (Kantorovich). hood of a0 in Rn , and f~ : U 7→ Rn a dierentiable mapping, with its derivative h i Df~(a0 ) invertible. Dene h i−1 ~h0 = − Df~(a0 ) f~(a0 ), a1 = a0 + ~h0 , U0 = B|~h0 | (a1 ) h i ~ If U0 ⊂ U and the derivative Df (x) satises the Lipschitz condition h i h i ~ ~ Df (u1 ) − Df (u2 ) ≤ M |u1 − u2 | for all points u1 , u2 ∈ U0 , and if the inequality h i−1 2 1 ~ ~ f (a0 ) Df (a0 ) M ≤ 2 is satised, the equation f~(x) = ~0 has a unique solution in the closed ball U0 , and Newton's method with initial guess a0 converges to it. Note that M above is the Lipschitz ratio, measuring the change in the derivative. By modifying our initial function to nd higher nth f (x) = xn − a, we can use Newton's method to roots as well. In this general case, the convergence criteria on the rst and second derivatives are met by any interval which does not contain use Kantorovich's Theorem to nd an initial guess x0 . x = 0, and we again The convergence criteria can also be adjusted to apply to complex valued functions [26], meaning that this method is not restricted to nding only real roots. Since we see some advantages to this method, we should continue investigating this method as it applies to matrices. Newton's method can be extended to matrices by starting with some initial guess X0 for a root, and expanding the iteration using Fréchet derivatives. Fréchet derivatives are used to nd the derivative of a function at a point within the domain of the function. A simple denition is: A function f is Fréchet dierentiable at a if CHAPTER 2. 10 ITERATIVE METHODS the limit f (x) − f (a) x→a x−a lim exists [47]. When applied to a matrix function, for example F : Cn×n → Cn×n , it is necessary to remember that matrix multiplication is not commutative for all matrices. working with matrix derivatives, for any A, B ∈ Cn×n Thus, when we may have something like [19] d(AB) = d(A)B + Ad(B). Now to apply Newton's Method to nd a square root of a matrix, we start with Cn×n and a matrix function F : Cn×n → Cn×n X, A ∈ dened as F (X) = X 2 − A To dene the iteration, we use a general function X0 G : Cn×n → Cn×n , taking as an initial approximation, we have: −1 Xk+1 = Xk − [G0 (Xk )] where G0 is the Fréchet derivative of G. G(Xk ), k = 0, 1, 2, . . . , Next identifying F (X + H) = X 2 − A + (XH + HX) + H 2 with the Taylor series for F, F (U ) = F (X) + F 0 (X)(U − X) + F 00 (X) (U − X)2 + · · · , 2 F (X + H) = F (X) + F 0 (X)H + F 00 (X) 2 H + ··· , 2 Xk ∈ Cn×n , and CHAPTER 2. 11 ITERATIVE METHODS so F (X) = X 2 − A, F 0 (X)H = XH + HX, F 00 (X) 2 H = H2 2 and so F 0 (X) is a linear operator, so F 00 = 2, F 0 : Cn×n → Cn×n . Thus the iteration can be written as [24]: X0 given, Xk+1 = Xk + Hk k = 0, 1, 2, . . . . where Hk is the solution of the equation Xk Hk + Hk Xk = A − Xk2 , which can be calculated using the Schur decomposition of If F 0 (X) is nonsingular and k X − X0 k converge quadratically to a square root X Xk [4]. is suciently small, the Newton iteration will of the original matrix A [24]. It is important to note that because this iteration as dened above requires calculating a Schur decomposition at each step, it is computationally expensive, especially as compared to other iterations. CHAPTER 2. 2.4 12 ITERATIVE METHODS Denman-Beavers Iteration The Denman-Beavers iteration [16] starts with both the identity matrix I and the matrix A, and iterates in two directions towards a root: Y0 = A, Z0 = I, 1 (Yk + Zk−1 ), 2 1 = (Zk + Yk−1 ). 2 Yk+1 = Zk+1 This iteration converges quadratically, with A, and Zk converging to its inverse X −1 . Yk converging to a square root X of matrix The iteration is not guaranteed to converge, even if a root exists. The computation is relatively expensive, requiring calculating two matrix inverses at each iteration. It should be noted that this iteration can be shown to be a simplication of the general form of the Newton's Method iteration [24]. 2.5 Weaknesses There are various methods which can be used to improve the likelihood of producing a root, but each of these methods and others like them share a few weaknesses. numerically based, they all calculate an approximated value. First, all being Second, they can all fail to converge to a solution, even if a root of a given matrix is known to exist. Lastly, they each only nd a single result, while there may be multiple roots for a given matrix. Given these limitations, it would desirable to nd some method that would always produce a root, or further, multiple roots, if such roots exist. Chapter 3 Complex Number Equivalents In order to nd a method assured of nding a root of any matrix, it makes sense to rst look at number systems where roots will be guaranteed to exist and nd matrix equivalents of these. 3.1 Complex Numbers The rst number system to investigate is the complex numbers. To nd any root in the complex number system, it is necessary only to use a generalization of de Moivre's formula. Given a complex number z in polar form z = r(cos θ + i sin θ) or in exponential form z = reiθ 13 CHAPTER 3. an nth root α 14 COMPLEX NUMBER EQUIVALENTS can be written α = r 1 n cos θ + 2kπ n + i sin θ + 2kπ n , or 1 α = r n ei where k is an integer with values from z n−1 0+i ei 2 0−i ei 2 1 2 + 1 2 − √ i 3 2 √ i 3 2 −1 + 0i , providing the unique roots of eiπ 1 1 to −1 + 0i z2 z3 0 θ+2kπ n π 3π π ei 3 2π ei 3 eiπ We can also of course nd the roots of the complex conjugate z̄ = re−iθ . z̄ = r(cos x − i sin x) or Quickly looking at a few sets of examples of the roots of a complex number and the roots of its conjugate values. z. 1 z̄ n , we notice that there do not appear to be any overlapping CHAPTER 3. z ei 2 1+i √ 2 ei 4 √ − 1+i 2 ei 4 1 1 π 0+i z2 z3 COMPLEX NUMBER EQUIVALENTS √ i 2 + i 2 − 3 2 √ 3 2 0−i z̄ π 1 z̄ 2 3π π 1 ei 6 z̄ 3 5π ei 6 π e−i 2 π 0−i e−i 2 1−i √ 2 e−i 4 1−i √ 2 e−i 4 − 2i + − 2i − π 3π √ 3 2 √ 3 2 0+i π e−i 6 5π e−i 6 π ei 2 15 CHAPTER 3. 1 1 z3 π 1+i √ 2 ei 4 + i sin π8 − cos π8 − i sin π8 √ √3 3 1 1 2 2 + 2√2 + i 2 − 2√2 2 √ √3 3 1 1 − 2 2 + 2√2 + i − 2 2 − 2√2 ei 8 z z2 cos π 8 1 1 z̄ 2 9π ei 8 π 1 ei 12 z̄ 3 9π ei 12 e π π π + i sin 16 π π −i cos 16 + sin 16 π π − cos 16 − i sin 16 π π i cos 16 − sin 16 cos π e−i 4 − i sin π8 − cos π8 + i sin π8 √ √3 3 1 1 2 2 + 2√2 + i − 2 + 2√2 2 e−i 8 √ − 1+i 2 √ √3 3 − 2 2 + 2√1 2 + i 2 2 + e−i 12 cos 1 ei 16 16 z̄ 4 9π ei 16 17π ei 16 25π ei 16 We would like to prove that the roots of z π 8 π 16 and the roots of its conjugate 0 ≤ k < n, 0 ≤ l < n , θ+2πk n θ + 2πk n = ei = −θ+2πl n , −θ + 2πl + 2πm, n θ + 2πk = −θ + 2πl + 2πmn, θ = π(l − k + mn), ⇒ eiθ ∈ R. z̄ π 9π e−i 8 π e−i 12 9π 1 √ 2 2 π − i sin 16 π π −i cos 16 − sin 16 π π − cos 16 + i sin 16 π π i cos 16 + sin 16 cos equal. We can start by seeing what would happen if they were equal. Let ei π 1−i √ 2 z̄ i 17π 12 √ − 1−i 2 z4 16 COMPLEX NUMBER EQUIVALENTS 17π e−i 12 π e−i 16 9π e−i 16 17π e−i 16 25π e−i 16 will never be k, l, n ∈ Z, n ≥ 1, CHAPTER 3. In other words, the roots of have n distinct 17 COMPLEX NUMBER EQUIVALENTS nth z and z̄ are only equal when z = z̄ . Otherwise, z and z̄ will each roots. This will be notable later in section 4.4. We would like to nd a ring isomorphism Searching yields the mapping φ from the complex numbers to a set of matrices. φ : C → R2×2 for a complex number to a real matrix [37]: z = a + bi, a b φ(z) 7→ Z = , −b a if then where z z ∈ C, Z ∈ R2×2 , and a, b ∈ R. We can then see that the roots of the complex number can be used to nd roots for the real matrix Z. This method has multiple advantages for the matrices in φ: it will nd exact roots, while the iterative methods all nd numerical approximations; it is very ecient, as roots of complex numbers are easy to calculate; and it will nd n roots, namely those which correspond to roots of the corresponding complex numbers. The weakness of this approach is that is it not clear that all possible roots for a matrix in φ will be found using this mapping. Further, this mapping cannot be used to represent any arbitrary 2×2 real matrix, leaving much of the matrix space of the 2×2 real matrices untouched. Thus, it appears to be necessary to try a larger complex space. 3.2 Quaternions A larger well-known complex number system is the set of quaternions. We also have a method for determining any root of a quaternion, using a variant of de Moivre's formula. This is achieved by taking a quaternion z ∈ H, z = a + bi + cj + dk , z = a + ω , where a is the real part of z , and ω If we now take q to be a unit quaternion (ie. and writing it as is what is called the pure quaternion part of |q| = 1), z. then we can write the unit quaternion CHAPTER 3. 18 COMPLEX NUMBER EQUIVALENTS q = a + bi + cj + dk as: q = cos θ + ω sin θ, where cos θ = a and 1 (bi + cj + dk) b2 + c 2 + d 2 1 (bi + cj + dk), = √ 1 − a2 ω = √ a form similar to the polar form of a complex number. Moivre's formula for a unit quaternion q It is then possible to express de as [8] q n = eωnθ = (cos θ + ω sin θ)n = cos nθ + ω sin nθ. A generalization of de Moivre's formula shows that nth roots of a unit quaternion can be written [31] α = cos where θ + 2kπ n + ω sin θ + 2kπ n , k, n ∈ Z, 0 ≤ k < n. The quaternions are also representable in matrix form, as either a complex 2×2 matrix, CHAPTER 3. or a real 4×4 19 COMPLEX NUMBER EQUIVALENTS matrix, with isomorphisms φ1 : H → C2×2 and φ2 : H → R4×4 [52]: a, b, c, d ∈ R, if z ∈ H = a + bi + cj + dk, a + bi c + di 2×2 φ1 (z) 7→ Z1 = ∈C , −c + di a − bi φ2 (z) 7→ Z2 a b c d −b a −d c = ∈ R4×4 . −c d a −b −d −c b a Even though we can use de Moivre's formula and the isomorphisms above to nd roots of matrices of the appropriate forms here, it should be noted that in neither of the matrix equivalences above are we able to represent any arbitrary matrix. We can say that we would like to nd is some isomorphism which forms a basis for the matrix space. 3.3 Cayley-Dickson construction We can continue looking at larger complex number elds through the same type of method used to yield the Quaternion space, namely the Cayley-Dickson construction [18]. This construction produces increasingly larger complex number spaces, doubling the dimension each time the construction is used on the previously resulting complex space. In this way, CHAPTER 3. 20 COMPLEX NUMBER EQUIVALENTS the complex numbers, quaternions, and larger spaces can be realized as constructions on the eld of real numbers. We start by looking at a complex number as an ordered pair (a, b). On such ordered pairs, we dene addition, multiplication, and the conjugate (or involution) as: (a, b) + (c, d) = (a + c, b + d), (a, b)(c, d) = (ac − db, ad + cb), (a, b)∗ = (a, −b). We take this to be a real algebra linear spaces A0 = A ⊕ A, A, and construct a new algebra A0 as a direct sum of with multiplication dened as (a, b)(c, d) = (ac − db∗ , a∗ d + cb) and conjugation extended to (a, b)∗ = (a∗ , −b). This construction can now be used to build an innite number of extensions to the complex numbers, known as Cayley-Dickson algebras, each with dimension twice the previous. However, with each further step, some properties of the algebra may be lost. Applying this construction to the real numbers yields the complex numbers. When applied to the complex numbers, we get the Quaternions, which are not commutative. Applied to the Quaternions this construction results in the Octonions, which are not associative [3]. Applying it further to the Octonions yields the Sedonions, which are not alternative [27]. Further iterations CHAPTER 3. COMPLEX NUMBER EQUIVALENTS 21 from the Sedonions do not lose any additional properties, and remain nicely normed and are power associative [2]. Given the loss of algebraic properties in the Cayley-Dickson construction, it is important to note in particular that the Octonions and beyond are no longer associative, meaning that it will probably not be possible to nd a direct matrix representation for them, as a mapping from a matrix space would be to an associative subgroup within the Octonions or above. This leaves only the complex numbers and quaternions with matrix representations known at this time. It appears necessary at this point to investigate more properties of matrix spaces in order to nd some alternate number system in which we might nd an equivalence or isomorphism to help us search for roots. Chapter 4 Matrix Forms and Properties It is not obvious that the iterative or series methods examined earlier are the only methods for looking for a root of a matrix. Nor is it obvious that the complex numbers and the quaternions are the only types of numbers that can be represented in a matrix space. Next we examine some transformations on matrices and dierent representations of a matrix, and the properties that can be found or calculations that can be made within each representation. 4.1 Powers of a Matrix and Diagonal Matrices A power of a matrix is well known, and has the same meaning with respect to matrix multiplication as a power of an element of a eld such as the real or complex numbers. When we take any positive integer power by itself n−1 n of any matrix A, we simply multiply that matrix times. An = A | ·A·A {z· · · · · A} n 22 CHAPTER 4. When we consider any diagonal real matrix Cm×m , 23 MATRIX FORMS AND PROPERTIES A ∈ Rm×m or diagonal complex matrix A ∈ multiplying it by itself looks like this: A2 = A ·A a1 0 0 a2 = . .. 0 0 2 a1 0 0 a2 2 = . .. 0 0 ··· ··· .. . ··· ··· ··· .. . ··· 0 a1 0 · · · 0 0 a2 · · · 0 0 · . . . . . . .. . . . . am 0 0 · · · am 0 0 . . . . 2 am By induction, we can see that to nd any positive integer power n of the matrix A, we simply have to raise each element on the diagonal to the same power. An We also note that the 0th n a1 0 · · · 0 0 an · · · 0 2 = . . . .. .. . . . n 0 0 · · · am power of a matrix A is dened to be the identity matrix I. It CHAPTER 4. 24 MATRIX FORMS AND PROPERTIES is clear that this extends the previous denition. A0 0 a1 0 = . .. 0 1 0 = . .. 0 0 ··· 0 a02 · · · 0 . .. . . . 0 0 · · · am 0 · · · 0 1 · · · 0 . .. . . . 0 ··· 1 = I. When considering a negative power of a matrix, we take such an exponent to be a power of the matrix inverse of A. The matrix inverse of a diagonal matrix yields another diagonal matrix containing the multiplicative inverse of each value on the diagonal of the original CHAPTER 4. 25 MATRIX FORMS AND PROPERTIES matrix in order. Thus, for any negative integer power n A−1 a1 0 0 a2 = . . . 0 0 −1 0 a1 0 a−1 2 = . .. 0 0 −n 0 a1 0 a−n 2 = . .. 0 0 −n, A−n = Thus, we can say that for any diagonal matrix diagonal to the nth ··· ··· .. . ··· A, ··· ··· .. . ··· ··· ··· .. . ··· −1 n 0 0 . . . am n 0 0 . . . −1 am 0 0 . . . . −n am nding power holds for any integer value of An by taking each element on the n. It can also be shown that the same is true for any rational power of a diagonal matrix. CHAPTER 4. 26 MATRIX FORMS AND PROPERTIES We start by looking at a diagonal matrix A0 where 0 n (a1 ) 0 = . .. 0 a1 0 0 a2 = . .. 0 0 0 n (A ) ··· 0 (a02 )n 0 ··· ··· .. (A0 )n = A. . ··· 0 ··· 0 . .. . . . 0 n · · · (am ) 0 0 . . . . am Taking the entries along the diagonal piecewise, we see ai = (a0i )n , i = 1, . . . , m Since each element in the arrays is either a real or a complex number (i.e. can say that a0i is an nth root of ai . matrix A0 . 1/n ai we We then can choose a0i = (ai )1/n , where we dene ai , a0i ∈ C), as the principal nth i = 1, . . . , m root of ai . This produces one example of such a CHAPTER 4. 27 MATRIX FORMS AND PROPERTIES Now, we can rewrite A0 in terms of values from 0 a1 0 0 a0 2 = . .. 0 0 1/n (a1 ) 0 = . .. 0 A0 Given this representation of A ··· 0 ··· 0 . .. . . . 0 · · · am ··· 0 (a2 )1/n · · · .. 0 0 . . . . · · · (am )1/n 0 then ai has n . A0 , it is reasonable to dene A1/n = A0 . of something we might call the principal root of a matrix. ai 6= 0, distinct nth roots, the matrix A This is our rst example Note that since, in general, if may have up to nm distinct of this type. We also note that it is not obvious that there are no other roots of nth A. roots Thus, we can see that if a matrix is diagonal, this implies that we can always nd roots of that matrix. In fact, this is also true if the matrix is diagonalizable, as we see in the next section. 4.2 Matrix Diagonalization To diagonalize a real matrix, we need to nd an invertible matrix a diagonal matrix [44]. If an m×m matrix has exactly m P such that P −1 AP is distinct eigenvalues, then that matrix is diagonalizable. Note however that the converse may not be true. Thus, if a matrix does not have m distinct eigenvalues, it may still be diagonalizable. In order to know if a given matrix is diagonalizable or not, the diagonalization theorem states that an matrix If A A is diagonalizable if and only if is diagonalizable and the m A has m m×m linearly independent eigenvectors [5]. eigenvalues of the matrix, counting multiplicity, are CHAPTER 4. λ1 , λ2 , . . . , λm , 28 MATRIX FORMS AND PROPERTIES then we can write λ1 0 · · · 0 0 λ2 · · · 0 D = P −1 AP = . . .. .. . . . 0 0 · · · λm Note that each column in the matrix column matrix n D P is equal to one of the eigenvectors of is an eigenvector associated with the eigenvalue contains the eigenvalues of columns of P A λn , A, namely and the resulting diagonal on its diagonal. The order of the eigenvectors in the is not important, as diering orders only changes the order of the eigenvalues along the diagonal of D. It can also be shown that we can use the diagonalized matrix a matrix. P −1 AP = D ⇐⇒ A = P DP −1 . D to nd the nth power of CHAPTER 4. Letting 29 MATRIX FORMS AND PROPERTIES 1 D0 = D n and A0 = P (D0 )P −1 , (A0 )n = (P (D0 )P −1 )n = (P (D0 )P −1 )(P (D0 )P −1 )(P (D0 )P −1 ) · · · (P (D0 )P −1 ) = P (D0 )(P −1 P )(D0 )(P −1 P ) · · · (P −1 P )(D0 )P −1 = P (D0 )n P −1 . So (A0 )n = A, and again we will have a principal nth root for which we can write 1 An = 1 P D n P −1 . 4.3 Examples: Diagonalizing Matrix Forms of Complex Numbers We already know that we can use de Moivre's theorem to directly nd the complex number z ∈ C, or of a quaternion q ∈ H. nth roots of a It would also be useful to see if the matrix forms of the complex numbers and quaternions are also diagonalizable, and further if the nth roots of such diagonal matrices have the correct form. 4.3.1 Example 1: 2×2 Matrix of a Complex Number The matrix form of a complex number contains only real numbers, so it makes sense to rst try to diagonalize as a real matrix. Let We want to nd a matrix Let X ∈ R2×2 C ∈ R2×2 such that be the matrix form of a complex number. X −1 CX a b C= . −b a is diagonal. CHAPTER 4. 30 MATRIX FORMS AND PROPERTIES Let w x X= . y z So, X −1 = 1 z −x . wz − xy −y w Now multiplying them together yields 1 z −x a b w x X −1 CX = wz − xy −y w −b a y z zb − xa w x 1 za + xb = wz − xy −ya − wb −yb + wa y z (za + xb)x + (zb − xa)z 1 (za + xb)w + (zb − xa)y = wz − xy (−ya − wb)w + (−yb + wa)y (−ya − wb)x + (−yb + wa)z which must be a diagonal matrix. This implies wb)w + (−yb + wa)y = 0. (za + xb)x + (zb − xa)z = 0 Simplifying these equations, we see 0 = (za + xb)x + (zb − xa)z = xza + x2 b + z 2 b − xza = x2 b + z 2 b = (x2 + z 2 )b = 0 and (−ya − CHAPTER 4. 31 MATRIX FORMS AND PROPERTIES and 0 = (−ya − wb)w + (−yb + wa)y = −wya − w2 b − y 2 b + wya = −w2 b − y 2 b = (w2 + y 2 )(−b) = 0. So, we nd that either b = 0 w, x, y, z ∈ R, w, x, y, z = 0. or both x2 + z 2 = 0 and w 2 + y 2 = 0, meaning that when Thus, the real matrix form of a non-real complex number is not diagonalizable as a real matrix. Since we note that the diagonalization failed in the real matrices, we should instead try to diagonalize again in the complex matrices. Let C ∈ C2×2 representation of a complex number. a b C= . −b a be a matrix in the form of a CHAPTER 4. MATRIX FORMS AND PROPERTIES 32 We verify that the matrix is diagonalizable by nding the eigenvalues. a − λ b det(C − λI) = −b a − λ = (a − λ)(a − λ) − (b)(−b) = λ2 − 2λa + a2 + b2 = 0 =⇒ (λ − a)2 = −b2 √ λ−a = −b2 = ±bi =⇒ λ = a ± bi. We have exactly two distinct eigenvalues when b 6= 0, which means that this form of 2×2 matrix is always diagonalizable. Also note that the eigenvalues of the matrix are the original complex number and its conjugate. We continue with the eigenvectors. For λ = a + bi, (A − λI)v = 0, −bi b x 0 = , −b −bi y 0 CHAPTER 4. 33 MATRIX FORMS AND PROPERTIES −bix + by = 0, −bx − biy = 0. Thus since we assume Similarly, for b 6= 0, y = ix and the eigenspace of a + bi λ = a − bi, (A − λI)v = 0, 0 bi b x = , 0 −b bi y bix + by = 0, −bx + biy = 0. Thus, the eigenspace of a − bi is spanned by 1 . −i So, placing the eigenvectors as columns in a matrix, we have 1 1 P = i −i and its inverse P −1 = 1 1 i . 2 1 −i is spanned by 1 . i CHAPTER 4. 34 MATRIX FORMS AND PROPERTIES So, we can use these to nd the diagonalized form of the matrix representation of any complex number: 0 1 i 1 a b 1 1 a + bi = , −b a i −i 0 a − bi 1 −i 2 where we note that the entries on the diagonal are the original complex number and its conjugate. The roots of the resulting diagonal matrix can now be easily calculated by the roots of the entries on the diagonal, and as noted earlier, these values will not be equal unless the original number is equal to its complex conjugate. These roots will be further investigated in section 4.4. 4.3.2 Example 2: For some quaternion 2×2 Matrix of a Quaternion q ∈ H, q = a + bi + cj + dk , quaternion. We nd the eigenvalues of let Q ∈ C2×2 be a matrix form of a Q: a + bi c + di Q = , −c + di a − bi c + di (a + bi) − λ det(Q − λI) = det −c + di (a − bi) − λ = (a − λ + bi)(a − λ − bi) − (c + di)(−c + di) = a2 − 2aλ + λ2 + b2 + (ba − bλ − ba + bλ) i − −c2 − d2 + (−cd + cd) i = λ2 − 2aλ + a2 + b2 + c2 + d2 . CHAPTER 4. 35 MATRIX FORMS AND PROPERTIES Thus, the characteristic equation is (λ − a)2 + b2 + c2 + d2 = 0, (λ − a)2 = −b2 − c2 − d2 , √ λ = a ± i b2 + c 2 + d 2 . √ We can choose to represent the quantity eigenvalues a + ωi, and a − ωi, b2 + c2 + d2 with ω, and we can diagonalize this matrix. eigenvectors, For giving us the two distinct λ = a + ωi, (Q − λI)v = 0, c + di x 0 bi − ωi = , 0 −c + di −bi − ωi y x (bi − ωi) + y(c + di) = 0, x(−c + di) − y (−bi − ωi) = 0. Now, to nd the CHAPTER 4. MATRIX FORMS AND PROPERTIES So (c + di)y = −(bi − ωi)x. And the resulting eigenvector: c + di v = . (−b + ω)i Similarly, for λ = a − ωi c + di x 0 bi + ωi = , 0 −c + di −bi + ωi y x (bi + ωi) + y(c + di) = 0, x(−c + di) − y (−bi + ωi) = 0. So (c + di)y = −(bi + ωi)x. 36 CHAPTER 4. 37 MATRIX FORMS AND PROPERTIES And the resulting eigenvector: c + di v = . (−b − ω)i Now the eigenvectors yield the matrix c + di c + di P = (−b + ω)i (−b − ω)i and its inverse P −1 = c + di 1 (b + ω)i . 2ωi(c + di) (−b + ω)i −(c + di) Note that for the inverse to exist, we must have (c + di) 6= 0. These are now used to nd the diagonalized form of the complex matrix representation of any quaternion, Qd ∈ C2×2 0 a + ωi P −1 QP = Qd = 0 a − ωi and we can now create nth roots as before. CHAPTER 4. 4.3.3 Example 3: We now look at the Q ∈ C4×4 38 MATRIX FORMS AND PROPERTIES 4×4 4×4 Matrix of a Quaternion matrix form of a quaternion, and see if it is diagonalizable. Let be a matrix form of a quaternion. We nd the eigenvalues of Q: b c d a −b a −d c Q = , −c d a −b −d −c b a a − λ b c d −b a − λ −d c det(Q − λI) = −c d a − λ −b −d −c b a − λ = λ2 − 2aλ + a2 + b2 + c2 + d2 λ2 − 2aλ + a2 − b2 + c2 + d2 = 0. CHAPTER 4. 39 MATRIX FORMS AND PROPERTIES Thus, (λ − a)2 + b2 + c2 + d2 = 0, (λ − a)2 = −b2 − c2 − d2 , √ λ = a ± −b2 − c2 − d2 , or (λ − a)2 − b2 + c2 + d2 = 0, (λ − a)2 = b2 − c2 − d2 , √ λ = a ± b2 − c 2 − d 2 . Here we see that when b 6= 0, we have four distinct eigenvalues, a+ √ −b2 − c2 − d2 , a − √ √ √ −b2 − c2 − d2 , a + b2 − c2 − d2 , and a − b2 − c2 − d2 , and when b = 0, the eigenvalues √ √ a + −c2 − d2 , a − −c2 − d2 each have multiplicity 2, indicating that this matrix is also diagonalizable. We can use the representations ω= √ b2 + c2 + d2 , ϕ = to write the eigenvalues in a more compact form: For −b2 + c2 + d2 , and γ = (c2 +d2 ) a + ωi, a − ωi, a + ϕi, a − ϕi. λ = a − ωi, √ −bd−cωi − γ − −bc+dωi γ v = . 1 0 CHAPTER 4. For 40 MATRIX FORMS AND PROPERTIES λ = a + ωi, −bd+cωi − γ − −bc−dωi γ v = . 1 0 For λ = a − ϕi, c( −bd+cϕi − −γ ϕi(bc−dϕi) b(−γ) ) d −bd+cϕi ϕi(bc−dϕi) − + b(−γ) −γ v = . ϕi −b 1 For λ = a + ϕi, c( −bd−cϕi + −γ ϕi(bc+dϕi) b(−γ) d −bd−cϕi − − −γ v = ϕi b 1 We then construct the matrix to generate a diagonal matrix. P and its inverse ) ϕi(bc+dϕi) b(−γ) . P −1 using the eigenvectors, and use them CHAPTER 4. P −1 − 2cϕi − 2dωiϕi γ b(γ) 2dϕi − 2cωiϕi γ b(γ) 4d2 ωiϕi 4c2 ωiϕi − − (γ)2 (γ)2 − 4c2 ωiϕi − 4d2 ωiϕi (γ)2 (γ)2 2cϕi 2dωiϕi − − 2dϕi − 2cωiϕi γ b(γ) γ b(γ) 2 ωiϕi 4d2 ωiϕi 4c2 ωiϕi 4d2 ωiϕi − − − − 4c(γ) 2 (γ)2 (γ)2 (γ)2 = − “ 2 2dωi 2 ” − “ 2 2cωi 2 ” (γ) − 4c ωiϕi − 4d ωiϕi (γ) − 4c ωiϕi − 4d ωiϕi (γ)2 (γ)2 (γ)2 (γ)2 “ 2dωi 2cωi “ ” ” 2 ωiϕi 2 − 4d ωiϕi (γ)2 (γ)2 (γ) − 4c 2 ωiϕi 2 − 4d ωiϕi (γ)2 (γ)2 (γ) − 4c 4 0 0 2 2bc2 ωi + 2bd 2ωi (γ)2 (γ) 2 2 − 4c ωiϕi − 4d ωiϕi (γ)2 (γ)2 2 2 − 2bc 2ωi − 2bd 2ωi (γ) (γ) 2 2 − 4c ωiϕi − 4d ωiϕi (γ)2 (γ)2 2 2 4 2c ϕi 4c d ϕi 2d ϕi − b(−γ)(γ) − b(−γ)(γ) − b(−γ)(γ) 2 2 ωiϕi − 4d ωiϕi (γ)2 (γ)2 2c4 ϕi 4c2 d2 ϕi 2d4 ϕi + b(−γ)(γ) + b(−γ)(γ) b(−γ)(γ) 2 2 − 4c ωiϕi − 4d ωiϕi (γ)2 (γ)2 2 2 − 2c ωiϕi − 2d ωiϕi (γ)2 (γ)2 2 2 − 4c ωiϕi − 4d ωiϕi (γ)2 (γ)2 2 2 − 2c ωiϕi − 2d ωiϕi (γ)2 (γ)2 2 2 − 4c ωiϕi − 4d ωiϕi (γ)2 (γ)2 − 4c . MATRIX FORMS AND PROPERTIES ϕi(bc−dϕi) ϕi(bc+dϕi) c( −bd+cϕi − b(−γ) ) c( −bd−cϕi + b(−γ) ) −γ −γ −bd−cωi −bd+cωi − γ d d − γ ϕi(bc−dϕi) ϕi(bc+dϕi) −bd−cϕi − −bc+dωi − −bc−dωi − −bd+cϕi + − − γ γ −γ b(−γ) −γ b(−γ) P = . ϕi ϕi 1 1 −b b 0 0 1 1 41 CHAPTER 4. 42 MATRIX FORMS AND PROPERTIES We write out the diagonal matrix Qd ∈ C4×4 0 0 0 a − ωi 0 a + ωi 0 0 P −1 QP = Qd = , 0 0 a − ϕi 0 0 0 0 a + ϕi and again we can nd roots of the diagonalized matrix. 4.4 Roots of Complex Numbers vs. Roots of Diagonal Matrices When de Moivre's formula is used to nd a root of a complex number the root 1 zn Considering z, the matrix form of can easily be seen to be one root of the matrix form of the complex number Z as the matrix form of the diagonalization of the matrix z = eiθ (using polar form for readability), z. Zd = P −1 ZP 1 1 Z , and z n or Zdn as the principal nth root of each, we have iθ 0 e Zd = , −iθ 0 e 1 in θ 1 e Zdn = 0 0 . 1 e−i n θ We know that these are not the only possible roots of the matrix each have roots of z n distinct roots, any of the n2 and z̄ on the diagonal of the Zd . In fact, since z and z̄ matrices formed by all possible permutations of the 2 × 2 matrix form a valid matrix root of Zd . Further, it is not obvious that the roots of a matrix found from diagonalization are the only roots of CHAPTER 4. MATRIX FORMS AND PROPERTIES the matrix. As an example, in the 2×2 real matrix space, we can construct an example of a root of a diagonal matrix which is not itself a diagonal matrix. 2 1 x 1 0 = . 0 −1 0 1 We can see that for any real matrix of this form 2 2 a b a ab + bd = , 2 0 d 0 d when a = −d, the square of the matrix is 2 a 0 In general, we can see for any 2×2 43 0 . d2 matrix whose square is diagonal, 2 2 a b a + bc ab + bd n 0 = , = 0 m ac + cd d2 + bc c d CHAPTER 4. 44 MATRIX FORMS AND PROPERTIES giving the set of equations a2 + bc = n, ab + bd = 0 ⇒ a = −d or b = 0, ac + cd = 0 ⇒ a = −d or c = 0, d2 + bc = m. So, b = 0, c 6= 0 ⇒ a = −d, a2 = d2 = n = m, c = 0, b 6= 0 ⇒ a = −d, a2 = d2 = n = m, b 6= 0, c 6= 0 ⇒ a = −d, a2 + bc = d2 + bc = n = m, b = c = 0 ⇒ a2 = n, d2 = m ∀a, d. And conversely, n = m ⇒ a2 + bc = d2 + bc ⇒ a2 = d 2 ⇒ a = ±d, CHAPTER 4. 45 MATRIX FORMS AND PROPERTIES so, a = d 6= 0 ⇒ b = c = 0 since ab + bd = 0 and ac + cd = 0, a = −d 6= 0 ⇒ ab + bd = 0, ac + cd = 0, a2 + bc = d2 + bc = n = m ∀b, c, a = d = 0 ⇒ ab + bd = 0, ac + cd = 0, bc = n = m ∀b, c. Thus, a diagonal 2×2 matrix A has non-diagonal square roots if and only if it is a scalar multiple of the identity matrix, ie. 4.5 A = nI . Jordan Canonical Form A matrix in Jordan canonical form, or Jordan normal form, is a block matrix of the form J1 J2 J = .. . Jm consisting of one or more Jordan blocks, where all entries outside of the blocks are zeros. Each Jordan block is a matrix having zeroes everywhere except the diagonal and the superdiagonal, CHAPTER 4. MATRIX FORMS AND PROPERTIES 46 and the elements on the superdiagonal all consist of 1. 0 ··· 0 0 λm 1 0 λ 1 0 0 m . .. . . 0 0 λm . . Jm = .. .. . . 1 0 0 0 0 λm 1 0 0 0 · · · 0 λm Each λm represents an eigenvalue of the Jordan matrix, and each may or may not have dierent values from other blocks. A 1×1 matrix can also be considered a Jordan block even though it lacks a superdiagonal [48]. For any square matrix A there is a unique Jordan matrix decomposition A = SJS −1 , where S −1 is the matrix inverse of S, Given any complex square matrix by nding a Jordan basis bi,j and J is a matrix in Jordan canonical form [49]. A, the matrix can be written in Jordan canonical form for each Jordan block, where each Jordan basis satises [39, 40] Abi,1 = λi bi,1 and Abi,j = λi bi,j + bi,j−1 . Much like diagonalized matrices, it is easy to relate a power of a matrix to a power of its CHAPTER 4. 47 MATRIX FORMS AND PROPERTIES Jordan canonical form, A = SJS −1 , An = SJS −1 n SJS −1 SJS −1 · · · SJS −1 = SJ S −1 S J S −1 S · · · S −1 S JS −1 = = SJ n S −1 , and a power of a Jordan matrix can also be easily calculated using the power of its Jordan blocks. n J1 J2n Jn = .. . n Jm . Also as with a diagonal matrix, any roots of a a Jordan matrix can be calculated using the roots of its Jordan blocks. Given a Jordan Matrix terms of the Jordan blocks from J (Ji0 )n = Ji . where (J 0 )n = J , we can write J0 in as: 0 J1 J20 J0 = where J0 .. . 0 Jm , This would potentially allow us to nd an nth provided we can nd roots of its corresponding Jordan blocks. root of any square matrix, The next sections explore CHAPTER 4. 48 MATRIX FORMS AND PROPERTIES cases where the roots may not exist, and a criteria for knowing when the roots do exist. 4.6 Nilpotent Matrices A nilpotent matrix is any matrix that when taken to some power For an m×m matrix n becomes the zero matrix. A, An = [0] n and we can call the minimum value of nilpotent matrix for which An is the zero matrix the index of the A. It can be shown that the eigenvalues of a nilpotent matrix are always 0. When nding an eigenvalue λ, A~x = λ~x, An~x = λn~x, and since An is the zero matrix, An~x = ~0 ⇒ λn~x = ~0. Because this will be true for any eigenvector ~x, we must have λ = 0. The most obvious example of a nilpotent matrix is any strictly triangular matrix, with all zeros along the diagonal. These are clearly not the only type of matrices that are nillpotent. In fact it is not necessary that any entries in the matrix be zeroes, as we can construct CHAPTER 4. 49 MATRIX FORMS AND PROPERTIES examples such as 1 1 1 1 1 1 1 1 1 2 2 2 2 , , 1 1 1 3 −1 −1 3 3 3 −2 −2 −2 −6 −6 −6 −6 which are all nilpotent with index 2. We see from the examples that it should be possible to construct a nilpotent matrix of index 2 for any m. m×m It would also be useful to show the maximum nilpotent index that could exist for a given size m. Let A be a nilpotent m×m then not the zero matrix, so we can say there exists a vector ~x matrix of index such that Ak−1~x k . Ak−1 is is non-zero. It can then be seen that the k non-zero vectors ~x, A~x, A2~x, . . . , Ak−1~x form a linearly independent set in values c0 , c1 , . . . , ck−1 Rm , which would imply k ≤ m. To wit, if we have scalar which satisfy c0~x + c1 A~x + · · · + ck−2 Ak−2~x + ck−1 Ak−1~x = ~0, CHAPTER 4. 50 MATRIX FORMS AND PROPERTIES then there will also be k−1 equations c0 A~x + c1 A2~x + · · · + ck−2 Ak−1~x + ck−1 Ak ~x = ~0, c0 A2~x + c1 A3~x + · · · + ck−2 Ak ~x + ck−1 Ak+1~x = ~0, . . . c0 Ak−2~x + c1 Ak−1~x + · · · + ck−2 A2k−4~x + ck−1 A2k−3~x = ~0, c0 Ak−1~x + c1 Ak ~x + · · · + ck−2 A2k−3~x + ck−1 A2k−2~x = ~0. Since Ak and all higher powers of c0 Ak−1~x = ~0, implies c1 = 0. implying that c0 = 0 . A are the zero matrix, the last equation above says Then, the previous equation c0 Ak−2~x + c1 Ak−1~x = ~0 Continuing in this manner through each equation, we nd that c0 , c1 , . . . , ck−1 are all zero, which establishes linear independence of the vectors above [30]. Thus, we have shown that for a nilpotent m×m matrix, the index k ≤ m. We can more easily see some properties of a nilpotent matrix by considering its Jordan canonical form. Since the Jordan canonical form of a nilpotent matrix always has zeros on the main diagonal, and ones or zeros on the superdiagonal, the only non-trivial Jordan canonical form of a 2×2 nilpotent matrix is 0 1 A= . 0 0 CHAPTER 4. For nilpotent 51 MATRIX FORMS AND PROPERTIES 3×3 2 matrices, we can have index 2 (B2 = 0) 3 or index 3 (B3 = 0). 0 1 0 0 0 0 , B2 = 0 0 1 . B3 = 0 0 1 0 0 0 0 0 0 And for nilpotent 4 4 ( C4 4×4 2 matrices, we can have index 2 (C2 = 0), 3 index 3 (C3 0 0 0 0 0 0 1 0 , C2 = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 . 0 0 1 0 0 0 = 0), or index = 0). 0 0 C4 = 0 0 1 0 0 0 0 0 1 0 , C3 = 0 0 0 1 0 0 0 0 By this representation, it seems clear that the index of the nilpotent is related to the number of Jordan blocks, and hence to the dimension of the eigenspace of the 4×4 (x ~1 = [1, 0, 0, 0]), C3 examples above, we note that has (x ~1 C4 0, ie. the null space. From has a nullspace of dimension = [1, 0, 0, 0], x~2 = [0, 1, 0, 0])), and C2 1, has (x ~1 generated by = [1, 0, 0, 0], x~2 = [0, 1, 0, 0]), x~3 = [0, 0, 1, 0])). We would also like to take a quick look at whether we can calculate roots of a nilpotent matrix. We start with the Jordan Canonical form of a 0 1 A= . 0 0 2×2 nilpotent matrix CHAPTER 4. MATRIX FORMS AND PROPERTIES We want to nd some matrix A0 where 52 A02 = A 2 2 a b a + bc ab + bd 0 1 A02 = = = . c d ca + dc cb + d2 0 0 So, we have the series of equations a2 + bc = 0, ab + bd = 1 ⇒ abc = c − bcd, ca + dc = 0 ⇒ abc = −bcd, cb + d2 = 0. Thus, c = 0, a2 = 0, a = 0, d2 = 0, d = 0, Contradiction, since ab + bd = 0. Thus, we see there is no solution to this system of four equations with four unknowns, and thus there are no square roots of a 2×2 nilpotent matrix. CHAPTER 4. 53 MATRIX FORMS AND PROPERTIES We next look at the Jordan Canonical forms of a 0 1 0 A= 0 0 1 0 0 0 where we want to nd some matrix A0 3×3 nilpotent matrix 0 0 0 or A = 0 0 1 , 0 0 0 where A02 = A or A03 = A. Utilizing the Mathematica software to solve the resulting systems of nine equations with nine unknowns also results in no solution, and thus there are no square roots or cube roots of a Looking further at the 4×4 1 0 0 1 0 0 0 0 0 0 , 1 0 AAAA = (AA)(AA) = [0], we can consider the matrix B = AA, where B 2 = [0]. the matrix B. nilpotent matrix. nilpotent matrices, we notice that for the matrix 0 0 A= 0 0 since 3×3 B is nilpotent, and the matrix A Thus, is considered to be a square root of the matrix This can be generalized based on the associative property of matrix multiplication for larger matrices as well. Theorem 4.6.1. Let A be a nilpotent m × m matrix of index m, and let p, n ∈ Z, where p, n > 1, such that pn = m. Then, there exists some nilpotent m × m matrix A0 of index p where An = A0 . Thus, we can say that A is an nth root of A0 . CHAPTER 4. 4.7 54 MATRIX FORMS AND PROPERTIES Existence of Roots We have already noted that there seems to be a relation between the dimension of the nullspace of a nilpotent matrix and its index. In fact, relations between dimensions of nullspaces can be used to determine the existence of of a matrix A, nth roots. We utilize the ascent sequence which is the sequence di = dim Null Ai − dim Null Ai−1 ; i = 1, 2, . . . . Psarrakos [36] shows the following: Theorem 4.7.1 (Psarrakos). Given a complex m × m matrix A ∈ Cm×m , A has an nth root if and only if for every integer ν ≥ 0, the ascent sequence of A has no more than one element between nν and n(ν + 1). and a corollary, Corollary 4.7.2 (Psarrakos). Let d1 , d2 , d3 , . . . be the ascent sequence of a sin- gular complex matrix A. (i) If d2 = 0, then for every integer n > 1, A has an nth root. (ii) If d2 > 0, then for every integer n > d1 , A has no nth roots. Now that we see a method for knowing if nth roots exist, we look for methods which might provide us with multiple roots, should they exist. Chapter 5 Hypercomplex Numbers Knowing that it is possible to nd multiple roots in the complex numbers and in the quaternions, it makes sense to next look for other number systems for which multiple roots can be found where we might be able to nd a ring isomorphism with the matrices. The area of Hypercomplex Numbers oers many number systems in which we can begin looking. A basic denition for a hypercomplex number is a number having properties which depart from those of the real or complex numbers [45]. More precisely, a hypercomplex number system in a nite dimensional algebra over the reals which is unital and distributive, but not necessarily associative, with elements generated by real number coecients for some basis 1, i0 , i1 , . . . , in , where each basis is conventionally normalized such that −1, 0, 1 [28]. 5.1 Multicomplex Numbers The multicomplex numbers are a sequence of number systems C0 is taken to be the real number system, and for each n > 0, in (a0 , a1 , . . . , an ) is some square root of −1. 55 Cn i2k ∈ dened inductively, where Cn+1 = {z = x + yin+1 : x, y ∈ Cn }, where A further requirement is that multiplication CHAPTER 5. 56 HYPERCOMPLEX NUMBERS on the imaginary units must be commutative, such that for any this denition, Cn C1 is the complex number system, is a multicomplex number system of order 5.1.1 n C2 n 6= m, in im = im in . Under is the bicomplex number system, and [35]. Bicomplex Numbers The bicomplex numbers were developed by Corrado Segre [42], using two commuting square roots of −1, h2 = i2 = −1, whose product would then have the square (hi)2 = (ih)2 = 1. The algebra also contains idempotent values g= 5.1.2 1 + hi 2 g∗ = 1 − hi . 2 Tessarines The tessarines were rst described by James Cockle [9, 10, 11, 12, 13, 14] as an algebra similar to the quaternions, but which have the form t = w + xα + yβ + zγ, where w, x, y, z ∈ R and α2 = −1, β 2 = +1, αβ = βα = γ, γ 2 = −1 CHAPTER 5. A tessarine t = w + xi + yj + zk i = α, j = β, k = γ , α, β, γ numbers, 57 HYPERCOMPLEX NUMBERS (using more modern notation for complex numbers: as listed above) can also be written in terms of two complex t = (w + xi) + (y + zi)j , since ij = k , so we can write t = a + bj , a, b ∈ C. We can use this representation to show a mapping into the f : T 7→ C2×2 2×2 complex matrices a b t = a + bj 7→ b a and c d s = c + dj → 7 , d c so we see a + c b + d a b c d t + s = (a + c) + (b + d)j 7→ + = d c b a b+d a+c and ac + bd bc + ad a b c d ts = (ac + bd) + (ad + bc)j 7→ = . bc + ad ac + bd b a d c We can easily see that for any complex 2×2 matrix of the appropriate form, a b , b a there will exist some all a, b ∈ C, t ∈ T , t = a + bj . So, the inverse mapping f 0 : C2×2 7→ T and we can see that this mapping is an isomorphism. is dened for CHAPTER 5. 5.2 58 HYPERCOMPLEX NUMBERS Split-Complex Numbers We can examine the non-real root of +1 seen above in the Tessarines and Bicomplex Numbers apart from the usual complex unit by examining the quantity z = a + bj , j 2 = +1, rst examined by Cockle [14], they are often referred to currently as a split complex numbers [38] or Hyperbolic numbers [43]. Much as seen earlier in the bicomplex numbers, the split complex numbers will have idempotent values z= 1+j , 2 z∗ = 1−j . 2 The split complex numbers also contain zero divisors. product of any two split complex numbers We can nd them by setting the z1 = a1 + b1 j , z2 = a2 + b2 j z1 z2 = (a1 + b1 j)(a2 + b2 j) = a1 a2 + a1 b 2 j + a2 b 1 j + b 1 b 2 = (a1 a2 + b1 b2 ) + (a1 b2 + a2 b1 )j = 0. Setting the real and imaginary parts both to zero, we have: a1 a2 + b1 b2 = 0, a1 b2 + a2 b1 = 0. to zero: CHAPTER 5. 59 HYPERCOMPLEX NUMBERS So a1 = −b1 b2 a2 and a1 = −a2 b1 b2 if a2 , b2 6= 0. Thus −b1 b2 −a2 b1 = , a2 b2 −b1 b22 = −a22 b1 , b22 = a22 . So we see, the split complex number z = a + bj where a = ±b is a zero divisor for any other split complex number. In the 2 × 2 matrix space, there are a few dierent matrices that can be used to represent a non-real root of +1. Looking at diagonal matrices, we can see that a matrix that has 1 and -1 on the diagonal will have a square that contains all 1's on the diagonal, and so is the image of the real +1. 2 2 1 0 1 0 −1 0 . = = 0 1 0 −1 0 1 Another matrix which squares to a diagonal matrix of ones is an anti-diagonal matrix of ones: 2 0 1 1 0 = . 1 0 0 1 Since this representation is orthogonal to the representation of a real number, we can use CHAPTER 5. 60 HYPERCOMPLEX NUMBERS this in mapping from the split complex numbers into the real 2×2 matrix space. If a, b ∈ R a b z = a + bj 7→ . b a 5.3 Dual Numbers As with the complex numbers (z a + bj, j 2 = +1), In the 2×2 = a + bi, i2 = −1) it is possible to dene a number and the split-complex numbers (z z = a + b where is nilpotent, matrix space, there are several matrices which could represent 2 = 0 = [6]. , namely any nilpotent matrix. The easiest example is any strictly triangular matrix, where all values on the diagonal are 0. 2 2 0 1 0 0 0 0 = = . 0 0 1 0 0 0 Then the mapping a b z = a + b 7→ 0 a is an isomorphism. 5.4 Split-Quaternions The split-quaternions, which were rst described as the coquaternions by James Cockle [10], are numbers of the form q = w + xi + yj + zk where ij = k = −ji, jk = −i = −kj, ki = j = −ik, ijk = 1, i2 = −1, j 2 = +1, k 2 = +1. CHAPTER 5. A split quaternion has conjugate w 2 + x2 − y 2 − z 2 . q̄ = w − xi − yj − zk , A split quaternion q q ∗ q −1 = q −1 ∗ q = 1, and multiplicative modulus q̄ , q q̄ which exists when the multiplicative modulus We can also write a split quaternion in terms of two complex numbers b = y + zi, so q = a + bj . q q̄ = will have a multiplicative inverse q −1 = such that 61 HYPERCOMPLEX NUMBERS q q̄ 6= 0. a = w + xi, When written this way, we can dene a mapping to the 2×2 complex matrix space a b q = a + bj 7→ , b̄ ā where a, b ∈ C, and ā, b̄ are the complex conjugates of a and b. Written out explicitly, this gives w + xi y + zi q = (w + xi) + (y + zi)j 7→ . y − zi w − xi Recall now, however, the real 2×2 matrix representations of each non-real component, where we have one real matrix which can represent a root of non-real root of +1. −1, and two that can can represent a We then consider the following mappings 0 1 i 7→ , −1 0 0 1 j 7→ , 1 0 CHAPTER 5. 62 HYPERCOMPLEX NUMBERS 1 0 k 7→ , 0 −1 and see that 0 1 1 0 0 −1 = , 1 0 0 −1 1 0 noting that this corresponds to jk = −i as seen in the coquaternions. We further note that these matrices together with the identity matrix form a basis for the Then for any arbitrary real 2×2 matrix, with a, b, c, d ∈ R, 2×2 real matrices. we have the mapping (b + c) (a − d) (a + d) (b − c) a b + i+ j+ k 7 q= → 2 2 2 2 c d which forms a ring isomorphism [29]. 5.4.1 Roots of Split-Quaternions The split quaternions can also be represented in a polar form [33] which can be used to exhibit a version of the de Moivre formula and subsequently used to nd roots[32]. The de Moivre formula for split quaternions is dened piecewise depending on the properties of a split quaternion N (q) = q = w + xi + yj + zk . p |w2 + x2 − y 2 − z 2 |. the split quaternion q is called The split quaternion q will have norm Considering the multiplicative modulus spacelike if q q̄ < 0, timelike if q q̄ > 0, N (q) dened as q q̄ = w2 +x2 −y 2 −z 2 , and lightlike if q q̄ = 0, with these labels stemming from how these quantities are used in physics. Note that under these labels, spacelike and timelike split quaternions will have multiplicative inverses, but lightlike quaternions will have no inverses. Further characterization of a split quaternion is taken by splitting it into its scalar part Sq = w, and its vector part V~q = xi + yj + zk , where the vector part is identied with the CHAPTER 5. 63 HYPERCOMPLEX NUMBERS Minkowski 3-space. The Lorentzian inner product of the vector part with itself, −x2 + y 2 + z 2 , is called spacelike if hV~q , V~q iL > 0, timelike if hV~q , V~q iL < 0, hV~q , V~q iL = or lightlike if hV~q , V~q iL = 0. It is then possible to write spacelike or timelike split quaternions in polar form, and extend de Moivre's formula for the split-quaternions as follows: We can write a timelike split quaternion with a spacelike vector part in the form q = N (q) (cosh θ + ~ε sinh θ) , where |w| cosh θ = , sinh θ = N (q) and ~ε 2 = ~ε ∗ ~ε = 1. p −x2 + y 2 + z 2 xi + yj + zk , ~ε = p N (q) −x2 + y 2 + z 2 De Moivre's formula is then written as q n = (N (q))n (cosh nθ + ~ε sinh nθ) . The equation wn = q will have only one root p θ θ n , w = N (q) cosh + ~ε sinh n n because the hyperbolic cosine and the hyperbolic sine are not periodic. For a timelike split quaternion with a timelike vector part, we write q = N (q) (cos θ + ~ε sin θ) , where w cos θ = , sin θ = N (q) p x2 − y 2 − z 2 xi + yj + zk , ~ε = p N (q) x2 − y 2 − z 2 CHAPTER 5. and 64 HYPERCOMPLEX NUMBERS ~ε 2 = ~ε ∗ ~ε = −1. De Moivre's formula is then written as q n = (N (q))n (cos nθ + ~ε sin nθ) . The equation wn = q will have n roots p θ + 2mπ θ + 2mπ n wm = N (q) cos + ~ε sin , n n where m = 0, 1, 2, . . . , n − 1. For a timelike split quaternion with a lightlike vector part, we write q = 1 + ~ε, where ~ε is a null vector. We can still write a form of de Moivre's formula as q n = 1 + n~ε. There will be innitely many roots of the equation wn = q . For a spacelike split quaternion, we write q = N (q) (sinh θ + ~ε cosh θ) , where w , cosh θ = sinh θ = N (q) and ~ε p −x2 + y 2 + z 2 xi + yj + zk , ~ε = p N (q) −x2 + y 2 + z 2 is a spacelike vector. De Moivre's formula is then written as q n = (N (q))n (sinh nθ + ~ε cosh nθ) , n is odd; CHAPTER 5. 65 HYPERCOMPLEX NUMBERS q n = (N (q))n (cosh nθ + ~ε sinh nθ) , The equation wn = q will have no roots if n n is even. is an even number, and if n is odd it will have one root p θ θ n w = N (q) sinh + ~ε cosh . n n So, for a split quaternion q = w + xi + yj + zk , w 2 + x2 − y 2 − z 2 < 0 multiple roots, when w 2 + x2 − y 2 − z 2 > 0 when we can nd a single nth root if n we can nd is odd, and otherwise we will not be able to nd roots. Re-stated in terms of the mapping to a real matrix, 2×2 a, b, c, d ∈ R (b + c) (a − d) (a + d) (b − c) a b + i+ j+ k, 7 q= → 2 2 2 2 c d the quantity w 2 + x2 − y 2 − z 2 = a+d 2 maps to 2 + root if 5.5 n 2 − b+c 2 2 − a−d 2 2 (a2 + 2ad + d2 ) + (b2 − 2bc + c2 ) − (b2 + 2bc + c2 ) − (a2 − 2ad + d2 ) 4 = So, when b−c 2 ad − bc > 0 4ad − 4bc = ad − bc. 4 we can nd multiple roots, when ad − bc < 0 we can nd a single nth is odd, and otherwise we will not be able to nd roots using this method. Biquaternions The biquaternions, or complexied quaternions, are numbers of the form where the elements i, j, k multiply as in the quaternions, and q = w+xi+yj +zk , w, x, y, z ∈ C are normal CHAPTER 5. 66 HYPERCOMPLEX NUMBERS complex numbers [15, 22, 23]. The normal complex unit is denoted h, and we have hi = ih, hj = jh, hk = kh since h is a scalar with respect to the quaternion elements in this context. A biquaternion has two forms of conjugate [41]. First is the complex conjugate, i.e. the complex conjugate of the complex components of the quaternion, q ∗ = w∗ + x∗ i + y ∗ j + z ∗ k, and for two biquaternions, (pq)∗ = p∗ q ∗ . Next is the quaternion conjugate q̄ = w − xi − yj − zk, where the conjugate rule for quaternions q̄ p̄ = pq and the generalization to more than two quaternions also applies, and a biquaternion commutes with its quaternion conjugate q q̄ = q̄q. The quaternion conjugate is used in nding the inverse of a biquaternion q −1 = q̄ q q̄ CHAPTER 5. 67 HYPERCOMPLEX NUMBERS which exists when the modulus q q̄ 6= 0, and is also seen when nding a polar form of a biquaternion which is used in the next section for nding roots. We can create a mapping from the biquaternions to the complex 2×2 matrices by rst mapping the elements h 0 i 7→ , 0 −h 0 1 j 7→ , −1 0 0 h k→ 7 , h 0 and see that h 0 0 1 0 h , = h 0 0 −h −1 0 which corresponds to ij = k in quaternion multiplication. We can then dene the mapping q = w + xi + yj + zk where for any complex 2×2 matrix 7→ a b w + xh y + zh A= , = −y + zh w − xh c d A ∈ C2×2 , there are values for (a + d) , 2 −h(a − d) x = , 2 (b − c) y = , 2 −h(b + c) z = , 2 w = w, x, y, z ∈ C, CHAPTER 5. 68 HYPERCOMPLEX NUMBERS such that the matrix ring is isomorphic to the biquaternion ring [17]. 5.5.1 Roots of Biquaternions There are two possible polar forms of a biquaternion, as the biquaternions have two possibilities for roots of -1: the complex root h, or a biquaternion root of -1. A biquaternion root of -1 is found by taking any pure biquaternion, ie. a biquaternion with its scalar part with modulus √ q q̄ 6= 0, w = 0, and dividing it by its modulus. q √ q q̄ 2 q2 q q̄ q = q̄ 0 + xi + yj + zk = 0 − xi − yj − zk = = −1. The rst polar form of a biquaternion is known as the Complex polar form, written as q = Q exp(hΨ) = Q(cos Ψ + h sin Ψ), where Q and Ψ are both quaternions. If we write the original biquaternion q = w+xi+yj+zk in its expanded form q = (wr + wi h) + (xr + xi h)i + (yr + yi h)j + (zr + zi h)k, where wr , wi , xr , xi , yr , yi , zr , zi ∈ R, we see that the biquaternion can be rewritten in the CHAPTER 5. 69 HYPERCOMPLEX NUMBERS form of two quaternions q = (wr + xr i + yr j + zr k) + (wi + xi i + yi j + zi k)h = qr + qi h, where qr , qi ∈ H. We can then see the complex modulus cos Ψ = qr , Q sin Ψ = Q= p qr2 + qi2 , and qi . Q We note that for this form, the modulus and exponential do not commute, meaning it is not evident that a form of de Moivre's formula is possible. The second form is known as the Hamilton polar form, written as q = R exp(ξΘ) = R(cos Θ + ξ sin Θ), where R = p w 2 + x2 + y 2 + z 2 , w , R p x2 + y 2 + z 2 sin Θ = , R xi + yj + zk ξ = p , x2 + y 2 + z 2 cos Θ = CHAPTER 5. 70 HYPERCOMPLEX NUMBERS and we see that ξ is a biquaternion root of −1. and also note that the form exists only when form of the complex modulus R We note that both R 6= 0. R and Θ are complex, We can also utilize the exponential to write the biquaternion as q = r exp(hφ) exp(ξΘ), where R = r exp(hφ), and the exponentials commute since h commutes with We can now take this form with two commuting exponentials to an nth ξ [41]. power, q n = (r exp(hφ) exp(ξΘ))n = (r exp(hφ) exp(ξΘ))(r exp(hφ) exp(ξΘ)) . . . (r exp(hφ) exp(ξΘ)) {z } | n = (r)n (exp(hφ))n (exp(ξΘ)n , since h and Θ commute with ξ = rn exp(hnφ) exp(ξnΘ) = rn (cos(nφ) + h sin(nφ))(cos(nΘ) + ξ sin(nΘ)), giving us an implementation of de Moivre's formula that can be used for biquaternions. Similarly, for nding a root of a biquaternion, we will have q where k1 , k2 0 φ + 2πk1 Θ + 2πhk2 = r exp h exp ξ n n 1 φ + 2πk1 φ + 2πk1 = r n cos + h sin n n Θ + 2πhk2 Θ + 2πhk2 cos + ξ sin n n 1 n are integers, and (q 0 )n = q . We can see that these are likely not the only possible CHAPTER 5. 71 HYPERCOMPLEX NUMBERS roots of the biquaternion q, since the Hamiltonian polar form is not the only possible polar form of the biquaternion. So, we know that we can nd roots of a biquaternion modulus p R = w2 + x2 + y 2 + z 2 6= 0, q = w + xi + yj + zk and in terms of a complex 2×2 matrix (b − c) −h(b + c) (a + d) −h(a − d) a b A= + i+ j+ k, 7 q= → 2 2 2 2 c d the modulus becomes s r = and we can a+d 2 2 + −h(a − d) 2 2 + b−c 2 2 + −h(b + c) 2 2 (a2 + 2ad + d2 ) − (a2 − 2ad + d2 ) + (b2 − 2bc + c2 ) − (b2 + 2bc + c2 ) 4 r 4ad − 4bc √ = = ad − bc, 4 √ nd roots if ad − bc 6= 0. when its Chapter 6 Conclusion There are multiple iterative methods of nding some nth root of a real or complex m×m matrix, though such methods may fail to nd a root even if such a root exists, and will only nd a single root. If an have m×m nth roots, for any integer n. For non-diagonalizable matrices, as seen specically in the nilpotent matrices, there may be no n. matrix is diagonalizable, it has been shown to always nth roots, or roots may exist only for specic values of Utilizing the isomorphism between the 2×2 real matrices and the split-quaternions, we have a form of de Moivre's formula which can be used to nd roots of a Similarly we have an isomorphism from the 2×2 2×2 complex matrices into the biquaternions, for which we can use de Moivre's formula to nd some of the possible roots of a matrix. real matrix. 2×2 complex The octonions have a form of de Moivre's formula [7], but have no isomorphism with a matrix space as they are not associative. We can use these results to point towards a potential method of nding roots for larger matrix spaces, where we attempt to nd some isomorphism into a hypercomplex space for which a form of de Moivre's formula can be found. 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