Matrix Theory over the Complex Quaternion Algebra Yongge Tian Department of Mathematics and Statistics Queen’s University Kingston, Ontario, Canada K7L 3N6 email:[email protected] Abstract. We present in this paper some fundamental tools for developing matrix analysis over the complex quaternion algebra. As applications, we consider generalized inverses, eigenvalues and eigenvec- tors, similarity, determinants of complex quaternion matrices, and so on. AMS Mathematics Subject Classification: 15A06; 15A24; 15A33 Key words: complex quaternion; matrix representation; universal similarity factorization; generalized inverse; eigenvalues and eigenvectors; determinant. 1. Introduction The complex quaternion algebra (biquaternion algebra) Q is well known as a four dimensional vector space over the complex number field C with its basis 1, e1, e2, e3 satisfying the multipli- cation laws 2 2 2 e1 = e2 = e3 = −1, e1e2e3 = −1. (1.1) e1e2 = −e2e1 = e3, e2e3 = −e3e2 = e1, e3e1 = −e3e3 = e2, (1.2) and 1 acting as unity element. In that case, any element in Q can be written as a = a0 + a1e1 + a2e2 + a3e3, (1.3) arXiv:math/0004005v1 [math.RA] 1 Apr 2000 where a0—a3 ∈ C. According to this definition, real numbers, complex numbers, and real quaternions all can be regarded as the special cases of complex quaternions. A well-known fun- damental fact on the complex quaternion algebra Q (see, e. g., [4, 6, 7]) is that it is algebraically isomorphic to the 2 × 2 total matrix algebra C2×2 through the bijective map ψ : Q −→ C2×2 satisfying 1 0 i 0 0 −1 0 −i ψ(1) = , ψ(e )= , ψ(e )= , ψ(e )= . 0 1 1 0 −i 2 1 0 3 −i 0 These four matrices are well-known as Pauli matrices. Based on this map, every element a = a0 + a1e1 + a2e2 + a3e3 ∈ Q has a faithful complex matrix representation as follows a + a i −( a + a i ) ψ(a) := 0 1 2 3 ∈ C2×2, (1.4) a2 − a3i a0 − a1i 1 In this article, we shall reveal a deeper relationship between a and ψ(a), which can simply be stated that there is an independent invertible matrix Q of size 2 over Q such that all a ∈ Q satisfy the following universal similarity factorization equality Q−1diag( a, a )Q = ψ(a), where Q has no relation with the expression of a. Moreover we also extend this equality to all m×n matrices over Q. On the basis of these results, we shall consider several basic problems re- lated to complex quaternion matrices, such as, generalized inverse, eigenvalues and eigenvectors, similarity, and determinant of complex quaternion matrices. Some known terminology on complex quaternions are listed below (see, e.g., [6]). For a = a0 + a1e1 + a2e2 + a3e3 ∈ Q, the dual quaternion of a is a = a0 − a1e1 − a2e2 − a3e3; (1.5) the complex conjugate of a is ∗ a = a0 + a1e1 + a2e2 + a3e3; (1.6) the Hermitian conjugate of a is † ∗ a = (a) = a0 − a1e1 − a2e2 − a3e3; (1.7) the weak norm of a is 2 2 2 2 n(a)= a0 + a1 + a2 + a3. (1.8) A quaternion a ∈ Q is said to be real if a∗ = a, to be pure imaginary if a∗ = −a, to be scalar if a = a, to be Hermitian if a† = a. m×n n×m For any A = (ast) ∈ Q , the dual of A is A = (ats) ∈ Q ; the Hermitian conjugate of † † n×m A is A = (ats) ∈ Q . A square matrix A is said to be self-dual if A = A, it is Hermitian if A† = A, it is unitary if AA† = A†A = I, the indentity matrix, it is invertible if there is a matrix B over Q such that AB = BA = I. Some known basic properties on complex quaternions and matrices of complex quaternions are listed below. Lemma 1.1[1][6]. Let a, b ∈ Q be given. Then (a) a = a, (a∗)∗ = a, (a†)† = a. (b) a + b = a + b, (a + b)∗ = a∗ + b∗, (a + b)† = a† + b†. (c) ab = ba, (ab)∗ = a∗b∗, (ab)† = b†a†. (d) aa = aa = n(a)= n(a); (e) a is invertible if and only if n(a) 6= 0, in that case a−1 = n−1(a)a. Lemma 1.2[6]. Let A ∈ Qm×n, B ∈ Qn×p be given. Then (a) A = A, (A†)† = A. (b) AB = BA, (AB)† = B†A†. 2 (c) (AB)−1 = B−1A−1, if A and B are invertible. (d) (A)−1 = (A−1), (A†)−1 = (A−1)†, if A is invertible. 2. A universal similarity factorization equality over complex quaternion algebra We first present a general result on the universal similarity factorization of elements over 2 × 2 total matrix algebra. Lemma 2.1. Let M2(F) be the 2 × 2 total matrix algebra over an arbitrary field F with its basis e11, e12, e21 and e22 satisfying the following multiplication rules esq, t = p estepq = , s,t,p,q = 1, 2. (2.1) 0, t 6= p Then for any a = a11e11 + a12e12 + a21e21 + a22e22 ∈ M2(F), where ast ∈ F, the corresponding diagonal matrix diag( a, a ) satisfies the following universal similarity factorization equality a 0 a a Q Q−1 = 11 12 ∈ F 2×2, (2.2) 0 a a21 a22 where Q has the independent form e e Q = Q−1 = 11 21 . (2.3) e12 e22 Proof. According to Eq.(2.1), it is easy to verify that the unity element in M2(F) is e = e11 + e22. In that case, the matrix Q in Eq.(2.3) satisfies 2 2 e11 e21 e11 + e21e12 e11e21 + e21e22 e11 + e22 0 Q = = 2 = = eI2, e12 e22 e12e11 + e22e12 e12e21 + e22 0 e11 + e22 −1 which implies that Q is invertible over M2(F) and Q = Q . Next multiplying the three matrices in the left-hand side of Eq.(2.2) yields the right-hand side of Eq.(2.2). ✷ Theorem 2.2. Let a = a0 + a1e1 + a2e2 + a3e3 ∈ Q be given. Then the diagonal matrix diag( a, a ) satisfies the following universal factorization similarity equality a 0 a + a i −( a + a i ) Q Q = 0 1 2 3 = ψ(a) ∈ C2×2, (2.4) 0 a a2 − a3i a0 − a1i where Q is an unitary matrix over Q 1 1 − ie e + ie Q = A−1 = Q† = 1 2 3 . (2.5) 2 −e2 + ie3 1+ ie1 Proof. According to Eq.(2.4), we choose a new basis for Q as follows 1 1 1 1 e = ( 1 − ie ) , e = ( −e + ie ) , e = ( e + ie ) , e = (1+ ie ) . (2.6) 11 2 1 12 2 2 3 21 2 2 3 22 2 2 3 Then it is not difficult to verify that the above basis satisfies the multiplication rules in Eq.(2.1). Under this new basis, any element a = a0 + a1e1 + a2e2 + a3e3 ∈ Q can be expressed as a = ( a0 + ia1 )e11 + ( −a2 − ia3 )e12 + ( a2 − ia3 )e21 + ( a0 − ia1 )e22. (2.7) Substituting Eqs.(2.6) and (2.7) into Eq.(2.2), we obtain Eqs.(2.4) and (2.5). ✷ The equality in Eq.(2.4) can also equivalently be expressed as a a a 0 Q 11 12 Q = ∈ Q2×2, (2.8) a21 a22 0 a where ast ∈ C is arbitrary, Q is as in Eq.(2.5), and a has the form 1 1 1 1 a = ( a + a )+ ( a − a )ie + ( a − a )e + ( a + a )ie . (2.9) 2 11 22 2 22 11 1 2 21 12 2 2 12 21 3 This equality shows that every 2 × 2 complex matrix is uniformly similar to an diagonal matrix with the form aI2 over the complex quaternion algebra Q. The complex quaternions and their complex matrix representations satisfy the following operation properties. Theorem 2.3 Let a = a0 + a1e1 + a2e2 + a3e3, b ∈ Q, λ ∈ C be given. Then (a) a = b ⇐⇒ ψ(a)= ψ(b). (b) ψ(a + b)= ψ(a)+ ψ(b), ψ(ab)= ψ(a)ψ(b), ψ(λa)= ψ(aλ)= λψ(a), ψ(1) = I2. 0 1 0 −1 (c) ψ(a)= ψT (a) . −1 0 1 0 0 1 0 −1 (d) ψ(a∗)= ψ(a) . −1 0 1 0 T (e) ψ(a†)= ψ(a) = ψ∗(a), the conjugate transpose of ψ(a). 2 2 2 2 (f) det ψ(a)= n(a)= a0 + a1 + a2 + a3; 1 † (g) a = 4 E2 ψ(a) E2, where E2 = [ 1 − ie1, e2 + ie3 ]. (h) a is invertible if and only if ψ(a) is invertible, in that case, ψ(a−1) = ψ−1(a) and −1 1 −1 † a = 4 E2 ψ (a) E2. For a noninvertible element over Q, we can define its Moore-Penrose inverse as follows. Definition. Let a ∈ Q be given. If the following four equations axa = a, xax = x, (ax)† = ax, (xa)† = xa (2.10) have a common solution x, then this solution is called the Moore-Penrose inverse of a, and denoted by x = a+. The existence and the uniqueness of the Moore-Penrose inverse of a complex quaternion a can be determined by its matrix representation ψ(a) over C. In fact, according to Theorem 4 2.3(a), (b) and (e), the four equations in Eq.(2.10) are equivalent to the following four equations over C ψ(a)ψ(x)ψ(a) = ψ(a), ψ(x)ψ(a)ψ(x) = ψ(x), [ ψ(a)ψ(x) ]∗ = ψ(a)ψ(x), [ ψ(x)ψ(a) ]∗ = ψ(x)ψ(a). According to the complex matrix theory, the following four equations ψ(a)Y ψ(a)= ψ(a), Y ψ(a)Y = Y, [ ψ(a)Y ]∗ = ψ(a)Y, [ Y ψ(a) ]∗ = Y ψ(a).