International Journal of Algebra, Vol. 2, 2008, no. 3, 131 - 142
Computing Real Logarithm of a Real Matrix
Nagwa Sherif and Ehab Morsy1
Department of Mathematics, Faculty of Science Suez Canal University, Ismailia 22541, Egypt
Abstract In this paper we will be interested in characterizing and computing for a nonsingular real matrix A ∈ Rn×n a real matrix X ∈ Rn×n that satisfies eX = A, that is, a logarithm of A. Firstly, we investigate the conditions under which such logarithm exists, unique, polynomial in A, or belongs to a particular class of matrices. Secondly, real Schur decomposition will be used to compute X.
Mathematics Subject Classification: 15A24
Keywords: Matrix functions, matrix logarithms, nonsingular matrices
1 Introduction
Logarithms of matrices arise in various contexts. For example [1, 3, 8], for a physical system governed by a linear differential equation of the form dy = Xy, dt where X is n-by-n unknown matrix. From observations of the state vector y(t), if y(0) = y◦ then we know that
tX y(t)=e y◦.
By taking n observations at t = 1 for n initial states consisting of the columns of the identity matrix, we obtain the matrix A = eX . Under certain conditions on A, we can then solve for X, that is X = log A. This raises the question of how to compute a logarithm of a matrix. We show that S(A), the solution set of this matrix equation, is nonempty if and only if A is nonsingular. In this paper we concerned with the real solvability of the matrix equation eX = A in case of real matrix A. Not every nonsingular real matrix have a real logarithm as the following example illustrates. 1ehabb [email protected] 132 N. Sherif and E. Morsy
Example 1 Let A = diag(1, −1). Then any logarithm of A is given by X = Udiag(log(1), log(−1))U −1 −1 = Udiag(2πij1,iπ(2j2 + 1))U where j1,j2 ∈ Z and U is any nonsingular matrix commuting with A. All these logarithms are matrices with noncomplex conjugate eigenvalues. Hence it can not be similar to a real matrix and no real logarithm of A can be obtained. The existence of a real logarithm of a real matrix is discussed in Section 2. In Section 2, we also characterize such logarithm, that is, set the conditions for which X is polynomial in A, symmetric, positive definite, or orthogonal. The computation of a real logarithm X of a real matrix A arises in many system identification, one of which is the mathematical modeling of dynamic systems [4]. In Section 3, we propose a technique to compute such X based on the real Schur decomposition.
2 Characterization of a real logarithm
In the following theorem we give a set of conditions on the matrix A that guarantees the existence of a real logarithm of A. We start by a lemma which constructs a real logarithm of a particular 2-by-2 matrices. Lemma 2 The 2-by-2 real matrices of the form −λ 0 ab A = ,λ>0 and A = ,b=0 1 0 −λ 2 −ba have real logarithms given by Logλ π θμ X = and X = , 1 −π Logλ 2 −μθ respectively, where eθ±iμ = a ± ib.
X1 X2 Proof. To prove this lemma, it is enough to show that e = A1 and e = A2. For the matrix X1, there exists a nonsingular matrix V such that −1 eX1 = eV diag(Logλ+iπ,Logλ−iπ)V −1 = V (−λI)V = −λI = A1.
For X2, it is clear that X2 is a normal matrix, and since A2 and X2 are diag- onalizable and commuting, they are simultaneously diagonalizable [7]. Then
−1 eX2 = eUdiag(θ+iμ,θ−iμ)U = Udiag(a + ib, a − ib)U −1
= A2. Computing real logarithm of a real matrix 133
Next we set up a sufficient and necessary condition for the existence of a real logarithm X of a real matrix A ∈ Rn×n . A proof of the following theorem can be found in Culver [5], and Ulig [6], however the proof we provide is different, it is based on the usage of real Jordan canonical form and the previous lemma.
Theorem 3 Let A ∈ Rn×n be a real matrix. Then there exists a real logarithm X of A if and only if A is nonsingular and each Jordan block of A belonging to negative eigenvalue occurs an even number of times.
Proof. Let X be a real logarithm of A, that is, eX = A, By using [7, Th 3.4.5] each complex Jordan block (if exists) of any size occur in the Jordan canonical form of a real matrix in conjugate pairs. Hence we may suppose that the Jordan canonical form of X is
JX = diag(Jm1 (x1),... ,Jmr (xr),B2mr+1 ,... ,B2mp ), (1) where x1,... ,xr are real, xr+1,... ,xp are complex, and x1,x2,... ,xp are not necessarily distinct and B2ms = diag(Jms (xs),Jms (xs)). The Jordan canonical X form JA of A = e has the form