An Explicit Formula for the Matrix Logarithm

We present an explicit polynomial formula for evaluating the principal logarithm of all matrices lying on the line segment $\{I(1-t)+At:t\in [0,1]\}$ joining the identity matrix $I$ (at $t=0$) to any real matrix $A$ (at $t=1$) having no eigenvalues on the closed negative real axis. This extends to the matrix logarithm the well known Putzer's method for evaluating the matrix exponential.


Introduction
Given a nonsingular matrix A ∈ IR n×n , any solution of the matrix equation e X = A, where e X denotes the exponential of the matrix X, is called logarithm of A. In general, a nonsingular real matrix may have an infinite number of real and complex logarithms. If A has no eigenvalues on the closed negative real axis then A has a unique real logarithm with eigenvalues in the open strip {z ∈ C : −π < Im z < π} of the complex plane (see, for instance, [5]). This unique logarithm may be written as a polynomial in A and is called the principal logarithm of A. It will be denoted by log A.
The problem of computing the principal matrix logarithm has received some attention in recent years (see, for instance, [1], [2], [3], [4] and [6]). In part, this interest has been motivated by the applications of the matrix logarithm in areas such as Systems Theory and Control Theory. The above cited papers list some applications.
As far as we know, most of the methods proposed for computing the principal logarithm are approximation methods . Unlike the matrix exponential case, for which several closed forms based on polynomial representations have been studied (see, for instance, [7], [8], [9] and [10]), little attention has been paid to closed forms for the matrix logarithm.
In this paper, we find for the matrix logarithm the analogue of the well known Putzer's method [9] for evaluating the matrix exponential. Assuming that for t ∈ IR the spectrum of I − At does not intersect IR − 0 , we consider the curve t → log(I − At) in IR n×n . Using the coefficients of a polynomial p(λ) of degree k such that p(A) = 0, every matrix in that curve will be written as a linear combination of the matrices I, A, · · · , A k−1 , in the following way: where the coefficients f 1 , · · · , f k are integrals of certain rational functions.
We find this simple method suitable for teaching purposes because the topics required for understanding it (basically, eigenvalues of matrices and integration of rational functions) are usually taught in the first years of undergraduate courses. We recall that, in contrast, other methods proposed for evaluating the matrix logarithm require advanced theory, such as Schur decompositions, matrix square roots and matrix Padé approximants.

A polynomial formula for the matrix logarithm
Given A ∈ IR n×n , let p(λ) = λ k +c 1 λ k−1 +· · ·+c k−1 λ+c k be a polynomial with real coefficients such that p(A) = 0 and let where I m denotes the m × m identity matrix, be the companion matrix of p(λ). Examples of polynomials p(λ) such that p(A) = 0 are the characteristic polynomial of A (k = n) and the minimum polynomial of A (k ≤ n). Before stating our main result, let us define the following subset of IR: where σ(X) stands for the spectrum of X and A is a given n × n matrix. For each t ∈ IR, the eigenvalues of I − At are of the form 1 − λt, with λ ∈ σ(A). Since non real eigenvalues of A always give rise to non real eigenvalues of I − At, it is enough to consider real eigenvalues of A to obtain a more clear description of the set D. Thus, we may write Theorem 2.1 Suppose that the above notation holds and that the vector function [f 1 (t), · · · , f k (t)] T is the solution in D of the initial value problem where e 2 = [0 1 0 · · · 0] T . Then for all t ∈ D.
Since the vector function [f 1 · · · f k ] T satisfies (1), a little calculation lead us to the system Using the equations of (4) and the identity A k = −c 1 A k−1 − · · · − c k−1 A − c k I, which follows from the Cayley-Hamilton theorem, we may write Since (3) has a unique solution, it follows that P (t) = log (I − At).
Since the coefficients functions in (2) are solutions of (4), we can obtain formulae foṙ f i , i = 1, · · · , k, by solving the first equation forḟ 1 and substituting it into the second equation, solving the second equation forḟ 2 and substituting it into the third equation and proceeding similarly until the last equation. The result iṡ We note that the constants arising in the integration process to find f i , i = 1, · · · , k, can be evaluated according to the identities f i (0) = 0, i = 1, · · · , k.
We now summarize the previous discussion in the next corollary.

Remark 2.4
The indefinite integrals in (5) may be obtained explicitly because we are dealing with rational functions. We note that many calculus textbooks provide methods for evaluating integrals of these kind of functions. Also, symbolic software packages like Mathematica, Maple or Derive are able to compute them. Obviously, this formula holds not only for all t ∈ [0, 1], but also for any t such that σ (I − (I − A)t)∩ IR − 0 = φ. In particular, for t = 1 we may compute directly log A: log A = f 1 (1)I + f 2 (1)(I − A) + · · · + f k (1)(I − A) k−1 .

Example
To illustrate the method proposed, we consider the matrix