Logarithm of a matrix
In mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix. It is thus a generalization of the scalar logarithm and in some sense an inverse function of the matrix exponential. Not all matrices have a logarithm and those matrices that do have a logarithm may have more than one logarithm. The study of logarithms of matrices leads to Lie theory since when a matrix has a logarithm then it is in a Lie group and the logarithm is the corresponding element of the vector space of the Lie algebra.
If B is sufficiently close to the identity matrix, then a logarithm of B may be computed by means of the following power series:
The rotations in the plane give a simple example. A rotation of angle α around the origin is represented by the 2×2-matrix
Thus, the matrix A has infinitely many logarithms. This corresponds to the fact that the rotation angle is only determined up to multiples of 2π.
In the language of Lie theory, the rotation matrices A are elements of the Lie group SO(2). The corresponding logarithms B are elements of the Lie algebra so(2), which consists of all skew-symmetric matrices. The matrix
The answer is more involved in the real setting. A real matrix has a real logarithm if and only if it is invertible and each Jordan block belonging to a negative eigenvalue occurs an even number of times. If an invertible real matrix does not satisfy the condition with the Jordan blocks, then it has only non-real logarithms. This can already be seen in the scalar case: no branch of the logarithm can be real at -1. The existence of real matrix logarithms of real 2×2 matrices is considered in a later section.
The logarithm of such a rotation matrix R can be readily computed from the antisymmetric part of Rodrigues' rotation formula, explicitly in Axis angle. It yields the logarithm of minimal Frobenius norm, but fails when R has eigenvalues equal to −1 where this is not unique.
A method for finding ln A for a diagonalizable matrix A is the following:
That the logarithm of A might be a complex matrix even if A is real then follows from the fact that a matrix with real and positive entries might nevertheless have negative or even complex eigenvalues (this is true for example for rotation matrices). The non-uniqueness of the logarithm of a matrix follows from the non-uniqueness of the logarithm of a complex number.
The algorithm illustrated above does not work for non-diagonalizable matrices, such as
The latter is accomplished by noticing that one can write a Jordan block as
where K is a matrix with zeros on and under the main diagonal. (The number λ is nonzero by the assumption that the matrix whose logarithm one attempts to take is invertible.)
This series has a finite number of terms (Km is zero if m is the dimension of K), and so its sum is well-defined.
Using the tools of holomorphic functional calculus, given a holomorphic function f defined on an open set in the complex plane and a bounded linear operator T, one can calculate f(T) as long as f is defined on the spectrum of T.
The function f(z)=log z can be defined on any simply connected open set in the complex plane not containing the origin, and it is holomorphic on such a domain. This implies that one can define ln T as long as the spectrum of T does not contain the origin and there is a path going from the origin to infinity not crossing the spectrum of T (e.g., if the spectrum of T is a circle with the origin inside of it, it is impossible to define ln T).
The spectrum of a linear operator on Rn is the set of eigenvalues of its matrix, and so is a finite set. As long as the origin is not in the spectrum (the matrix is invertible), the path condition from the previous paragraph is satisfied, and ln T is well-defined. The non-uniqueness of the matrix logarithm follows from the fact that one can choose more than one branch of the logarithm which is defined on the set of eigenvalues of a matrix.