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Concept
Logarithm of a matrix
Summary
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 an element of a Lie group and the logarithm is the corresponding element of the vector space of the Lie algebra.
The exponential of a matrix A is defined by
Given a matrix B, another matrix A is said to be a matrix logarithm of B if eA = B. Because the exponential function is not bijective for complex numbers (e.g. ), numbers can have multiple complex logarithms, and as a consequence of this, some matrices may have more than one logarithm, as explained below.
If B is sufficiently close to the identity matrix, then a logarithm of B may be computed by means of the following power series:
Specifically, if , then the preceding series converges and .
The rotations in the plane give a simple example. A rotation of angle α around the origin is represented by the 2×2-matrix
For any integer n, the matrix
is a logarithm of A.
⇔
where
...
qed.
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
is a generator of the Lie algebra so(2).
The question of whether a matrix has a logarithm has the easiest answer when considered in the complex setting. A complex matrix has a logarithm if and only if it is invertible. The logarithm is not unique, but if a matrix has no negative real eigenvalues, then there is a unique logarithm that has eigenvalues all lying in the strip .
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