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. 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 one-to-one for complex numbers (e.g. e π i = e 3 π i = − 1 {displaystyle e^{pi i}=e^{3pi i}=-1} ), 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 ‖ B − I ‖ < 1 {displaystyle left|{B-I} ight|<1} , then the preceding series converges and e l o g ( B ) = B {displaystyle e^{mathrm {log} (B)}=B} . 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. 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.