Logarithm of a matrix

In mathematics, the logarithm of a matrix is a generalization of the scalar logarithm to matrices. It is in some sense an inverse function of the matrix exponential.

Definition

A matrix B is a logarithm of some matrix A if A is the matrix exponential of B:

${\displaystyle e^{B}=A.\,}$ 

Properties

A matrix has a logarithm if and only if it is invertible. However, this logarithm may be complex even if all the entries in the matrix are real numbers. In any case, the logarithm is not unique.

Numerical method: matrix diagonalization

A method for finding ln A is the following:

Find the matrix V of eigenvectors of A (each column of V is an eigenvector of A).

Find the inverse V⁻¹ of V.

Let

${\displaystyle A'=V^{-1}\cdot A\cdot V.}$ 

Then A′ will be a diagonal matrix whose diagonal elements are eigenvalues of A.

Replace each diagonal element of A′ by its (natural) logarithm in order to obtain ${\displaystyle \ln A'}$ .

Then

${\displaystyle \ln A=V\cdot \ln A'\cdot V^{-1}.}$ 

See also

• Square root of a matrix