Logarithm of a matrix

The logarithm of a matrix A is a matrix B such that

$\lim _{n\rightarrow \infty }\left({B \over n}\right)^{n}=A$.

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

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

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 $\ln A'$.

Then

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

See also

• Square root of a matrix