Logarithm of a matrix

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

Definition

A matrix B is a logarithm of a given matrix A, where A is the Matrixexponential B:

Properties

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

Calculate the logarithm of a diagonalizable matrix

In the following, a method will be described, In A a diagonalizable matrix A to calculate:

That is the logarithm of A can be complex, even though A is real, arises from the fact that a real matrix may have complex eigenvalues ​​(this applies for example to rotation matrices ). The non-uniqueness of the logarithm follows from the non-uniqueness of the logarithm of a complex number.

The logarithm of a matrix nichtdiagonalisierbaren

The above algorithm will not work for nichtdiagonalisierbare matrices such as

For such matrices, one must first determine the Jordan normal form. Instead of the logarithm of the diagonal entries you must calculate here the logarithm of the Jordan blocks.

The latter is achieved in that to write the Jordan matrix

Where K is a matrix with zeros below and on the main diagonal. ( The number λ is equal to zero, assuming that the matrix, which one wants to calculate the logarithm is invertible. )

By the formula

Obtained

This series does not converge for a general matrix K, as they would do it for real numbers with sum less than 1. However, this particular matrix K is a matrix nilpotent, so that the series is a finite number of terms has (Km is zero when m is of rank K hereinafter).

Through this approach, one obtains

From the point of view of functional analysis

A square matrix represents a linear operator on the Euclidean space. Since this space is finite, each operator is bounded.

Let be a holomorphic function on an open set in the complex plane, and is a bounded operator. To f ( T) directions, if f (z) is defined in the range of t.

The function f ( z) = ln z can on every simply connected open set in the complex plane that does not contain zero, are defined and is holomorphic on this set of definitions. It follows that In T is defined as the range of T does not include zero and there is a path from zero to infinity, which does not intersect the range of T ( Forms for example, the range of T is a circle whose center is zero, then it is not possible to define T ln ).

For the special case of the Euclidean space, the spectrum of a linear operator is the set of eigenvalues ​​of the matrix, that is finite. As long as zero is not included in the spectrum (the matrix therefore invertible ), and thus apparently the above path condition is met, it follows that In T is well-defined. The ambiguity follows from the fact that one can select more than one branch of the logarithm of which is defined on the set of eigenvalues ​​of the matrix.