Roots and Logarithms of Matrices.

Question

Ok, this question may be too broad or fuzzy, if it is please let me know and I'll try and sharpen or narrow it down a bit.

Hi. I am aware of some of the difficulties of defining roots and logarithms of matrices, often there are several or even infinite number of ways to do this. We need to choose a "branch" so to speak. Famous examples include square root functions of complex numbers and many types of logarithms of matrices.

So my question is... would it be possible to reduce the possibilities by imposing extra restrictions on the operation? Are there any typically popular ways to do this and for what applications are they popular?

Answer 1

The Omno's definition of $\log(.)$ has two drawbacks: its domain is very small and practically, we cannot derive the series which defines it.

Let $\log(.)$ be the principal logarithm over $\mathbb{C}$. In the sequel, $A\in M_n(\mathbb{C})$ denotes any matrix with no eigenvalues in $(-\infty,0]$.

STEP 1. If $A$ is diagonalizable: $A=Pdiag(\lambda_i)P^{-1}$, then $\log(A)=Pdiag(\log(\lambda_i))P^{-1}$.

STEP 2. If $A=D+N=D(I+D^{-1}N)=D(I+M)$ (decomposition of Jordan-Chevalley), then $\log(A)=\log(D)+\sum_{k=1}^{n-1}(-1)^{k+1}/k.M^k$.

Properties. 1. $\log(A)$ is a polynomial in $A$.

2. $\exp(\log(A))=A$.

3. $\log(A)=\int_0^1(A-I_n)(t(A-I_n)+I_n)^{-1}dt$. This form is interesting because we know how to derive the function $f:A\rightarrow \log(A)$.

4. $Df_A(H)=\int_0^1(t(A-I_n)+I_n)^{-1}H(t(A-I_n)+I_n)^{-1}dt$.

Answer 2

It is common to define the matrix logarithm via the logarithm's Taylor series at $x=1$. That is, for a matrix lying in a certain domain (more on that in a bit), we have $$ \log(X+I) = X-X^2/2+X^3/3-\cdots $$ The issue here, however, is that we must then worry about the convergence of the series. Of course, when the series does converge, the function it describes is a unique "choice of branch".

A common restriction in order to ensure convergence is state that $X$ is only in the domain of $\ln(X)$ if $\|X-I\|<1$, where $\|\cdot\|$ denotes some choice of matrix norm, (usually the Frobenius norm). Note that this condition is sufficient, but not necessary, for convergence of the series.

This definition is commonly used in the study of Lie Groups and Lie Algebras.