# Logarithmic solution of the matrix differential equation

I am looking for a solution of the matrix ODE $$\dot{\mathbf{x}}(t) = \mathbb{A} \,\mathbf{x}(t),$$ where $\mathbf{x}(t)$ is a vector and $\mathbb{A}$ is a matrix. I am curious if it is possible to transform this equation in terms of logarithm of a vector, i.e. $\ln{\mathbf{x}(t)}$.

It is known that the solution can be formally written in the following form $$\mathbb{x}(t)=\exp{(\mathbb{t\,A})}\,\mathbf{x}(0).$$ So, probably this can help somehow? Probably there exists some algorithm of taking a logarithm of a matrix vector product $\ln{(\exp{(\mathbb{A})}\mathbf{x}(0)})$?

• Do you mean this: n = 10; x0 = RandomReal[{-1, 1}, n]; A = RandomReal[{-1, 1}, {n, n}]; x[t_] := MatrixExp[t A, x0];? – Henrik Schumacher Aug 31 '18 at 16:37
• Yes, this is the solution. Now you can calculate a logarithm of the result. My question is if it is possible to propagate the logarithm of a vector explicitly without calculating a vector. – QuantumNik Aug 31 '18 at 16:44
• Nah, this will work only is special situations, for example when A is a diagonal matrix. Actually, I am under the impression that this is the only possible case, but I don't have a quick argument for that (it involve the argument thatA needs to commute with all diagonal matrices). I think you are mixing up Exp and MatrixExp as well as Log and MatrixLog, respectively. – Henrik Schumacher Aug 31 '18 at 16:55
• Well, I understand it. Actually, I do not expect that there exists a general analytical solution. But probably some approximations can help? Let's say, using approximation for matrix exponentiation by Taylor series or something similar. – QuantumNik Aug 31 '18 at 17:22
• I'm not aware of any way of defining a logarithm of a vector. – mikado Aug 31 '18 at 18:32