Solving the matrix equation: $e^{\frac{1}{\ln(A)}}=A$

Question

Looking to solve the matrix equation: $$ e^{\frac{1}{\ln(A)}}=A $$

where $A$ is a matrix with a logarithm.

I am interested if this equation has a solution. I read about the logarithm of a matrix and the square root of a matrix. Still having trouble solving it though.

Answer 1

This equation does indeed have a solution:

Let's define $1=Id$ as the identity matrix.

First, take the natural logarithm of both sides to get $\frac{Id}{\log(A)}=\log(A).$

Next, multiply both sides by $\log(A).$ So, $Id=\log^2(A).$

$\log(A)=\begin{bmatrix} -1 & 0 \\ 0 &-1 \end{bmatrix}$

for $A=\begin{bmatrix} 1/e & 0 \\ 0 &1/e \end{bmatrix}.$ So it follows that $\log^2(A)=Id.$

$\square$