How to find this equation's solution?

Question

How to find the answer below equation?

$$\sum_{i=1}^\infty A^n = \ \begin{pmatrix} 0 & 1/2 & 1/2 \\ 1/2 & 0 & 1/2 \\ 1/2 & 1/2 & 0 \\ \end{pmatrix}$$

At first, I thought it is related to infinite geometric series. However, I could not find the answer.

Answer 1

Hint: Adding the identity on both sides you get $$\exp(A) = \pmatrix{1&\frac{1}{2}&\frac{1}{2}\\\frac{1}{2}&1&\frac{1}{2}\\\frac{1}{2}&\frac{1}{2}&1}\ .$$ Take the logarithm (diagonalizing might help).

Answer 2

Telling my friend Octave about the problem

>> B = [1, 1/2, 1/2; 1/2, 1, 1/2; 1/2, 1/2, 1]
B =

   1.00000   0.50000   0.50000
   0.50000   1.00000   0.50000
   0.50000   0.50000   1.00000

and asking for the matrix logarithm he told me:

>> A = logm(B)
A =

  -0.23105   0.46210   0.46210
   0.46210  -0.23105   0.46210
   0.46210   0.46210  -0.23105

Testing:

>> expm(A)
ans =

   1.00000   0.50000   0.50000
   0.50000   1.00000   0.50000
   0.50000   0.50000   1.00000

The numbers seem to be $(2/3) \ln(2)$ and $-(1/3) \ln(2)$.