Matrix Decomposition $B = B^* + \sum_{i>1}\lambda_i B_i$

I recently encountered the following decomposition in a paper, may I know whether anyone on CV happened to know the name of the decomposition and a proof of it?

$B$ is a regular stochastic matrix and can be decomposed as:

$$B = B^* + \sum_{i>1}\lambda_i B_i$$ where $\lambda_i$ is the eigenvalue, $B_i = f_i \pi_i$ ($f_i$ and $\pi_i$ are the left and right eigenvectors respectively).

Further details:

B is a $mn \times mn$ matrix and has strictly positive entries and $lim_{k \to \infty} B^k = B^*$ where $B^*$ is the unique stationary distribution.

The address to the paper is: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2970694

and i'm referring to Theorem 2 on page 9. The author didn't specify how the above equation is derived.

• This is meaningless without a description or definition of $B^{*}$: what is it? – whuber Nov 26 '17 at 23:14
• Can you please provide a reference to paper you mention? – usεr11852 Nov 27 '17 at 0:21
• Thanks a lot, I just edited the question with more details on the definition of B and $B^*$. – CobbDG Nov 27 '17 at 1:47