I have a known matrix $A$ with entries $a_{ij}$ and would like to know matrix $B$ with entries $b_{ij}$ of the same dimension which entries obey:
\begin{equation} b_{ij} = 1 + \sum_{k \neq j} a_{ik} b_{kj} \qquad\text{for $i \neq j$} \end{equation} and \begin{equation} b_{ii} = 0 \qquad\text{for all $i$.} \end{equation}
Is there a simple matrix form for these equations?
The approach I tried only works with out restrictions as explained below.
If it weren't for the restriction of the zero diagonal and the restriction in the sum $k \neq j$. I would have the set of equations:
\begin{equation} b_{ij} = 1 + \sum_{k} a_{ik} b_{kj} \end{equation}
which by definition of matrix product can be expressed as the matrix equation:
$$ B = J + AB $$
where $J$ denotes the matrix with all entries equal to $1$ and of same dimensions as $A$ and $B$. One could obtain matrix $B$ if matrix $(I - A)$ is invertible by:
$$ B = (I - A)^{-1}J $$
where $I$ is the identity matrix.
Is there a similar matrix equation that corresponds the problem with restrictions in the first equation?
Thanks in advance.
Since $b_{j,j}=0$ for each$~j$, one has $\sum_{k \neq j} a_{i,k} b_{k,j} =\sum_k a_{i,k} b_{k,j}$, and your analysis then applies to the problem. Quite likely $(I - A)^{-1}J$ has some nonzero entries on the main diagonal, in which case your problem has no solutions.