Suppose $A$ is a doubly stochastic matrix, that is each row and column have the sum of their entries as $1$ and the entries are all nonnegative. Now, suppose $A=BC$ where $B$ is idempotent and $C$ is unitary. Then, it is easy to see that $B$ and $C$ have the row and column sum of entries in each row and/ or column as $1$. But, does this also imply that $B, C$ are also doubly stochastic? Any hints? Thanks beforehand.
Decomposition of doubly stochastic matrices
linear algebramatricesstochastic-matrices
Best Answer
No. Counterexample: $$ A=B=\frac13\pmatrix{1&1&1\\ 1&1&1\\ 1&1&1},\ C=\frac13\pmatrix{-1&2&2\\ 2&-1&2\\ 2&2&-1}. $$