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For a matrix $P$, suppose $P^2=P$ and all entries of $P$ are positive, and $\sum_j P_{ij}=1$. Prove $P_{ij}=P_{ii}$ for all $i,j$. The original problem is from stochastic processes. I reduced it into this linear algebra problem, and wonder if there is a result in linear algebra about it.
The original problem is exercise 2.22 of Sidney Resnick's book "Adventures in Stochastic Processes". It says suppose an irreducible Markov chain, not neccessarily finite many states, has the property that $P^2 = P$. Show that $p_{ij} = p_{jj}$ for all $i, j\in S$.

Connor
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    Under the conditions you cite, the conclusion doesn't hold. It does hold that every row of $P$ is the same. But, for example, the matrix [ 1/3 2/3 ; 1/3 2/3 ] satisfies your requirements. Perhaps you also want your columns to sum to $1$? In fact, any matrix all whose rows are equal and sum to $1$ satisfy your requirement (and the solutions to the posed problem.) If you further ask that the columns sum to $1$, then every entry is $1/n$, as you want. – Pedro Sep 17 '16 at 05:01
  • I post the original problem. Thanks for your example. – Connor Sep 17 '16 at 05:08
  • And I am curious about how to prove they must have same rows? – Connor Sep 17 '16 at 05:10

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You have misunderstood the original problem; note that "$p_{ij} = p_{ii}$" is different from "$p_{ij} = p_{jj}$."

Because $P^2 = P$, each column $P_{(i)}$ of $P$ is an eigenvector of $P$, associated with the eigenvalue $1$. By the Perron-Frobenius Theorem, all $P_{(j)}$ for $1 \leq j \leq n$ must be positive multiples of $P_{(1)}$, denoted as $$ P_{(j)} = \alpha_j\cdot P_{(1)} $$ where $\alpha_j > 0$ and $\alpha_1 = 1$. We have $$ \sum_{j} P_{ij} = \sum_{j} \alpha_jP_{i1} = P_{i1}\sum_{j}\alpha_j = 1 $$ for all $i$. It is hence not hard to conclude that all elements of $P_{(1)}$ are the same.

PSPACEhard
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  • It seems we need to show 1 is the eigenvalue with maximum absolute value. How do we prove it? – Connor Sep 17 '16 at 17:45
  • @Connor You may refer to http://math.stackexchange.com/questions/40320/proof-that-the-largest-eigenvalue-of-a-stochastic-matrix-is-1. – PSPACEhard Sep 17 '16 at 17:48