I must show that:

Let $A$ be a nonnegative $n\times n$ matrix, show that $A$ is reducible iff there exists a proper subset $\{e_{j_1},e_{j_2},...,e_{j_k}\}$ of the standard basis of $\mathbb{R^n}$ such that $$<Ae_{j_1},Ae_{j_2},...,Ae_{j_k}>\subset <e_{j_1},e_{j_2},...,e_{j_k}>$$

I have used the fact that there exists a permutation matrix $P$ such that $$M=\begin{bmatrix} B& C \\ 0& D \\ \end{bmatrix}=P^tAP$$

also $P=[e_{j_1},...,e_{j_n}]$ where those are column vectors and the indexes represent a permutation of the ones on the regular basis. $AP=[Ae_{j_1},...,Ae_{j_n}]$ and I defined the span as follows $$<Ae_{j_1},Ae_{j_2},...,Ae_{j_k}>=\{x\in\mathbb{R^n}/x=b_1Ae_{j_1}+b_2Ae_{j_2}+...+b_kAe_{j_k}=A(b_1e_{j_1}+b_2e_{j_2}+...+b_ke_{j_k})=Ay, y\in <e_{j_1},e_{j_2},...,e_{j_k}>\}$$. But I dont know how to prove this.


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