Intuition behind criterion for an irreducible Markov chain to be transient

I have been looking over my notes for Markov chains, and I have come across the following:

Theorem: An irreducible Markov chain is transient iff for some state $i$ there exists a nonzero vector $y$ such that $y_j = \sum_{k\neq i} p_{jk}y_k$, for all $j\neq i$ and $|y_j| \leq 1$for all $j$.

Could you please explain why this is true? I hope to gain some intuition into the result.

Many thanks.

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