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Taken from “Introduction to Linear Algebra” by Gilbert Strang: Perron-Frobenius

I am getting stuck on this proof, specifically at the inequality reasoning in the area I’ve hi-lighted. Why is the strict inequality produced when multiplying by A? I need to get unstuck at this step in order to understand the rest of the proof I believe.

Also, I’m feeling a bit unsure about the statement “and tmax could be increased.” I think this may be cleared up though if I better understand the inequality produced.

Hanzy
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1 Answers1

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Suppose $y \geq z$ so that $Ay \geq Az$. If possible let $Ay=Az$. Then $A(y-z)=0$. Write this out explicitly in terms of the entries of $A$: $\sum_{j=1}^{n} a_{ij} (y_j-z_j)=0$ for each $i$. If a sum of non-negative numbers is $0$ then each term must be $0$. Conclude that $y=z$ necessarily. Can you take it from here?

Kavi Rama Murthy
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