Why is the higlighted equivalence true?

Question

I am not able to see why the equivalence highlighted is true? Is it possible to move the u from the right hand side of A on to the left hand side?

Any help would be much appreciated!

Answer 1

Note that

• $Au$ is a column vector wich components are the sum of the columns components that is $(Au)_i=\sum_{j=1}^N A_{ij}$
• $u\cdot Au$ is the sum of all $A_{ij}$ that is $\sum_{ij} A_{ij}$
• $u^2=u\cdot u=1+1+\dots+1=N$