Assume that $A$ is a symmetric idempotent $n\times n$ real matrix. Prove that sum of all entries of $A$ is less than $n$.
I have proven it is less than $n^2$ by considering each row and column as a vector and the fact that $\langle A_1, A_1 \rangle = a_{11}$ , $\langle A_2, A_2 \rangle = a_{22}$ , $\ldots$
Best Answer
One could represent the sum of all elements as ${v}^TAv$, where $v$ is a vector with $n$ components all equal to $1$. Now, $$ {v}^TA v= {v}^TAAv ={(Av)}^T A v= \|{ Av}\|^2\leq\|A\|^2\|v\|^2 \leq \| v\|^2=n $$ Here, $A=A^2=AA=A^TA$ implies the first equality and also the inequality since it guarantees that the induced norm of the matrix $A$ (eigenvalues $0$ or $1$) is $\leq1$.