2

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 $\langle A_1, A_1 \rangle = a_{11}$ , $\langle A_2, A_2 \rangle = a_{22}$ , $\ldots$

Zoran Loncarevic
  • 1,263
mahrap
  • 324

1 Answers1

3

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 $\leq1$.

ewcz
  • 329
  • 1
    I hope you found my edit helpful. My opinion is that your use of the "." to indicate matrix multiplication is more confusing than it is helpful. – Ben Grossmann Sep 28 '15 at 20:33
  • yes, thank you, I couldn't agree more – ewcz Sep 28 '15 at 20:34
  • what is the induced norm of a matrix? – mahrap Sep 28 '15 at 20:38
  • here, I meant just the $2-$norm - it is bounded from above by the largest eigenvalue of the matrix and since $A$ is idempotent, its largest eigenvalue can not exceed $1$. But otherwise - https://en.wikipedia.org/wiki/Matrix_norm#Induced_norm – ewcz Sep 28 '15 at 20:41
  • Is there any book that I can learn all these theorems? I do not know enough application about eigenvalues – mahrap Sep 28 '15 at 20:51
  • the concept is discussed at length, e.g., here http://math.stackexchange.com/questions/603375/norm-of-a-symmetric-matrix-equals-spectral-radius – ewcz Sep 28 '15 at 21:10