# Inequality – Cauchy-Schwarz Master Class Exercise 1.13

This is the question in Michael Stelle's book, exercise 1.13:

Show that yes $${a_ {jk}: 1 leq j leq m, 1 leq k leq n }$$ is a series of real numbers, then one has
$$m sum_ {j = 1} ^ m left ( sum_ {k = 1} ^ n a_ {jk} right) ^ 2 + n sum_ {k = 1} ^ n left ( sum_ { j = 1} ^ m a_ {jk} right) ^ 2 leq left ( sum_ {j = 1} ^ m sum_ {k = 1} ^ n a_ {jk} right) ^ 2 + mn sum_ {j = 1} ^ m sum_ {k = 1} ^ n (a_ {jk}) ^ 2$$
On the other hand, showing equality is maintained if they exist. $$alpha_j$$ Y $$beta_k$$ such that $$a_ {jk} = alpha_j + beta_k$$ for all $$1 leq j leq m$$ Y $$1 leq k leq n$$.

Based on the solution at the end of the book, using Cauchy-Schwarz to show that:

$$left ( sum_ {j = 1} ^ m sum_ {k = 1} ^ n x_ {jk} right) ^ 2 leq mn sum_ {j = 1} ^ m sum_ {k = 1 } ^ n (x_ {jk}) ^ 2 tag {1}$$

and configuration $$x_ {jk} = a_ {jk} – r_j / n – c_k / m$$, where $$r_j = sum_ {k = 1} ^ n a_ {jk}$$ Y $$c_k = sum_ {j = 1} ^ m a_ {jk}$$ one finds the desired inequality.

However, to show inequality, we observe that equality holds if equation (1) is equal. I mean, yes $$x_ {jk} = c$$, for some constant $$c$$ by Cauchy-Schwarz.

The problem I have is that the author says then that $$alpha_j = c + r_j$$ Y $$beta_k = c_k$$ so that $$a_ {jk} = alpha_j + beta_k$$ For equality. Should not it be in its place? $$alpha_j = c + r_j / n$$ Y $$beta_k = c_k / m$$ instead? I'm not sure how the author made this leap.

Any advice or explanation would be of great help.