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.