# Cauchy-Schwarz Master Class Exercise 1.13

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

Show that if $$\{a_{jk} : 1\leq j \leq m, 1 \leq k \leq n\}$$ is an array of real numbers then one has $$m \sum_{j=1}^m \left( \sum_{k=1}^n a_{jk} \right)^2 + n \sum_{k=1}^n \left( \sum_{j=1}^m a_{jk} \right)^2 \leq \left( \sum_{j=1}^m \sum_{k=1}^n a_{jk} \right)^2 + mn\sum_{j=1}^m \sum_{k=1}^n (a_{jk})^2$$ Moreover, show equality holds iff there exist $$\alpha_j$$ and $$\beta_k$$ such that $$a_{jk} = \alpha_j + \beta_k$$ for all $$1 \leq j \leq m$$ and $$1 \leq k \leq n$$.

Based on the solution at the back of the book, by 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 setting $$x_{jk} = a_{jk} - r_j / n - c_k / m$$, where $$r_j = \sum_{k=1}^n a_{jk}$$ and $$c_k = \sum_{j=1}^m a_{jk}$$ one finds the desired inequality.

However, to show equality we note that the equality holds iff equation (1) is equal. That is, iff $$x_{jk} = c$$, for some constant $$c$$ by Cauchy-Schwarz.

The trouble I'm having is that the author states then that $$\alpha_j = c + r_j$$ and $$\beta_k = c_k$$ so that $$a_{jk} = \alpha_j + \beta_k$$ for equality. Shouldn't it be instead $$\alpha_j = c + r_j/n$$ and $$\beta_k = c_k/m$$ instead? I am uncertain how the author made this leap.

Any hints or explanation would be greatly helpful.

EDIT 1: The book's answer seems to imply that each $$a_{jk} = d$$ where $$d$$ is some constant. Whereas the answer I think seems to include that and more. For example, take a 2x2 matrix $$A$$ with $$a_{12} + a_{21} = a_{11} + a_{22}$$.