1
$\begingroup$

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 $a_{12} + a_{21} = a_{11} + a_{22}$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.