The Cauchy-Schwarz master class习题1.13

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Michael Stelle The Cauchy-Schwarz master class
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 $1 \leq k \leq n$.

书后的解答(p242)是由Cauchy-Schwarz不等式证明：$$\left( \sum_{j=1}^m \sum_{k=1}^n x_{jk} \right)^2 \leq mn\sum_{j=1}^m \sum_{k=1}^n (x_{jk})^2 \tag{1}\label1$$然后令$x_{jk} = a_{jk} - r_j / n - c_k / m$,其中$r_j = \sum_{k=1}^n a_{jk}$,$c_k = \sum_{j=1}^m a_{jk}$就得到了要证的不等式.
然而,我们注意到该等式仅当 \eqref{1} 取等时才成立,也就是说,$x_{jk} = c$,$c$是常数.

书上设$\alpha_j = c + r_j$和$\beta_k = c_k$使$a_{jk} = \alpha_j + \beta_k$取等.我觉得应该是$\alpha_j = c + r_j/n$和$\beta_k = c_k/m$才对.

好像是说 $a_{jk} = d$ 其中 $d$ 是常数. 而我认为除此之外在其他情况下也能取等.例如,取2×2矩阵$A$使$a_{12} + a_{21} = a_{11} + a_{22}$.