We have matrix $D$ as follows $$\begin{bmatrix} d_{1,1} & d_{2,1} & d_{3,1}\\ d_{1,2} & d_{2,2} & d_{3,2}\\ d_{1,3} & d_{2,3} & d_{3,3} \end{bmatrix}.$$ For this matrix a research paper shows that the elements of the matrix follow following properties
- Every entry is non-negative.
- For every $1\leq j\leq 3$, $$\sum_{\tilde{k}=1}^3 d_{j,\tilde{k}}+\sum_{\tilde{j}=1,\tilde{j}\neq j}^3\max_k\{ d_{\tilde{j}k}\}\leq 1 \tag{1}.$$
After two pages the same research paper writes the second property as follows: $$\sum_{\tilde{k}}^{3}d_{j,\tilde{k}}+\max_k\left\{ \sum_{\tilde{j}=1,\tilde{j}\neq j}^3 d_{\tilde{j}k} \right\} \leq 1 \qquad \forall\ j\in\{1,2,3\}. \tag{2}$$
I tried to verify that left hand sides of (1) and (2) produce same results for $j = 1$, but (1) produces $$\sum_{\tilde{k}=1}^3 d_{1,k}+\max\{d_{2,1},d_{2,2},d_{2,3}\}+\max\{d_{3,1},d_{3,2},d_{3,3}\}$$ while the left hand side of (2) produces $$\sum_{\tilde{k}=1}^3 d_{1,k}+\max\{d_{2,1}+d_{3,1},d_{2,2}+d_{3,2},d_{2,3}+d_{3,3}\}$$ which I think are not same. For example, consider the following matrix $$D=\frac{1}{11}\begin{bmatrix} 1 & 3 & 2\\ 2 & 1 & 3\\ 3 & 2 & 1 \end{bmatrix}.$$ For this matrix the (1) inequality is invalid while the (2) inequality is valid. Am I wrong somewhere or the left hand sides of (1) and (2) are not equal to each other. Any help in this regard will be much appreciated. Thanks in advance.
The second inequality is implied by the first.
This is because for each fixed $j$ ($1 \leq j \leq 3$), and every $\tilde{j},k$ ($1 \leq \tilde{j}, k \leq 3$) with $\tilde{j} \neq j$, we have $$ \begin{alignat}{10} & & d_{\tilde{j}k} &\leq \max_{1 \leq k \leq 3}\{ d_{\tilde{j}k} \}\\ \implies & & \sum_{\tilde{j} = 1, \tilde{j} \neq j}^3 d_{\tilde{j}k} &\leq \sum_{\tilde{j} = 1, \tilde{j} \neq j}^3 \max_{1\leq k \leq 3}\{d_{\tilde{j}k}\} \\ \implies & & \max_{1 \leq k \leq 3} \left \{\sum_{\tilde{j} = 1, \tilde{j} \neq j}^3 d_{\tilde{j}k} \right\} &\leq \sum_{\tilde{j} = 1, \tilde{j} \neq j}^3 \max_{1\leq k \leq 3}\{d_{\tilde{j}k}\} \end{alignat} $$
The first inequality is not implied by the second, as you have shown with the counterexample.