What is meant by a "$S$-strictly diagonally dominant matrix" in the book 'Geršgorin and His Circles' What is meant by a "$S$-strictly diagonally dominant matrix" in the book Geršgorin and His Circles. 
The definition of strictly diagonally dominant is easy to find, but the definition of  "$S$-strictly diagonally dominant matrix"  is needed.
 A: Recall that a matrix $A=(a_{ij})_{i=1, \, j=1}^{n,n}$ is called strictly diagonally dominant if $|a_{ii}| \gt \sum_{j=1, \, j\neq i}^n |a_{ij}|$ for each $i$. In words, for each row the diagonal entry is in absolute value larger than the sum of the absolute values of all other entries in the same row.
The "strictly" refers to the inequality being "strict." 
Now, for a subset $S \subset \{1, \dots, n\}$ one calls the matrix $S$-strictly diagonally dominant if


*

*$|a_{ii}|> r_i^S(A)$ for all $i \in S$ and 

*$(|a_{ii}-r_i^S(A))(|a_{jj}|-r_j^{\overline{S}}(A))>  r_i^{\overline{S}}(A)r_j^S(A)$ for all $i \in S$ and $j \in \overline{S}$


where  $r_i^T(A)= \sum_{j \in T, \, j \neq i}|a_{ij}| $ and $\overline{S}$ is the compleemnt of $S$. 
The special case $S= \{1, \dots, n \}$ yields the first definition.
The relevance of this notion is that it is weaker than strictly diagonally dominant (every SSD matrix is $S$-SSD for every $S$), yet it still allows to derive some of the same conclusions on the matrix. 
An example of a matrix that is $\{1,2\}$-SSD yet not SSD is: 
$$
\begin{pmatrix} 2.6 & -0.4 & -0.7 & -0.2 \\
-0.4 & 2.6 & -0.5 & -0.7 \\
-0.6 & -0.7 & 2.2 & -1.0 \\
-0.8 & -0.7 & -0.5 & 2.2  
\end{pmatrix} 
$$
taken from Bru,  Pedroche,  Szyld  "Subdirect sums of S-strictly diagonally dominant matrices."
