A pinching has the form $M \mapsto T * M$, where $*$ is the entrywise product and $T$ is a $0/1$-matrix. I have the impression that (1)-(3) are all equivalent to
and that it can be proved via the following: if a $0/1$-matrix $T$, with $1$ on the diagonal, avoids the pattern $\left( \begin{array}{ccc} 1 & 1 & 0 \\ 1 & 1 & 1 \\ 0 & 1 & 1 \end{array} \right)$, right)$, then$T$must be block-diagonal. Either (1) or (2) imply that$T$avoids this pattern: for (1) via your example, for (2) because it implies (4) (apply the pitching to the matrix with all entries equal to 1). 1 A pinching has the form$M \mapsto T * M$, where$*$is the entrywise product and$T$is a$0/1$-matrix. I have the impression that (1)-(3) are all equivalent to (4) T is positive, and that it can be proved via the following: if a$0/1$-matrix$T$, with$1$on the diagonal, avoids the pattern$\left( \begin{array}{ccc} 1 & 1 & 0 \ 1 & 1 & 1 \ 0 & 1 & 1 \end{array} \right)$, then$T$must be block-diagonal. Either (1) or (2) imply that$T\$ avoids this pattern: for (1) via your example, for (2) because it implies (4) (apply the pitching to the matrix with all entries equal to 1).