For positive block-matrix we write $M=\begin{pmatrix} A & X\\ {X^*} & B\end{pmatrix} \in {\mathbb{M}}_{n+m}^+$, with $A\in {\mathbb{M}}_n^+$, $B \in {\mathbb{M}}_m^+.$ The main result is first to study the consequences of a decomposition lemma due to C.~Bourrin and second to extend the class of these P.S.D. matrices $M$ written by blocks that satisfies the inequality: $\|M\|\le \|A+B\|$ for all symmetric norms and to give examples whenever it is necessary.