On transformation of the stable matrices to a block-diagonal form

We consider square matrices $A$ such that $||exp(tA)||\leqslant K$ for all $t\geqslant 0$. We show that each matrix possessing this property can be transformed to a block-diagonal form such shat condition numbers of all the digaonal blocks and the transformation matrix depend only on $K$ and matrix size. The obtained result is applied to the analysis of long-time simulation accuracy of difference schemes.