Compatible

Let $\vert\vert{\hbox{\sf A}}\vert\vert$ be the Matrix Norm associated with the Matrix ${\hbox{\sf A}}$ and $\vert\vert{\bf x}\vert\vert$ be the Vector Norm associated with a Vector ${\bf x}$. Let the product ${\hbox{\sf A}}{\bf x}$ be defined, then $\vert\vert{\hbox{\sf A}}\vert\vert$ and $\vert\vert{\bf x}\vert\vert$ are said to be compatible if

$$\vert\vert{\hbox{\sf A}}{\bf x}\vert\vert\leq \vert\vert{\hbox{\sf A}}\vert\vert\,\vert\vert{\bf x}\vert\vert.$$

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1115, 1980.