Natural Norm

Let $\vert\vert{\bf z}\vert\vert$ be a Vector Norm of ${\bf z}$ such that

$$\vert\vert{\hbox{\sf A}}\vert\vert=\max_{\vert\vert{\bf z}\vert\vert=1} \vert\vert{\hbox{\sf A}}{\bf z}\vert\vert.$$

Then $\vert\vert{\hbox{\sf A}}\vert\vert$ is a Matrix Norm which is said to be the natural norm Induced (or Subordinate) to the Vector Norm $\vert\vert{\bf z}\vert\vert$. For any natural norm,

$$\vert\vert{\hbox{\sf I}}\vert\vert=1,$$

where ${\hbox{\sf I}}$ is the Identity Matrix. The natural matrix norms induced by the L1-Norm, L2-Norm, and L(infinity)-Norm are called the Maximum Absolute Column Sum Norm, Spectral Norm, and Maximum Absolute Row Sum Norm, respectively.

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1115, 1979.