Norm

Given a $n$-D Vector

$${\bf x} = \left[{\matrix{x_1\cr x_2\cr \vdots\cr x_n\cr}}\right],$$

a Vector Norm $\vert\vert{\bf x}\vert\vert$ is a Nonnegative number satisfying

1. $\vert\vert x\vert\vert>0$ when ${\bf x}\not={\bf0}$ and $\vert\vert{\bf x}\vert\vert=0$ Iff ${\bf x}={\bf0}$,

2. $\vert\vert k{\bf x}\vert\vert=\vert k\vert\,\vert\vert{\bf x}\vert\vert$ for any Scalar $k$,

3. $\vert\vert{\bf x}+{\bf y}\vert\vert\leq \vert\vert{\bf x}\vert\vert+\vert\vert{\bf y}\vert\vert.$

The most common norm is the vector L2-Norm, defined by

$$\vert\vert{\bf x}\vert\vert _2=\vert{\bf x}\vert = \sqrt{{x_1}^2+{x_2}^2+\ldots+{x_n}^2}.$$

Given a Square Matrix ${\hbox{\sf A}}$, a Matrix Norm $\vert\vert{\hbox{\sf A}}\vert\vert$ is a Nonnegative number associated with ${\hbox{\sf A}}$ having the properties

1. $\vert\vert{\hbox{\sf A}}\vert\vert>0$ when ${\hbox{\sf A}}\not={\hbox{\sf0}}$ and $\vert\vert{\hbox{\sf A}}\vert\vert=0$ Iff ${\hbox{\sf A}}={\hbox{\sf0}}$,

2. $\vert\vert k{\hbox{\sf A}}\vert\vert=\vert k\vert\,\vert\vert{\hbox{\sf A}}\vert\vert$ for any Scalar $k$,

3. $\vert\vert{\hbox{\sf A}}+{\hbox{\sf B}}\vert\vert\leq \vert\vert{\hbox{\sf A}}\vert\vert+\vert\vert{\hbox{\sf B}}\vert\vert$,

4. $\vert\vert{\hbox{\sf A}}{\hbox{\sf B}}\vert\vert\leq \vert\vert{\hbox{\sf A}}\vert\vert\,\vert\vert{\hbox{\sf B}}\vert\vert.$

See also Bombieri Norm, Compatible, Euclidean Norm, Hilbert-Schmidt Norm, Induced Norm, L1-Norm, L2-Norm, L(infinity)-Norm, Matrix Norm, Maximum Absolute Column Sum Norm, Maximum Absolute Row Sum Norm, Natural Norm, Normalized Vector, Normed Space, Parallelogram Law, Polynomial Norm, Spectral Norm, Subordinate Norm, Vector Norm

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, pp. 1114-1125, 1979.