Matrix Norm

Given a Square Matrix ${\hbox{\sf A}}$ with Complex (or Real) entries, 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 ,
: 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. $\vert\vert{\hbox{\sf A}}{\hbox{\sf B}}\vert\vert\leq \vert\vert{\hbox{\sf A}}\vert\vert\,\vert\vert{\hbox{\sf B}}\vert\vert.$

For an $n\times n$ Matrix ${\hbox{\sf A}}$ and an $n\times n$ Unitary Matrix ${\hbox{\sf U}}$ ,

$\begin{displaymath} \vert\vert{\hbox{\sf A}}{\hbox{\sf U}}\vert\vert=\vert\vert{... ...}}{\hbox{\sf A}}\vert\vert=\vert\vert{\hbox{\sf A}}\vert\vert. \end{displaymath}$

Let $\lambda_1$ , ..., $\lambda_n$ be the Eigenvalues of ${\hbox{\sf A}}$ , then

$\begin{displaymath} {1\over\vert\vert{\hbox{\sf A}}^{-1}\vert\vert}\leq \vert\lambda\vert\leq \vert\vert{\hbox{\sf A}}\vert\vert. \end{displaymath}$

The Maximum Absolute Column Sum Norm $\vert\vert{\hbox{\sf A}}\vert\vert _1$ , Spectral Norm $\vert\vert{\hbox{\sf A}}\vert\vert _2$ , and Maximum Absolute Row Sum Norm $\vert\vert{\hbox{\sf A}}\vert\vert _\infty$ satisfy

$\begin{displaymath} (\vert\vert{\hbox{\sf A}}\vert\vert _2)^2\leq \vert\vert{\hb... ... A}}\vert\vert _1\,\vert\vert{\hbox{\sf A}}\vert\vert _\infty. \end{displaymath}$

For a Square Matrix, the Spectral Norm, which is the Square Root of the maximum Eigenvalue of ${\hbox{\sf A}}^\dagger{\hbox{\sf A}}$ (where ${\hbox{\sf A}}^\dagger$ is the Adjoint Matrix), is often referred to as ``the'' matrix 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.