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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 $k$,

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

\vert\vert{\hbox{\sf A}}{\hbox{\sf U}}\vert\vert=\vert\vert{...
...}}{\hbox{\sf A}}\vert\vert=\vert\vert{\hbox{\sf A}}\vert\vert.

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

{1\over\vert\vert{\hbox{\sf A}}^{-1}\vert\vert}\leq \vert\lambda\vert\leq \vert\vert{\hbox{\sf A}}\vert\vert.

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

(\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.

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.

See also Compatible, Hilbert-Schmidt Norm, Maximum Absolute Column Sum Norm, Maximum Absolute Row Sum Norm, Natural Norm, Norm, Polynomial Norm, Spectral Norm, Spectral Radius, Vector Norm


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

© 1996-9 Eric W. Weisstein