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Spectral Norm

The Natural Norm induced by the L2-Norm. Let ${\hbox{\sf A}}^\dagger$ be the Adjoint of the Square Matrix ${\hbox{\sf A}}$, so that ${\hbox{\sf A}}^\dagger=a_{ji}^*$, then

$\displaystyle \vert\vert{\hbox{\sf A}}\vert\vert _2$ $\textstyle =$ $\displaystyle (\hbox{maximum eigenvalue of } {\hbox{\sf A}}^\dagger{\hbox{\sf A}})^{1/2}$  
  $\textstyle =$ $\displaystyle \max_{\vert\vert x\vert\vert _2\not=0}{\vert\vert{\hbox{\sf A}}{\bf x}\vert\vert _2\over\vert\vert{\bf x}\vert\vert _2}.$  

See also L2-Norm, Matrix Norm


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

Strang, G. §6.2 and 7.2 in Linear Algebra and Its Applications, 4th ed. New York: Academic Press, 1980.

© 1996-9 Eric W. Weisstein