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Spectral Norm


The natural norm induced by the L2-norm. Let A^(H) be the conjugate transpose of the square matrix A, so that (a_(ij))^(H)=(a^__(ji)), then the spectral norm is defined as the square root of the maximum eigenvalue of A^(H)A, i.e.,

||A||_2=(maximum eigenvalue of A^(H)A)^(1/2)
(1)
=max_(|x|_2!=0)(|Ax|_2)/(|x|_2),
(2)

This matrix norm is implemented as Norm[m, 2].


See also

L2-Norm, Matrix Norm, Maximum Absolute Column Sum Norm, Maximum Absolute Row Sum Norm

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References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1115, 2000.Horn, R. A. and Johnson, C. R. "Norms for Vectors and Matrices." Ch. 5 in Matrix Analysis. Cambridge, England: Cambridge University Press, 1990.Strang, G. §6.2 and 7.2 in Linear Algebra and Its Applications, 4th ed. New York: Academic Press, 1980.

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Spectral Norm

Cite this as:

Weisstein, Eric W. "Spectral Norm." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SpectralNorm.html

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