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Spectral Norm

The spectral norm of an m×n real matrix or complex matrix A, with m,n>=1, is the induced matrix norm for the Euclidean vector norms. It is the largest singular value of A and satisfies

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

where A^H is the conjugate transpose and the norms of vectors are Euclidean (Higham 1992, p. 539). For a normal matrix, it equals the spectral radius. This equality need not hold for a matrix that is not a normal matrix. 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.Higham, N. J. "Estimating the Matrix p-Norm." Numer. Math. 62, 539-555, 1992. https://doi.org/10.1007/BF01396242.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 Resource. https://mathworld.wolfram.com/SpectralNorm.html

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