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Schatten Norm

The Schatten p-norm of an m×n complex matrix A is the vector norm of its singular values. If r=min(m,n) and the singular values are sigma_1(A),...,sigma_r(A), then for 1<=p<infty,

||A||_p=(sum_(j=1)^rsigma_j(A)^p)^(1/p),

while

||A||_infty=max_(1<=j<=r)sigma_j(A).

The Schatten 1-norm is the sum of the singular values, the Schatten 2-norm is the Frobenius norm or Hilbert-Schmidt norm, and the Schatten infinity-norm is the spectral norm. Multiplication on either side by a unitary matrix leaves a Schatten norm unchanged.

See also

Hlawka's Inequality, Matrix Norm, Singular Value Decomposition

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References

Bhatia, R. Matrix Analysis. New York: Springer-Verlag, 1997.

Cite this as:

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

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