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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.

The Schatten class S_p consists of compact operators on a Hilbert space whose singular values have finite p-norm. Long (2026) reported that the Hilbert transform on L^p(R;S_p) has sharp operator norm cot(pi/(2p^*)), where 1<p<infty and p^*=max(p,p/(p-1)).

See also

Hlawka's Inequality, Matrix Norm, Singular Value Decomposition, Trace Norm

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References

Bhatia, R. Matrix Analysis. New York: Springer-Verlag, 1997.Long, Z. "Sharp Hilbert Transform Norms on Schatten Classes: Classification and Optimal Spectral Corrections." 7 Oct 2026. https://arxiv.org/abs/2610.10158.

Cite this as:

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

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