Matrix Norm - Schatten Norms

Schatten Norms

For more details on this topic, see Schatten norm.

The Schatten p-norms arise when applying the p-norm to the vector of singular values of a matrix. If the singular values are denoted by σ_i, then the Schatten p-norm is defined by

These norms again share the notation with the induced and entrywise p-norms, but they are different.

All Schatten norms are sub-multiplicative. They are also unitarily invariant, which means that ||A|| = ||UAV|| for all matrices A and all unitary matrices U and V.

The most familiar cases are p = 1, 2, ∞. The case p = 2 yields the Frobenius norm, introduced before. The case p = ∞ yields the spectral norm, which is the matrix norm induced by the vector 2-norm (see above). Finally, p = 1 yields the nuclear norm (also known as the trace norm, or the Ky Fan 'n'-norm), defined as

(Here denotes a matrix such that . More precisely, since is a positive semidefinite matrix, its square root is well-defined.)