Trace Norm

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A Trace Norm is a p-norm which is defined as the sum of all singular values of a matrix.



References

2015

[math]\displaystyle{ \|A\|_p = \left( \sum_{i=1}^{\min\{m,\,n\}} \sigma_i^p \right)^{1/p}. \, }[/math]
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
[math]\displaystyle{ \|A\|_{*} = \operatorname{trace} \left(\sqrt{A^*A}\right) = \sum_{i=1}^{\min\{m,\,n\}} \sigma_i. }[/math]
(Here [math]\displaystyle{ \sqrt{A^*A} }[/math] denotes a positive semidefinite matrix [math]\displaystyle{ B }[/math] such that [math]\displaystyle{ BB=A^*A }[/math]. More precisely, since [math]\displaystyle{ A^*A }[/math] is a positive semidefinite matrix, its square root is well-defined.)

2013