Distance Norm between matrices

Trace Norm

Let $A,B\in {\mathbb{R}}^{m\times n}$ be two matrices. The trace norm (or nuclear norm) of their difference can be computed as follows:

1. Compute the difference matrix $D=A-B$.

2. Calculate the singular value decomposition (SVD) of $D$, i.e., $D=U\mathrm{\Sigma }{V}^T$, where $U$ and $V$ are orthogonal matrices and $\mathrm{\Sigma }$ is a diagonal matrix with the singular values of $D$ on its diagonal.

3. The trace norm of $D$, denoted as $||D|{|}_{\ast }$, is the sum of the singular values of $D$, which are the diagonal entries of $\mathrm{\Sigma }$. Formally, 

$$||D|{|}_{\ast }=||A-B|{|}_{\ast }=\sum _{i=1}^{min(m,n)}{\sigma }_i(D)$$

where ${\sigma }_i(D)$ is the $i^{th}$ singular value of $D$.