Trace Norm / Nuclear Norm: How to verify?

Question

The nuclear norm is defined by this [from wikipedia]:

$$\|A\|_* = \text{trace} \left( \sqrt{A^*A} \right) = \sum_{i=i}^{\min\{m,n\}}\sigma_i(A)$$

I get the derivation of this equation. However, I wanted to test it in MATLAB. So used this script:

clc;
clear;
close all;

P = rand([3,4]);
PTP = P'*P;

%compute trace(P'*P)
B = sqrt(P'*P);
S1 = trace(B)

%Compute sum of sigma_i(P)
E = svd(P);
S2 = sum(E)

%Do the same for eigenvalues
E3 = sqrt(eig(P*P'));
S3 = sum(E3)

But for some reason, the values inside S1 and S2 does not match. I do not understand where I did wrong. Could anybody help?

Answer 1

On the line

B = sqrt(P'*P);

"sqrt" in Matlab calculates element-wise square root.

You probably want to use "sqrtm" : matrix square root instead.