I did this in a jupyter notebook:

# %matplotlib inline
import numpy as np
import matplotlib.pyplot as plt

trace_norms = [np.sqrt(np.eye(i).trace()) for i in range(100)]

plt.plot(trace_norms)

I get this logarithmic curve for all matrices so far

I got a similar curve for the below matrices:

• hadamard
• hankel
• pascal
• inverse pascal
• hilbert
• inverse hilbert
• fourier
• circulant

I may put together something more rigorous in future, as I am still curious as to why this happens.