Trace inequality

[Shadow]

Hi there,

I'm having trouble understanding a step in a textbook I'm currently reading (MIT Lectures on Dynamic Systems and Control); I just took a picture of the text to save myself the trouble of copying it over. W is a unitary matrix, [math]\sigma_i[/math] are the singular values of A.

 

 

I haven't the slightest clue as to why this might be, and seeing as I've search the internet and found no refference to this and the author's don't elaborate I assume it's fairly simple and should be well within my reach, but for some reason I just don't get it. Thanks in advance for any help.

[MotleyNoumenon]

This is on the level of Hilbert space theory, which (if you have seen it) isn't that difficult to understand, but usually one learns of this from Fourier analysis. This and the more general inequality can be found in the section "Relations between p-norms" : https://www.wikiwand.com/en/Lp_space#The_p-norm_in_finite_dimensions.

You can find a proof of it at https://math.stackexchange.com/questions/245052/showing-that-l-2-norm-is-smaller-than-l-1.