Frobenius versus operator norm for random matrices

NumericalIntermediate⏱ 30 minRandom Matricesoperator normfrobeniusconditioning

Problem

Let $A\in\R^{n\times p}$ have i.i.d. standard Gaussian entries.

Compute $\E\|A\|_F^2$ exactly.
State the order of $\E\|A\|$ from the spectral-norm theorem.
For $n=p=1000$ , compare the two numerically and compute the ratio.
What would $\|A\|$ be if all the "mass" were in a single singular value? Compare.

Compute the requested quantities and say what they mean. A number without an interpretation is not an answer.

Opening this before you have a real attempt on paper is the fastest way to feel like you understand something you cannot reproduce.

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