# Statistical intepretation of ratio of spectral norm of covariance matrix to its Frobenius norm

In statistics, given a covariance matrix $$\Sigma$$ with singular values $$\sigma_1 \ge \sigma_2 \ge \ldots \ge \sigma_p$$, is the ratio of its spectral norm to the Frobenius norm, i.e the ratio $$\dfrac{\sigma_1}{\sqrt{\sum_{i=1}^p \sigma_j^2}}$$ of any interest ? That is, does this ratio appear naturally in the analysis of certain procedures, algorithms, convergence limits, etc. ?

• OK, thanks for mentioning MANOVA (never heard of that!). Any explicity reference of the how this ratio makes its way into MANOVA ? About low ranksity, more generally, it's not hard to prove that $1 / rank(\Sigma) \le \sigma_1 / \|\Sigma\|_F \le 1$. For example, see math.stackexchange.com/a/3638563/168758. Apr 22, 2020 at 17:40
• In 1-way MANOVA one wants to test equality of means when the response is multivariate. The statistics are functions of the eigenvalues of $E^{-1}H$ where $H = n\sum_i(Y_{i.} - Y_{..})(Y_{i.} - Y_{..})^T$. Now, if the centroids of the groups are nearly aligned, a test statistic such as the Roy's maximum root will have better power than alternative statistics, such as Pillai's or Lawley-Hotelling. Too long to explain in detail in a comment. Apr 22, 2020 at 18:01