When I was going through a proof, I saw the following step:

$$\frac{1}{p}\operatorname{trace}(\Sigma) \leq \frac{1}{\sqrt{p}} \|\Sigma\|$$

where, $p$ is the number of variables, $\sigma$ is the population covariance matrix and $\|\cdot\|$ stands for Frobenius norm.

They stated that this from jensen's inequality. Can anyone explain me how they got this result.

Thanks alot


It follows from (3) here with $\phi(t)=t^2$ which implies that for $x=[x_1,\ldots,x_p]$, we have $$ \left(\frac{\sum_i x_i}{p}\right)^2\leq \sum_i\frac{x_i^2}{p}. $$ You just need to replace $x$ with the vector of eigenvalues of $\Sigma$ to get the result.

EDIT: Let $\lambda_1,\ldots,\lambda_p$ be the positive eigenvalues of $\Sigma$ (since it is a covariance matrix, I suppose it is symmetric and positive definite; it works also for semi-definite matrices). Then $$ \mathrm{trace}(\Sigma)=\sum_{i=1}^p\lambda_i, \qquad \|\Sigma\|=\left(\sum_{i=1}^p\lambda_i^2\right)^{1/2}. $$ So putting $\lambda$'s instead of $x$'s of the inequality above gives $$ \left(\frac{\sum_i\lambda_i}{p}\right)^2 \leq \sum_i\frac{\lambda_i^2}{p} \quad\Leftrightarrow\quad \frac{1}{p^2}[\mathrm{trace}(\Sigma)]^2\leq\frac{1}{p}\|\Sigma\|^2. $$ Then take the square root.

EDIT: The Jensen's inequality (3) on the linked page states that $$ \phi\left(\frac{\sum_i x_i}{n}\right)\leq \frac{\sum_i \phi(x_i)}{n} $$ for $n$ numbers $x_1,\ldots,x_n$ in the domain of a convex function $\phi$. Just put $\phi(t)=t^2$ (which is convex).

  • $\begingroup$ Still I did not get. It would be great if you could explain it. $\endgroup$ – shani Oct 3 '13 at 13:19
  • $\begingroup$ Bit confused with this given formula. How did you get it from Jensen's inequality $\endgroup$ – shani Oct 3 '13 at 13:23
  • $\begingroup$ Ah OK, let me edit again :) $\endgroup$ – Algebraic Pavel Oct 3 '13 at 13:28
  • $\begingroup$ Yes, I got that point.But how you got the formula with x's from Jensen;s inequality? $\endgroup$ – shani Oct 3 '13 at 13:30
  • $\begingroup$ It's in the second edit. $\endgroup$ – Algebraic Pavel Oct 3 '13 at 13:31

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