Is this inequality on Schatten p-norm and diagonal elements true? Let $A=[a_{ij}]\in\mathbb R^{m\times n}$ be a matrix with $m\ge n$, and $\Vert A\Vert_p$ be the Schatten p-norm of $A$.
It is known that $\Vert A\Vert_1\ge \sum_i \vert a_{ii}\vert$ and $\Vert A\Vert_2^2=\Vert A\Vert_F^2\ge \sum_i \vert a_{ii}\vert^2$.
Can we show that $\Vert A\Vert_p^p\ge \sum_i \vert a_{ii}\vert^p$ for general $p\ge 1$?
Thanks!

This is true, and is a consequence of the following majorization result. Let $d_1,\dots,d_n$ be the main diagonal entries of $A$ ordered so that $|d_1|\ge |d_2| \ge \dots\ge |d_n|$. Also let $\sigma_1\ge \sigma_2 \ge \dots \ge \sigma_n$ be the singular values of $A$. The aforementioned (weak) majorization inequality is
$$
\sum_{j=1}^k |d_j| \le \sum_{j=1}^k \sigma_j\qquad \forall \ k =1,2,\dots,n \tag{1}
$$
Weak majorization implies that 
$$
\sum_{j=1}^n \phi(|d_j|) \le \sum_{j=1}^n \phi(\sigma_j)\tag{2}
$$
for every increasing convex function $\phi$. Taking $\phi(t)=t^p$ and recalling that $\Vert A\Vert_p^p = \sum_{j=1}^n \sigma_j^p$, the result follows. 
References for inequality (1): 


*

*Problem 21 of section 3.3 in Topics in Matrix Analysis by Horn and Johnson; 

*Theorem 2.4 of Chi-Kwong Li's lecture notes. Here the result is stated for square matrices, but also includes the more difficult converse direction, describing all the possible combinations  of singular values and diagonal entries.  This is a theorem proved independently by Thompson and Sing in 1970s.


References for inequality (2): 


*

*Section I.3.C of Inequalities: Theory of Majorization and Its Applications by Marshall, Olkin, and Arnold, 2nd edition.

*Section 3.17 of Inequalities by Hardy, Littlewood, and Polya (they consider strong majorization, which requires equality to hold in (1) when $k=n$; but for increasing $\phi$ the weakly-majorized case reduces to strongly majorized).

