Can we bound $\mathbf{A}$ with $\mathbf{A^\*}$ as ${\bf{A}} \preceq {{\bf{A}}^\*}$ where 
\begin{equation}
{\bf{A}} = \left[ {\begin{array}{\*{20}{c}}
{{{\bf{A}}\_{11}}}&{...}&{{{\bf{A}}\_{1N}}}\\
 \vdots & \ddots & \vdots \\
{{\bf{A}}\_{1N}^H}&{...}&{{{\bf{A}}\_{NN}}}
\end{array}} \right],\quad {{\bf{A}}^\*} = \left[ {\begin{array}{\*{20}{c}}
{{{\bf{A}}\_{11}}}&{...}&{\bf{0}}\\
 \vdots & \ddots & \vdots \\
{\bf{0}}&{...}&{{{\bf{A}}\_{NN}}}
\end{array}} \right]
\end{equation}
actually all $\mathbf{A}\_{i,i}$ and $\mathbf{A}$ are positive semidefinite matrices. If the above inequality cannot hold, please remark if with some extra assumptions on $\mathbf{A}\_{i,i}$, it can be made to hold?
what other facts can be stated about the eigen-values of the two matrices?



Bounding the positive semi-definite matrix with its block diagonal matrix
Actually, there is a great question hiding in there that the OP could have asked. Since he does not care about norm inequalities or other majorization results, here's the trivial most counterexample.


\begin{equation\*}
A = \begin{pmatrix} 
1 & 1\\
1 & 1
\end{pmatrix}.
\end{equation\*}


It follows from results on matrix pinchings that 
\begin{equation\*}
\lambda(A^\*) \prec \lambda(A)
\end{equation\*}
where $\lambda(\cdot)$ is the eigenvalue map.


Other more interesting things can also be said about this question, but I'll write them out only if there is more interest. 




---


**EDIT.** Just adding some material for the interested. Positive definite matrices with Hermitian blocks (not necessarily psd as in the OP), arise in quantum information theory and related areas. Indeed, Bourin, Lee, and Lin show in **[[1](http://arxiv.org/abs/1208.6494)]** that if $A$ is a positive semidefinite matrix partitioned into $N\times N$ blocks, then for all unitarily invariant norms 
\begin{equation\*}
 \|A\| \le \left\|\sum\_{i=1}^N A\_{ii}\right\|.
\end{equation\*}
Their proof is based on generators of a Clifford algebra. Of course, this subject has connections to the operation of taking partial traces.


**[[1](http://arxiv.org/abs/1208.6494)]** J.-C. Bourin, E.-Y. Lee, and M. Lin. [Positive definite matrices with Hermitian blocks and their partial traces](http://arxiv.org/abs/1208.6494), arXiv, 2012.


