Let $ \begin{bmatrix}
A& B \\\\ B^\* &C
\end{bmatrix}$ be positive semidefinite, $A,C$ are of size $n\times n$. 


Are the following plausible inequalities true? I have seen a lot of similar results, but for the following inequalities, I cannot locate them in the literature or find that they have been pointed out to be false.


1 $$\quad \sum\limits\_{i=1}^k\lambda\_i\begin{bmatrix}
A& B \\\\ B^\* &C
\end{bmatrix}\le \sum\limits\_{i=1}^k\left(\lambda\_i(A)+\lambda\_i(C)\right)\quad, $$
where $1\le k\le n$. 


2 **Modified**
$$\quad \prod\limits\_{i=1}^{2k}\lambda\_i\begin{bmatrix}
A& B \\\\ B^\* &C
\end{bmatrix}\le \prod\limits\_{i=1}^k \lambda\_i(A)\lambda\_i(C) \quad, $$
where $1\le k\le n$.


3 $$ 2\lambda\_i^{1/2}(B^\*B)\le \lambda\_i(A+C),$$ where $1\le k\le n$.


Here, $\lambda\_i(\cdot)$ means the $i$th largest eigenvalue of $\cdot\quad$. Any references or counterexamples are appreciated.



Ask some matrix eigenvalue inequalities.
**Item 1** is true. This is part of Problem 22 (b) in Section 3.5 of Horn and Johnson [HJ94], which states that for [Ky Fan norm](http://en.wikipedia.org/wiki/Singular_value_decomposition#Ky_Fan_norms) ||⋅|| (and in fact for any unitarily invariant norm) and a positive semidefinite block matrix $\begin{pmatrix}A & B \\ B^\* & C\end{pmatrix}$, it holds that $\left\|\begin{pmatrix}A & B \\ B^\* & C\end{pmatrix}\right\| \le \left\|\begin{pmatrix}A & 0 \\ 0 & 0\end{pmatrix}\right\| + \left\|\begin{pmatrix}0 & 0 \\ 0 & C\end{pmatrix}\right\|$.


([Aud06] contains a proof of a slight generalization of this inequality among other results.)


**Item 2 in the original question** is false by considering the case where *A*=*C*=*I*/2, *B*=0, and *k*=1, where *I* is the identity matrix. (Did you mean to square the left-hand side?)


**Modified item 2** is false; see Willie Wong’s comment on this answer.


**Item 3** is false. A simple counterexample is *n*=2, *i*=1,
$A=\begin{pmatrix}1 & 0 \\ 0 & 0\end{pmatrix}$,
$B=\begin{pmatrix}0 & 1 \\ 0 & 0\end{pmatrix}$,
$C=\begin{pmatrix}0 & 0 \\ 0 & 1\end{pmatrix}$.
Then $2\sqrt{\lambda\_1(B^\*B)}=2$ but *λ*1(*A*+*C*)=1.


References


[Aud06] Koenraad M. R. Audenaert. A norm compression inequality for block partitioned positive semidefinite matrices. *Linear Algebra and its Applications*, 413(1):155–176, Feb. 2006. <http://dx.doi.org/10.1016/j.laa.2005.08.017>


[HJ94] Roger A. Horn, Charles R. Johnson. *Topics in Matrix Analysis*. Cambridge University Press, 1994.


