Question

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.

Answer 1

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 ||⋅|| (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 λ₁(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.