I am seeking for help about whether the following inequality is true:
I have k rank-one real matrices in $R^{d\times d}$, $x_1x_1^T, x_2x_2^T,...,x_kx_k^T$, where each $x_i$ is a vector in $R^d$ and $||x_i||_2\leq 1$, and I have k real vectors $y_1,y_2,...,y_k$ in $R^d$ and $||y_i||_2\leq c$ for each i. $\lambda>0$. Is the following inequality true?
$||(\sum_{i=1}^k x_ix_i^T+\lambda I)^{-1}\sum_{i=1}^k x_ix_i^Ty_i||_2\leq c\sqrt{d}$.
I have asked another question and get $||\sum_{i=1}^k x_ix_i^Ty_i||_2\leq c\sqrt{d}||\sum_{i=1}^k x_ix_i^T||_2$, previously I thought if I can get this, I will get the result by:
$||(\sum_{i=1}^k x_ix_i^T+\lambda I)^{-1}\sum_{i=1}^k x_ix_i^Ty_i||_2\leq ||(\sum_{i=1}^k x_ix_i^T+\lambda I)^{-1}||_2 ||\sum_{i=1}^k x_ix_i^Ty_i||_2\\\leq c\sqrt{d}||(\sum_{i=1}^k x_ix_i^T+\lambda I)^{-1}||_2 ||\sum_{i=1}^k x_ix_i^T||_2\leq c\sqrt{d}$.
But this is not ture since I can not get $||(\sum_{i=1}^k x_ix_i^T+\lambda I)^{-1}||_2 ||\sum_{i=1}^k x_ix_i^T||_2 \leq 1$. Then, can we prove the inequality holds? I have tried to do some programming to check whether this bound is possible, and I found that in all the random generated examples, this bound holds, but I don't know whether we can prove it. If it does not hold or we can not prove this bound, is it possible to prove a looser bound like $cd$ in the RHS? I just want to get an upper bound that only depends on c and d, independent of k. Could you please help me?
Million thanks!
A counterexample is $$x_1 = \frac{1}{\sqrt 5}\left(\begin{array} 1 2\\1 \end{array}\right)= -y_1,\qquad x_2 = \frac{1}{\sqrt 5}\left(\begin{array} 1 0\\1 \end{array}\right),\qquad y_2 = \sqrt 5 x_2,\qquad c=1 $$ and $\lambda$ small enough.