Let the tridiagonal matrix ${\bf C} \in {\Bbb R}^{n \times n}$ and the diagonal matrix ${\bf D} \in {\Bbb R}^{n \times n}$ be
\begin{equation}\label{matrix-C,D} {\bf C} := \begin{bmatrix} 0 & 1 & & & &\\ -1 & 0 & 1 & & &\\ &-1 & 0 & 1 & &\\ & & \ddots&\ddots&\ddots\\ & & &-1 & 0 & 1\\ & & & &-1 & 1 \end{bmatrix}, \qquad {\bf D} :=\begin{bmatrix} 1& & & & &\\ & 1& & & &\\ && 1& & &\\ &&&\ddots&&\\ & & & & 1&\\ & & & & &\frac{1}{2} \end{bmatrix} \end{equation}
I want to prove the following lemma: for $\lambda>0$, $\left\lVert ({\bf C} + 2\lambda {\bf D})^{-1} \right\rVert_2\leq n$.
Can you please give me some tips? I wonder whether the upper bound $n$ in this lemma is sharp? is there any sharp bound for matrix $({\bf C} + 2\lambda {\bf D})^{-1}$?
