$1$-norm of the inverse of lower Toeplitz-like triangular matrix

Question

In recent days, I need to estimate the 1-norm (or $\infty$-norm) of the inverse of the following lower Toeplitz-like triangular matrix, i.e., \begin{equation} C = \begin{bmatrix} 1 &\\ -2 &\frac{3}{2} \\ \frac{1}{2}&-2 &\frac{3}{2} \\ &\ddots & \ddots &\ddots\\ & &\frac{1}{2}&-2&\frac{3}{2} \end{bmatrix}\in\mathbb{R}^{N\times N}. \end{equation} After I use the MATLAB to compute the inverse of the above matrix with different size $N$, it seems that we have

1. $\|C^{-1}\|_1\leq \frac{3N}{2}$;

2. $\|C^{-1}\|_{\infty} = N$.

where $\|C\|_1 = \max\limits_{1\leq j\leq N}\sum\limits^{N}_{i=1}|C_{ij}|$ and $\|C\|_{\infty} = \max\limits_{1\leq i\leq N}\sum\limits^{N}_{j=1}|C_{ij}|$.

Can we prove these conclusions?

Answer 1

You have the following two closed form formulas: $$ \|C^{-1}\|_1=\sum_{i=1}^N \frac{(3^i-1)/2}{3^{i-1}}, $$ and $$ \|C^{-1}\|_\infty=\frac{(3^N-1)/2}{3^{N-1}}+\sum_{i=2}^N \frac{3^{N-i+1}-1}{3^{N-i+1}}. $$