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

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., $$$$C = \begin{bmatrix} 1 &\\ -2 &\frac{3}{2} \\ \frac{1}{2}&-2 &\frac{3}{2} \\ &\ddots & \ddots &\ddots\\ & &\frac{1}{2}&-2&\frac{3}{2} \end{bmatrix}\in\mathbb{R}^{N\times N}.$$$$ After I use the MATLAB to compute the inverse of the above matrix with different size $$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?

• Many thanks for your comments, I will add the definition of the matrix norm. – Hsien-Ming Ku Oct 30 '19 at 13:33

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