Norm 2 of a lower trinangular matrix let  $\text{A}$ be a lower triangular matrix that elements on every subdiagonal are same! like this:
\begin{align*}
A=\left[\begin{array}[l l l l l l]  &a\\b&a\\c&b&a&\\.&.&.&.&\\.&.&.&.&.\\.&.&.&.&.&.\\d&e&f&g&c&b&a\\h&d&e&f&g&c&b&a\\i&h&d&e&f&g&c&b&a\end{array}\right]
\end{align*}
Is there any explicit relation for the norm 2 of this matrix?
 A: [Hint]
There are some inequalities that are easy to get:
$||A||_1 = ||A||_\infty = |a|+|b|+|c|+|d|+|e|+|f|+|g|+|h|+|i|$
$||A||_2 \leq \sqrt{||A||_1||A||_\infty}$
$\boxed{||A||_2 \leq |a|+|b|+|c|+|d|+|e|+|f|+|g|+|h|+|i|}$
$||A||_F = \sqrt{9\,|a|^2+8\,|b|^2+7\,|c|^2+6\,|d|^2+5\,|e|^2+4\,|f|^2+3\,|g|^2+2\,|h|^2+|i|^2}$
$||A||_2 \leq ||A||_F$
$\boxed{||A||_2 \leq \sqrt{9\,|a|^2+8\,|b|^2+7\,|c|^2+6\,|d|^2+5\,|e|^2+4\,|f|^2+3\,|g|^2+2\,|h|^2+|i|^2}}$
Now, since $||A||_2 = \sqrt{\lambda_{max}(A^H\,A)}$, to get the exact value, we should take a look at the matrix $B = A^H\,A$. Let's write it for a smaller case:
$B = \begin{bmatrix}
|a|^2+|b|^2+|c|^2+|d|^2 & a\,\bar{b}+b\,\bar{c}+c\,\bar{d} & a\,\bar{c}+b\,\bar{d} & a\,\bar{d} \\
b\,\bar{a}+c\,\bar{b}+d\,\bar{c} & |a|^2+|b|^2+|c|^2 & a\,\bar{b}+b\,\bar{c} & a\,\bar{c} \\
c\,\bar{a}+d\,\bar{b} & b\,\bar{a}+c\,\bar{b} & |a|^2+|b|^2 & a\,\bar{b} \\
d\,\bar{a} & c\,\bar{a} & b\,\bar{a} & |a|^2
\end{bmatrix}$
We know the product of all eigenvalues of $B$ is $|a|^{2\,n}$. Can we find the maximal eigenvalue?
We also know that $||A||_2 \geq \sqrt{\frac{||B\,v||}{||v||}},\;\forall v\in\mathbb{C}^n$.
Matrix Norm (Wolfram)
Matrix Norm (wiki)
