Problem
I am trying to numerically verify the fact that "the orthogonal lower (upper) triangular matrix has to be diagonal". However, I have difficulty finding general matrices that satisfies both orthogonality and triangularity since brute force search is of no hope.
Instead, I am wondering if I could manually specify such matrices. More formally, find matrices $\mathbf{A}=[\mathbf{v}_1,\mathbf{v}_2, \cdots, \mathbf{v}_n]$ such that $$ \begin{cases} \mathbf{v}_i^T\mathbf{v}_j=0, i\neq j\\ \mathbf{v}_i^T\mathbf{v}_j=1, i=j\\ \mathbf{v}_i[i-k]=0, i\geq 2,k=1,2\cdots,(k-1) \end{cases} $$