I'm asking this just out of curiosity because a brief googling failed to give me the answer.
By skew-triangular matrices I mean matrices with this
$$
\begin{bmatrix}
\times & \times & \times & \times \\
\times & \times & \times \\
\times & \times \\
\times \\
\end{bmatrix}
$$
or this
$$
\begin{bmatrix}
& & & \times \\
& & \times & \times \\
& \times & \times & \times \\
\times & \times & \times & \times \\
\end{bmatrix}
$$
sparsity pattern. Simple experiments with Mathematica show that the inverse of the first type is a matrix of the second type (and vice versa, of course).
1. Do these matrices have their real name and where do they occur?
2. What other interesting properties do they possess?
Best Answer
I don't know a name for these matrices, but they are invertible only if all of the entries along the main anti-diagonal are non-zero. And yes, you can prove that the inverse of one has the form of the other. These collections of matrices are translates of the traditional Borel subgroup of upper (or lower triangular matrices). The traditional notation is $B$ for the subgroup of upper triangular matrices, and $w_0$ for the matrix $$ \begin{pmatrix} 0 & 0 & \cdots & 1 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 1 & \cdots & 0 \\ 1 & 0 & \cdots & 0 \end{pmatrix}.$$ Your collections of matrices are $w_0 B$ and $B w_0$, respectively. The lower triangular matrices are denoted $B^-$ and $B^- = w_0 B w_0$.
These facts are worked out as an example in most texts on algebraic groups. In fact, the notion of Borel subgroup has a far-reaching generalization to reductive algebraic groups.