Skew-triangular (?) matrices and their properties

Question

By skew-triangular matrices, I mean matrices with the following sparsity patterns.

$$\begin{bmatrix} \times & \times & \times & \times \\ \times & \times & \times \\ \times & \times \\ \times \\ \end{bmatrix} \qquad \text{or}\qquad \begin{bmatrix} & & & \times \\ & & \times & \times \\ & \times & \times & \times \\ \times & \times & \times & \times \\ \end{bmatrix}$$

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?

I am asking this just out of curiosity because a brief googling failed to give me the answer.

Answer 1

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.

Answer 2

I was wondering this myself because I noticed that every "antitriangular" or "skew triangular" or "peritriangular" matrix can be converted to a normal triangular matrix by simply swapping the first and last rows, 2nd to first and 2nd to last, etc and so their determinants would be equal in magnitude (but would change in sign if an odd number of rows were swapped). So in other words, what i have discovered is that the determinant of an antitriangular matrix is equal to the absolute value of the product of its main "antidiagonal" entries.

Also if it is an n by n matrix, then if n is even, the determinant will have a minus sign if n/2 is odd and have a plus sign if n/2 is even for my above mentioned formula (getting rid of the absolute value sign). And if n is instead odd, then the determinant will have a minus sign if (n-1)/2 is odd and have a plus sign if (n-1)/2 is even.

What is this significance? I think it would save time in computing the determinant if we noticed it was an "antitriangular" matrix. It wouldn't save that much time though unless the matrix was incredibly large and required a great many amount of rows to get swapped to convert it into a normal triangular matrix.

I am using of course some theories of determinants which I didn't explain and trust the readers would know or be able to find out.