Is the group $Lx+b$ amenable where $L$ is Cholesky?

Question

Let $L$ be any real lower triangular matrix with positive diagonal entries (a Cholesky matrix). Let $x$ and $b$ be real vectors. Is the group of actions $(L, b)$ on $x$, $$L x + b$$ amenable?

Answer 1

Yes.

Scaling $Lx$: The group of invertible lower triangular matrices under matrix multiplication is a Borel subgroup of the general linear group. This implies that it is solvable. The group of lower triangular matrices with positive diagonal entries is therefore also solvable since subgroups of solvable groups are solvable.

Shifting $x+b$: The group of vectors under addition is solvable since it is abelian.

The group of actions $Lx+b$ on $x$ is the semidirect product of the shifting and scaling groups. The semidirect product of solvable groups is solvable. Every solvable group is amenable.