What is the geometrical meaning of matrix $\mathbf{A}\in\mathbb{R}^{n\times n}$ in the form \begin{align*} \begin{bmatrix} 1 & 0 & 0 & \cdots & 0 \\ a_{21} & a_{22} & a_{23} & \cdots & a_{2n} \\ \vdots & & & \ddots & \\ a_{n1} & & \cdots & & a_{nn} \end{bmatrix} \end{align*} $a_{ij}\in\mathbb{R}$.
Is it some type of an affine transform?
Well it means that the first unit vector $e_1=(1,0,...,0)$ is an eigenvector of the matrix to the eigenvalue 1. In other words, when you consider the action of the matrix on the vector space, the subspace spanned by $e_1$ remains precisely the same. If it would be a 2 instead, this subspace would be scaled by a factor of 2...