What are these matrices called?

A paper I'm writing heavily uses block diagonal matrices with the property that each block is upper triangular and constant along its diagonal. Like this: $$\begin{bmatrix} A_1 \\ & \ddots \\ && A_k \end{bmatrix},$$ where each $$A_i$$ is of the form $$\begin{bmatrix} \lambda_i && * \\ & \ddots \\ && \lambda_i \end{bmatrix}$$ for some scalar $$\lambda_i$$. For instance, every Jordan matrix has this form. Is there a standard name for matrices of this type?

(Adding an "operator algebras" tag because that is the setting in which these matrices seem interesting. The unital matrix algebra generated by a matrix of this type only contains matrices of this type, for example.)

• Would you be OK if I edited to change your example, which breaks on my comparatively narrow viewing window, to the less demanding "$\begin{pmatrix} A_1 \\ & \ddots \\ && A_k \end{pmatrix}$, where each $A_i$ is of the form $\begin{pmatrix} \lambda_i & & * \\ & \ddots \\ && \lambda_i \end{pmatrix}$"? – LSpice Jan 11 at 17:14
• Of course it's not a name, but these matrices are precisely those that lie in the unipotent radical of the 'standard' Borel subgroup of the adjoint quotient of a 'standard' Levi subgroup of $\operatorname{GL}_n$. ('Standard' is in scare quotes because it doesn't mean anything on its own, only relative to a maximal torus; I mean it with respect to the 'standard' maximal torus, consisting of diagonal matrices.) – LSpice Jan 11 at 17:15
• Sure to the edit. And thank you for the other comment as well. – Nik Weaver Jan 11 at 17:21