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.)