Is there a name for this type of “almost diagonal” matrix

Question

I have a symmetric matrix $A\in\Bbb{R}^{n\times n}$, for which the only $a_{ij}\neq0$ have $i=1$, $j=1$ or $i=j$. That is: $$A=\begin{pmatrix}d_1 & a_2 & a_3 & \cdots & a_n\\a_2 & d_2 & 0 & \cdots & 0\\a_3 & 0 & d_3 & \cdots & 0\\\vdots & \vdots & \vdots & \ddots & \vdots\\a_n & 0 & 0 & \cdots & d_n\end{pmatrix}\,.$$ Is there a name for this class of real-valued matrices (or the obvious extension to Hermitian matrices)? The closest I've found in my search is the class of Frobenius matrices, mentioned in an answer to this question, but they aren't symmetric.

Answer 1

Yes: it is called an arrowhead matrix (for obvious, appearance-related reasons).

Edit: what you specifically want is a symmetric arrowhead matrix.