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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.

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

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

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  • $\begingroup$ Argh! beat me to it by 16 seconds. $\endgroup$
    – Drew Brady
    Mar 1, 2022 at 3:23
  • $\begingroup$ That's not a lot of seconds :) $\endgroup$
    – Clement C.
    Mar 1, 2022 at 3:23
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What you are discussing is referred to as an Arrowhead matrix, or sometimes just "arrow" matrix. These are matrices with zeros everywhere except the first row, column and main diagonal.

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