Is there a name for symmetric matrices with exactly one non-zero eigenvalue which also equals the trace?

Question

Specifically, matrices of this form.

In 2 dimensions:

\begin{bmatrix} a^2 & ab\\ ab & b^2 \end{bmatrix}

In 3 dimensions:

\begin{bmatrix} a^2 & ab & ac\\ ab & b^2 & bc \\ ac & bc & c^2 \\ \end{bmatrix}

Thank you!

Answer 1

just called rank one symmetric. There is a nonzero column vector $v$ so that your matrix is $$ v v^T $$ For any column vector $w$ orthogonal to $v,$ we get $v v^T w = v(v^Tw) = 0$ while the nonzero eigenvalue comes from $v v^T v = v |v|^2,$ and the nonzero eigenvalue is $|v|^2 = v \cdot v$