# Eigenvector of a diagonal matrix times a rank-1 matrix

Is there a simple expression for the eigenvector of $D \mathbf a \mathbf a^H$ ?

Assuming $\lVert \mathbf a \rVert = 1$ and $D$ is positive definite and diagonal.

By inspection, the vector $x = Da$ satisfies $Daa^Hx = Daa^HDa = (a^HDa)Da = (a^HDa)x$.
Hence, $Da$ is an eigenvector with eigenvalue $a^HDa$.
The other $\text{dim}(a)-1$ eigenvalues will be $0$, and the corresponding eigenvectors are any vectors orthogonal to $a$.