Let $x$ be a non zero vector in the complex vector space $\mathbb C^n$ and $A=xx^H$.Find all the Eigenvalues and their Eigen spaces.

Question

Let $x$ be a non zero vector in the complex vector space $\mathbb C^n$ and $A=xx^H$.Find all the Eigenvalues and their Eigen spaces.[where $x^H=$ conjugate transpose of $x$]

Here rank of $A$ is $1$ and $A$ is a Hermitian matrix so all the Eigenvalues are real.since rank $<n $ it implies that at least one eigenvalue is $0$.Now considering the null space of $A$ it has dimension $(n-1)$.So algebraic multiplicity of $0$ is either $(n-1)$ or $n$

But I can not proceed further.Is there any other Eigenvalues and what are their Eigen spaces?

Answer 1

Note that $\lambda = x^{H}x$ is an eigenvalue associated to the eigenvector $x$. Next note that $A$ satisfies $f(t) = t^{2} - \lambda t$.

Answer 2

Hint: Can you show that $Ax$ is a non-zero constant multiple of $x$? If so, then $x$ is an eigenvector, and that non-zero constant is the corresponding eigenvalue.