Given $\mathbf v=\begin{bmatrix}v_1\\v_2\\\vdots\\v_n\end{bmatrix}$ and $\mathbf A=\mathbf{vv}^\top$, find the eigenvectors and eigenvalues of $\mathbf A+\lambda\mathbf I$.
My current work progress is:
Since $(\mathbf A+\lambda\mathbf I)\mathbf v =\mathbf{Av}+\lambda\mathbf v=\mathbf v(\mathbf v^\top\mathbf v) + \lambda\mathbf v=\mathbf v \|\mathbf v\|+ \lambda\mathbf v=(\|\mathbf v\|+\lambda)\mathbf v$ ,
so one of the eigenvalue is $\lambda_1=(\|\mathbf v\|+\lambda)$ with corresponding eigenvector $\mathbf v_1=\mathbf v$.
I'd like to find out other pairs of eigenvalues and eigenvectors. I first calculate the trace.
$$\mathrm{Tr}\,(\mathbf A)=n\lambda+\|\mathbf v\|=\lambda_1+(n-1)\lambda$$
Hence, the rest of $n-1$ eigenvalues has sum $(n-1)\lambda$.
From here, I do not know how to proceed.
- Can I assume the rest of eigenvalues are $\lambda$ with multiplicities $n-1$? Why?
- How to find corresponding eigenvectors?
[Note1] This is like a follow up question regarding this one.
[Note2] Found later on there is a question but people focus on linking it to Note1.