Q:

Eigenvalues of the following matrix

I have to find the eigenvalues of the matrix
$$
I+xx^T/||x||^2
$$
With x being a column vector of size n and I an identity matrix of size nxn. The eigenvalues are supposed to be easily found and be $\lambda_n= 2$ and $\lambda_1=1$, with the rest of eigenvalues inside that range
Any help would be highly appreciated

A:

Note that $xx^T/||x||^2$ is a matrix with rank $1$, so there is exactly a non-zero eigen-value and zero eigen-values with multiciplity $n-1$. Since $(xx^T/||x||^2)x=x$ then the non-zero eigen value of $xx^T/||x||^2$ is $1$. Moreover, since for any matrix $A$, $\lambda$ is eigen value of $A$ iff $1+\lambda$ is eigen value of $I+A$ then the eigen values of $I+xx^T/||x||^2$ are $2$ with multiplicity $1$ and $1$ with multiplicity $n-1$.

