Eigenvalues and eigenvectors of $I+uv^T$

Question

Let u and v be two non-zero vectors in $R^n$. Let A=I+u$v^T$.

First we show that u$v^T$ is symmetric iff u=cu.

Second question: Assume $v^T$u=-1, Find the spectrum of A=I+u$v^T$ in addition to its corresponding eigenvectors and conclude that the matrix A is not invertible.

How to solve the second question?

I started with (I+u$v^T$)(I-u$v^T$)=I+u$v^T$ after using the assumption..

Answer 1

What is the spectrum of $uv^T$? Study its column space. Study its null space.

Then spectrum of $I+uv^T$ is that of $uv^T$ plus one.

Answer 2

First of all note that $u$ is an eigenvector of $A$ $$Au=(I+uv^T)u=(1+v^Tu)u$$ Now think about what happens when $v^Tu=-1$.

Note. Other eigenvectors are the vectors that are orthogonal to $v$ with eigenvalue $1$.