How to prove that $I+A^{T}A$ is invertible 
Let $A$ be any $m\times n$ matrix and $I$ be the $n\times n$ identity.
  Prove that $I+A^{T}A$ is invertible.


The matrix $I+A^TA$ is invertible because it is positive definite: if $v\in\mathbb{R}^n$ is nonzero, then
$$
v^T(I+A^TA)v=v^Tv+v^TA^TAv=|v|^2+|Av|^2\geq |v|^2>0.
$$
In particular, all the eigenvalues are positive and their product must too be positive (i.e. nonzero).
