Ridge Linear Regression Estimation Invertible

Reference: https://math.stackexchange.com/questions/2447060/prove-that-the-regularization-term-in-rls-makes-the-matrix-invertible

$\hat{\theta} = (X^TX + \lambda I)^{-1}Xy$
For any vector $v \in \mathbb{R}^d$, we have $v^TX^TXv = (Xv)^T(Xv) >= 0$, so it is a positive semi-definite. Also, we know $\lambda >0$ so that $\lambda I$ is already a positive definite matrix.
Therefore,
$v^T(X^TX + \lambda I)v=(Xv)^T(Xv) + \lambda v^TIv >= v^T(\lambda)v>0$
is a positve definite and hence invertible.