How does multicollinearity affect the eigenvalues of a matrix? I have been looking into ridge regression as a method to address multicollinearity in data.
I am aware that multicollinearity can cause high variance in coefficient estimates. I have seen equations such as this:
$var(\hat{\beta}) = \sigma^2(X'X)^{-1}$
I have read that when perfect multicollinearity is present, the matrix is singular and hence no inverse exists. When multicollinearity is present (but not perfect multicollinearity) than the matrix becomes ill-conditioned. This apparently causes the $(X'X)^{-1}$ term to become very large, inflating the variance of $\beta$. 
Seeing as the condition score of a matrix is the ratio is $ \sqrt{\frac{\lambda_{max}}{\lambda_{min}}}$ this suggests that multicollinearity causes a larger difference between the eigenvalues of $X'X$.
Based on the above i have 2 questions:
1) Why, when $X'X$ is ill-conditioned, does $(X'X)^{-1}$ become very large?
2) Please can you explain how multicollinearity cause the eigenvalues of X'X to change, as well as why 
  there is a greater difference in their magnitudes between eachother? 
 A: *

*Because the inverse of a small number is large. The inverse of a Grammian matrix
$K = Q\Lambda Q^T$ where $Q$ is the eigenvectors matrix and $\Lambda$ the eigenvalue matrix, is effectively the $K^{-1} = Q\Lambda^{-1} Q^T$. As such when we inverse a very small eigenvalue from the diagonal matrix $\Lambda$, we get a very large number in the inverse of it as well as subsequently on the $K^{-1}$. Wikipedia commonly is great for such topics so checking the section: Matrix inverse via eigendecomposition is a good first step to get some further background.

*The multicollinearity is caused by a linear dependence between between the columns of $X$. In that sense we already had a problem with $X$ just this was highlighted in $X^TX$. Note that by taking $X^TX$ we are squaring its respective eigenvalues (if $X$ was square) or its respective singular values (in a more general case); the square of 0 is still 0 and the square of a number less than 1 is something even smaller. 

*For the multicollinearity itself: it means that despite having a $p$
dimensional data ($p$ being our number of features), the data in our
design matrix contains enough information for $q < p$ dimensions.
For example, think of us having both imperial (pounds) and metric
(kilograms) weight measurements; realistically we have information
across a single dimension (weight), not two. Because we only have
variance across a single dimension, the variance across the second dimension is  zero. As that variance maps directly to the eigenvalues, then we get
that zero-th (or very small) eigenvalue. (That is only natural as the eigenvalues of a $X^TX$ matrix are the variances of its of the matrix independent coordinate. Unless you have already read it CV.SE has an epic thread on the matter here: Making sense of principal component analysis, eigenvectors & eigenvalues to assist your understanding of eigenvectors and eigenvalues.) 

