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Ridge Regression Eigenvalue Shift

Medium · Linear Algebra & Machine Learning · Quant Trader interview question · ridge-regression, eigenvalues, linear-algebra, machine-learning

In ridge regression, the ordinary least squares (OLS) estimate is modified by adding a penalty term, resulting in the estimator: $ (X^T X + \lambda I)^{-1} X^T Y $ where $X$ is the feature matrix, $Y$ is the target variable, and $ \lambda $ is the regularization parameter. Consider a scenario where you're tuning a ridge regression model. How does increasing $ \lambda $ affect the eigenvalues of the matrix $X^T X$?