Can you explain the concept of Principal Component Analysis (PCA) in dimensionality reduction? Describe how PCA works to transform high-dimensional data into a lower-dimensional space while retaining most of its variance. Discuss the steps involved in PCA, including covariance matrix computation, eigendecomposition, and the selection of principal components. Additionally, highlight the applications of PCA in feature extraction, noise reduction, and visualization of high-dimensional data. Could you also discuss any limitations or scenarios where PCA might not be suitable or might encounter challenges in effectively capturing the data's underlying structure?

Principal Component Analysis (PCA) is a powerful technique used for dimensionality reduction, primarily employed to simplify high-dimensional data while preserving its essential structure.

Understanding PCA

PCA aims to transform a dataset with possibly correlated variables into a set...