This chapter discusses eigenvalues and eigenvectors with an emphasis on their application to covariance matrices and principal component analysis (PCA). First, we discuss the fundamentals and properties of eigenvalues and eigenvectors and then develop the eigendecomposition in the context of square, symmetric matrices. We discuss eigenvector bases for the nullspace and range of a square, symmetric matrix. We then change gears and discuss the concepts of correlation, covariance, covariance matrices per se, and covariance matrices of stationary random processes. The above topics are then combined into principal component analysis (PCA). We present an application of eigen-analysis from the field of array signal processing and three separate examples demonstrating the effectiveness of the PCA method. A major aim of this presentation is to de-mystify the concepts of eigenvalues and eigenvectors by presenting several examples and interpretations that are relevant to the field of signal processing.