Sample covariance matrix

The sample covariance matrix is a tool that summarizes how multiple variables change together based on observed data. Each element in the matrix shows whether two variables tend to increase or decrease together. For example, if one variable goes up when another does, their covariance is positive; if one goes up while the other goes down, it's negative. The matrix helps identify relationships among variables, such as in finance for understanding how different asset returns move relative to each other. It provides a compact way to understand the interconnected variability of several data features simultaneously.