covariance matrices

A covariance matrix is a table that shows how different variables change together. Each number in the matrix measures how two variables vary in relation to each other—positive values mean they tend to increase or decrease together, negative values indicate they move inversely, and zero suggests no relationship. The diagonal elements show how each variable varies on its own. Covariance matrices are useful in fields like statistics and machine learning to understand the relationships and spread among multiple variables simultaneously, aiding in data analysis and decision-making.