A covariance matrix is a mathematical tool used in statistics to measure how different variables change together. It provides a summary of the relationships between multiple variables, indicating whether they tend to increase or decrease together. Each element in the matrix represents the covariance between a pair of variables, helping to understand their joint variability. In a covariance matrix, the diagonal elements represent the variances of each variable, while the off-diagonal elements show the covariances between pairs of variables. This matrix is essential in various fields, including finance, machine learning, and data analysis, as it helps in understanding the structure of data and making informed decisions.