A positive definite matrix is a special type of square matrix that has important properties in mathematics and statistics. It is defined such that for any non-zero vector, the product of the vector and the matrix, followed by the vector's transpose, results in a positive value. This means that the matrix always produces positive outcomes when applied to certain vectors. Positive definite matrices are commonly used in various fields, including optimization, machine learning, and statistics. They ensure that certain mathematical functions, like quadratic forms, have a unique minimum, making them essential for solving problems that require stability and reliability in their solutions.