Positive Definite Matrices

A positive definite matrix is a special kind of square matrix that, when used in certain calculations, always produces positive results unless the input vector is zero. Specifically, for any non-zero vector, multiplying it by the matrix and then taking the dot product of the result with the original vector yields a positive number. This property ensures the matrix is associated with concepts like stability and meaningful measurements, often appearing in optimization, physics, and machine learning. In simple terms, positive definite matrices help us model systems where things behave "positively" or "well-behaved" mathematically.