positive definite matrix

A positive definite matrix is a symmetric matrix whose eigenvalues are all strictly positive. It is the matrix world's version of a positive number: just as a positive number always scales things in the same direction, a positive definite matrix never flips a vector's sense in a way that makes the 'energy' x^T*A*x go negative or zero.

That energy expression is the heart of the intuition. For any nonzero vector x, you get x^T*A*x > 0 always. Picture a bowl-shaped landscape (a convex paraboloid) with a single lowest point: that shape is exactly what a positive definite matrix describes, which is why these matrices are the friendly case in optimization, where you want a unique minimum to slide down to.

They also show up as covariance matrices in statistics (variances cannot be negative) and let you use fast, stable methods like the Cholesky factorization. A close cousin, positive semi-definite, allows zero eigenvalues, giving x^T*A*x >= 0 with the bowl possibly flat in some directions.