# Why are positive definite matrices useful?

I've recently been learning about how to tell if a matrix is positive definite and how to create a positive definite matrix, but I haven't been given a reason why they're useful yet. I'm sure there are plenty of reasons, I just haven't seen them yet. In what ways do the properties of a positive definite matrix make them advantageous to have? Thanks for your time!

One reason is that if a matrix A is positive definite, the quadratic form

$$f(x) = \frac{1}{2} x A^T x + b^Tx + c$$

has a unique minimum (expressions like these crop up in a number of places). A positive definite matrix A can be visualized as a paraboloid (look at the graph of f) that is stretched in the directions of A's eigenvectors. If A is indefinite, the graph will have a saddle point instead of a nice minimum (or be degenerated further).

An article that explains this (and some other linear algebra key ideas) nicely is "Introduction to the Conjugate Gradient Method Without the Agonizing Pain" by JR Shewchuck.

AlephZero