A popular krylov subspace method for solving linear systems of equations, particularly those that exhibit symmetric positive definiteness.

Assume a system of equations of the form:


where $A$ is a given positive definite matrix, $b$ is the given right-hand side, and $x$ is unknown.

The Conjugate Gradient method exploits the idea that we can write the solution x as a linear combination of mutually $A$-conjugate basis vectors. This allows us to exploit a scheme similar to gradient descent, but with $A$-conjugate vectors instead of residual vectors.