I am trying to internalize this video and perhaps work an example or two. I understand the professor's point that quadratic form minimization can be used to solve Ax = b from linear algebra, using gradient descent from calculus (seemingly an exercise for computers moreso than humans), but does it work the other way as well, allowing would-be calculus optimization problems to be solved via linear algebra?

For example, I'm imagining taking a given quadratic equation such as z = x^2 + 2x + 3y^2 -xy + 9, and solving it by first rewriting the RHS in the form (1/2)[x, y]A[x, y]^T - [x,y]b + c, where A is a matrix, and b and c are vectors of constants, and then solving A[x y]^T = b for [x y]^T where the resulting x and y are respectively the x- and y-coordinates of the minimum that would have been obtained using optimization from MV calculus. Then the z-coordinate could be found trivially. Is that the idea?

I haven't found any such exercises online, and I don't know how to make up a quadratic equation which yields a positive definite A, which I understand to be essential for quadratic form minimization. Any help here would be appreciated.