The key of Gauss-Jordan Elimination is to transfer matrix A to the identity matrix:
$Ax=b$
If A is non-singular, then:
Deﬁnition: Let $A \in \mathbb{R}^{n \times n}$. Matrix A is said to be symmetric positive definite(SPD) if
Consider the quadratic polynomial $$p(\chi) = \alpha \chi^2 + \beta \chi + \gamma = \chi \alpha \chi + \beta \chi + \gamma$$
The graph of this function is a parabola that is “concaved up” if $\alpha > 0$. In that case, it attains a minimum at a unique value $\chi$.
Now consider the vector function $f: \mathbb{R}^n \to \mathbb{R}$ given by $$f(x) = x^T A x + b^T x + \gamma$$ where $A \in \mathbb{R}^{n \times n}, b \in \mathbb{R}^n, \text{ and } \gamma \in \mathbb{R}$ are all given. If A is a SPD matrix, then this equation is minimized for a unique vector x. If $n = 2$, plotting this function when A is SPD yields a paraboloid that is concaved up: