is there a theory for real positive (semi-)definite matrices? Here, a matrix $A\in\mathbb R^{n\times n}$ is called positive definite (resp. positive semidefinite) if $x^\top A x > 0$ (resp. $x^\top A x\geq 0$, for all $x\in\mathbb R^n\setminus\{0\}$. Note that $A$ needs not be symmetric.
For instance, it is known that a positive definite matrix is a $P$-matrix (i.e., all its principal minors are positive), cf. Generating method 4.2 in "Generating and detecting matrices with positive principal minors", Tsatsomeros, 2002. However, all $P$-matrices are not necessarily positive definite.
My question: Is there an existing theory for positive definite matrices? Such as Cholesky-like decompositions, matrix inequalities, etc.
Thanks!