In my notes for my Linear Algebra class I wrote the following:
Matrix A is positive definite if $A=LU$ with positive pivots
where L and U are lower and Upper triangular matrices.
Now, it's unclear to me whether this means $A$ has positive pivots or $LU$ has positive pivots, and the textbook is equally ambiguous. All the definitions I found online use eigenvalues, which I have not learned about yet, or they talking about multiplying by certain vectors.
Can someone clarify the definition I wrote down?
They are talking about the pivots (i.e. diagonal entries) of $U$ -- note that if you have $A$ in echelon form, its pivots may switch from positive to negative by simply rescaling the rows, so it's not meaningful to talk about the sign of the pivots of a general matrix. On the other hand, it is meaningful to talk about the sign of the diagonal elements of a matrix.
Every symmetric positive definite matrix $A$ has a Cholesky factorization, which basically states that it has an LU decomposition where $U=L^T$ and the diagonal entries (pivots of $U$) are positive.
Actually your starting point is rather interesting, because we usually define a (symmetric) positive definite matrix $A$ by saying that for every nonzero $x \in \mathbb{R}^n$, we have $x^T A x > 0$. Then we prove things about it like the existence of the Cholesky factorization. Your definition seems to be for general (possibly nonsymmetric) matrices. You should nail down that ambiguity first.