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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?

Zachary F
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  • It's unclear to me why it is that you consider those statements to be distinct. What's the difference, as you see it? – Ben Grossmann Sep 10 '16 at 23:24
  • It could be that A has positive pivots but L and U don’t have them until you multiply them, or L and U each have them but when you multiply they go away, somehow. I’ll admit I didn’t know much about matrices at this point – Zachary F Feb 17 '22 at 18:05
  • I like topics about positive definite matrices. So I googled your note at searchonmath.com/, and found this StachExchange topic that might help this topic. – José Claudinei Ferreira Feb 17 '22 at 17:58

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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.

zeno
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