I computed a (the?) Cholesky factorization of a 3x3 real, symmetric matrix. No problem, just a pretty standard computation.
However part (b) asks how could I determine that the original matrix $A$ is positive-definite, by inspecting individual entries of $L$, where $A= LL^t$.
Well, both factors, $L$ and $L^t$, that I computed have positive diagonal entries; of course, taking the transpose keeps the diagonal fixed. So, I know that both factors have the same, positive eigenvalues, counting multiplicity, etc.
Is this enough to conclude that $A$ is positive-definite? I.e., is the product of two positive-definite matrices again positive definite? Basically I can't say 100%, because, although I know, by the multiplicativity of the determinant, $A$ will have positive determinant, but I feel like it's possible for $A$ to have pairs of negative eigenvalues, too.
Unless...the spectrum of $L$ (and of $L^t$) somehow also determines the spectrum of $A$? Then we could say for sure, that by inspecting the factors $L$ and $L^t$, we could conclude that $A$ is positive-definite.
What do you think?
Thanks,