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Can I find and use $U$ such that

$$A = U U^{T}$$

where $U$ is an upper triangular matrix, to find a solution instead of finding $L$ such that $ A = L L^{T}$ (where $L$ is a lower triangular matrix) to solve $Ax=b$ using Cholesky factorization? If not, what is the correct way of using Cholesky factorization?

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migrated from mathoverflow.net Dec 1 '16 at 20:54

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  • $\begingroup$ Maybe I'm missing something, but I'm wondering who gave this a +1? $\endgroup$ – suvrit Dec 1 '16 at 17:46
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    $\begingroup$ Reverse the order of rows and columns, and you turn a lower triangular factorization to an upper... $\endgroup$ – Robert Israel Dec 1 '16 at 19:08
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You can use $A=UU^T$ to solve linear equations. However it is not what people call Cholesky decomposition. Also the algorithm is less intuitive.

Note: Both decompositions are equivalent. Let $P$ the permutation which reverses the order, which is symmetric. Then, $\tilde A = PAP$ is symmetric and positive definite iff $A$ is. In particular we have the cholesky decomposition $$ \tilde A = PAP = (PUP)(PUP)^T. $$

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  • $\begingroup$ Is it possible to apply proposed approach in order to find a matrix $B$ such that: $A=BB^T$? where $B$ is the positive definite matrix? $\endgroup$ – Nick Dec 6 '16 at 8:25
  • $\begingroup$ @Nick I don't think so. Is $B$ supposed to symmetric? For the symmetric root, you could consider the eigen value decomposition of $A$. $\endgroup$ – user251257 Dec 6 '16 at 8:36
  • $\begingroup$ I have only one assumption on $B$, it's should be the positive definite matrix. $\endgroup$ – Nick Dec 6 '16 at 8:40
  • $\begingroup$ How do you get the Cholesky factorization from $A=UU^T$? $\endgroup$ – Ion Sme Jun 11 '18 at 4:15

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