Why people make and use LDU factorization?

I think LU factorization and PA = LU are enough to solve equation.

Anyone know why?


Expanding on what J W linked, let the matrix be positive definite be such that it can be represented as a Cholesky decomposition, $A = LL^{-1}$, we can determine that $L_{11} = \sqrt{A_{11}}$

If we instead let $A = LDL^{-1}$, we can instead set the diagonal of $L$ to unity and let $D_{11} = A_{11}$

This produces the same result but allows us to avoid having to compute $\sqrt{A_{11}}$


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