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How do I calculate $A=UL$ factorization where $U$ is upper triangular matrix with 1's along the diagonal and $L$ is lower triangular matrix?

How is this similar to the $LU$ factorization?

hardmath
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Bob
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    Start at the bottom and work your way up? The last row of $U$ has to be $(0,0,\ldots,1)$, so the last row of $L$ is.... the same as the last row of $A$. What can you do with the last column of $L$? – hardmath Jul 24 '15 at 06:00

1 Answers1

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If you already know how to get an $LU$ factorization, then one approach to getting your $UL$ factorization is by similarity transformation.

Let $B = PAP$ where $P$ is the permutation matrix with 1's on the anti-diagonal and 0's elsewhere. Thus $P = P^T = P^{-1}$, and $B$ is orthogonally similar to $A$.

If $B = LU$ is a factorization with lower triangular $L$ having 1's along the diagonal, and $U$ an upper triangular matrix, then by similarity:

$$ A = (PLP^T)(PUP^T) $$

Note that $PLP^T$ is an upper triangular matrix with 1's along the diagonal, and $PUP^T$ is a lower triangular matrix, so the above is a factorization of the desired form.

hardmath
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