We have a matrix $A$ and $A'$ is a submatrix of $A$. first we make LU decomposition of $A$, $$A = L_a \cdot U_a.$$ Now, I want to make LU decomposition of $A'$, $$A' = L_a' \cdot U_a'.$$ is there any relation between $L_a, U_a, L_a',U_a'$? Maybe some similar operations between the two LU decompositions?
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1Is $A'$ just an arbitrary submatrix? Or is it the leading submatrix? – 5xum Feb 14 '14 at 10:14
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A' is an arbitrary submatrix of A. – lhc1988 Feb 14 '14 at 15:14
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For an arbitrary submatrix of $A$, the $LU$ decomposition may not exist. This is because $LU$ decompositions only exist for invertible matrices, and arbitrary submatrices of a matrix need not be invertible. Therefore, a general formula will not be possible.
5xum
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Could you please give me some reference of this? I've a very similar problem and any reference might be useful – SSC Napoli Dec 02 '14 at 11:20
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I will have a look thank you, anyway my problem is summarized in the following question: https://math.stackexchange.com/questions/1047983/lu-decomposition-for-the-solution-of-two-linear-systems#1047983 Maybe you want to have a look at it – SSC Napoli Dec 02 '14 at 11:53