Given a sequence of matrices $A_{i}$ which we know converges to a matrix $A$, are we able to say anything about the convergence of the LU decompositions for the $A_{i}$? In other words, knowing that $A_i$ converges, can we say that $L_i$ and $U_i$ converge, where $L_iU_i = A_i$?
What can we say about the convergence of $LU$ given the convergence of $A$?
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linear-algebra
1 Answers
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The LU decomposition is not unique in general. If you assume, for example, that the element on the diagonal of L are all 1, then it is unique. The problem is that you can choose $A_i$ which have LU decompositions, but converge to some $A$ which doesn't have an LU decomposition. For example, take
$$\left(\begin{array}{cc} a & 1\\ 1 & 1 \end{array}\right)= \left(\begin{array}{cc} 1 & 0\\ \frac{1}{a} & 1 \end{array}\right)\cdot \left(\begin{array}{cc} a & 1\\ 0 & \frac{a-1}{a} \end{array}\right) $$ As $a\to 0$, both of the $L$ and $U$ matrices have entries that go to infinity.