When is lu decomposition possible

Not all square matrices have an LU decomposition, and it may be necessary to permute the rows of a matrix before obtaining its LU factorization. Definition A matrix is said to have an LU decomposition if and only if there exist a lower triangular matrix and an upper triangular matrix such that. Example The matrix has the following LU factorization:.

Example The matrix can be written as. Proposition Not all square matrices have an LU factorization. It is sufficient to provide a single counter-example. Take the invertible matrix Suppose has an LU factorization with factors and Compute the product Now, implies which in turn implies that at least one of and must be zero.

As a consequence, at least one of and is not invertible because triangular matrices are invertible only if their diagonal entries are non-zero. This is in contradiction with the fact that is invertible and, as a consequence, and must be invertible see the proposition about the invertibility of products.

Thus, we have proved by contradiction that cannot have an LU decomposition. Proposition If a matrix has an LU decomposition, then it is not unique. Suppose a matrix has an LU decomposition Take any diagonal matrix whose diagonal entries are all non-zero.

Then, is invertible, its inverse is also diagonal and we can write A diagonal matrix is lower triangular, and the product of two lower triangular matrices is lower triangular. Therefore is lower triangular. The inverse , being diagonal, is upper triangular. Since the product of two upper triangular matrices is upper triangular, we have that is upper triangular. Thus, by changing the matrix , we are able to get infinite factorizations of into a lower triangular matrix and an upper triangular matrix.

We now provide some sufficient conditions for the existence of an LU decomposition of a matrix. Proposition Let be a matrix. But the product of L and U is not coming out to be equal to the orig matrix. What am I doing wrong? Subjects All categories General Aptitude 2. Follow gateoverflow. GATE Overflow. Recent Blog Comments I believe you are too much in respect of IITD has no institutional support regarding Arjun gatecse where has this link been moved Its updated Connect and share knowledge within a single location that is structured and easy to search.

Note: decomposition and factorization are equivalent in this article. From the Wikipedia article on LU decompositions :. The problem is this third statement here. This leads us back to the question: is there a way of truly knowing whether a matrix has an LU decomposition?

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