# Questions tagged [lu-decomposition]

Questions regarding the numerical method LU decomposition to decompose a matrix into the multiplication of two triangular matrices: A lower triangle matrix and an upper triangular matrix

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### Can a matrix not admitting an LU decomposition be shifted so that it does admit an LU decomposition?

Almost all square matrices have an (unpivoted) LU decomposition, but some don't. The question I have is whether - assuming $M$ is a square matrix over $\mathbb R$ or $\mathbb C$ which does not admit ...
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### Does $M^2$ have an LU decomposition?

It's well known that matrices of the form $M^T M$ have an (unpivoted, of course) LU decomposition. In fact, because they are positive semidefinite, they have a Cholesky decomposition. The field here ...
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### generalization of LU decomposition?

I've just begun studying numerical approaches to LU decomposition and it got me thinking. Is there a more "general" (not sure if this is the right term for what I'm describing) form of LU ...
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### How to find a solution other than the vector $0$ of Linear System $AX=b$ with $b$ belonging to $0$

Hello I am currently trying to find the solution of a spring system without condition on border, so naturally my vector $b \in 0_{M\{n,1\}}$, I am resolving this system with an LU algorithm but ...
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### Understanding the proof the iterative improvement method for the result of linear equation solving using LU decomposition in numerical recipes

The question is from section 2.5 Iterative Improvement of a Solution to Linear Equations in Numerical Recipes book. When we solve $\mathbf{Ax = b}$ using LU decomposition numerically, the result is ...
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### Proving by induction that $L = I + \sum_{ k=1}^{n-1} u ^{(k)} e ^T _k$

Let $L = \prod_{k=1}^{n-1}(I + u^{(k)}e^T_k)$ where $u^{(k)} (i) = 0$ for $i = 1 : k$. Prove that $L = I + \sum_{k=1}^{n-1}u^{(k)} e^T_k$ by induction. (where $L$ is the lower triangular matrix of ...
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### Unit lower triangular matrices multiplication

We know that product of two unit lower triangular matrices is a unit lower triangular matrix. However, if product of two lower triangular matrices is unit lower triangular then is it necessary for the ...
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### Counter example of LU decomposition uniqueness

LU decomposition Theorem: If $A \in \mathbb{R}^{n \times n}$ is such that each principal minor $A_k$ has $det(A_k) \neq 0, \, k = 1, 2, \dots, n-1$, then $A = LU$, beeing $L$ a lower triangular unit ...
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Find the permutation matrix $P$, the lower triangular matrix $L$ and the upper triangular matrix $U$ such that $$PA=LU$$ Given  A= \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ -2 ...