# 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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### Matlab. LU decomposition Crout's Method. And usage of decomposition on accurate Block Matrix.

I have to implement such a program()Look at picture I attached I mostly implemented everything: Crout's Algorithm, solving linear equations, I created this block matrix, but I don't know to use that ...
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### Block matrices in MATLAB. [closed]

I am completely new to the matlab programming. Could please tell me algorithm/implementation of the block matrix M(PICTURE). How to code it in Matlab? Thank you.[This Block Matrix M ][1] Block Matrix ...
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### Matlab.Block matrices

I have implemented algorithm of Crouts method. But I don't have any idea how to create this M function in Matlab and implement in my algorithm .Please help me. CODE of algorithm: function [L,U] = ...
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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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### Why A is required to be invertible in $PA=LU$ decomposition

my quetsion is on $PLU \ decomposition$ of matrix and is from Introduction to Linear Algebra, $5^{th}$ edition by Gilbert Strang. In the chapter 2.7 Transposes and Permutations, it's said: If $A$ is ...
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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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### Is $A$ ill conditioned matrix?

Suppose we have a matrix $A$ with is its $LU$-decomposition such that $A=LU$ and suppose that $U$ is ill conditioned ($\left \| U \right \|\left \| U^{-1} \right \|$ is large) , does it mean that $A$ ...
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### Is the permutation matrix P of PLU decomposition unique?

Let $A$ be a square matrix. Then there exists a permutation matrix $P$ such that $A=PLU$, where $L$ is a lower triangular matrix and $U$ is an upper triangular matrix. To further ensure the uniqueness,...
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### $PA = LU$ decomposition

Consider a matrix $A= \begin{pmatrix} 1 & 2 & 1\\ 3 & 6 & 1\\ 0 & 4 & 1 \end{pmatrix}$ I am applying the transformations on matrix $A$ to convert it to $U$ using the ...
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### Solve many linear equations of similar structure

Given G: real and symmetric square matrix v: real column vector I need to solve n linear systems of the form \begin{align} A = \begin{pmatrix} G & v \\\ v^T & 0 \end{pmatrix}\end{align} \...
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### Is a symmetric matrix positive definite iff $D$ in its LDU decomposition is positive definite?

Given $$A=LDU$$ where $A$ is a real symmetric matrix $L$ is a lower unitriangular matrix $D$ is a diagonal matrix $U$ is an upper unitriangular matrix can we say that $$A>0 \iff D>0$$ ? Edit:...
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### Determining whether a matrix is positive definite from its LU decomposition

Given that $A=LU$ where $L$ and $U$ are (known) lower and upper triangular matrices, is there any simple way to determine whether $A$ is positive definite? Background I have been using this algorithm ...
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### Given A=LU factorization, prove that the basis of column space A is the columns of L that correspond to the pivot columns of U

I understand that the basis of column space A is just the columns of A that correspond to the pivot columns of U. This is because U is just the reduced row echelon form. However, as mentioned in the ...
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### Finding a matrix that can be represented with only single LU decomposition

I'm trying to disprove the following statement: Let $M$ be a singular matrix $3\times 3$ that can be represented with LU decomposition ($M=LU$), then the decomposition is unique (only one ...
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### $PA=LU$ decomposition for the special matrix

The cost of decomposition $LU=PA$ for the matrix $A_{N\times N}$ is $O(N^3)$. However if we know about some special properties of matrix $A$ then we can reduce this cost but I wonder how to do it. In ...
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### Showing $LU$ is impossible... [closed]

Show that $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}=LU$ is impossible where $L$ is lower triangular and $U$ is upper triangular.
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### How to prove that LU decomposition is unique?

given the following matrix how could I prove that LU decomposition of it is unique? A= 1 3 1 2 9 2 1 3 1 L= 1 0 0 2 1 0 1 0 1 U= 1 3 1 0 3 0 0 0 0
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### How can I find an LU factorisation of this $3 \times 3$ matrix?
$$A=\begin{bmatrix}1&2&-3\\-2&-4&8\\-3&-4&14\end{bmatrix}$$ This is what I found: $$U=\begin{bmatrix}1&2&-3\\0&0&2\\0&0&8\end{bmatrix}$$ L=\begin{...