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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LU decomposition with partial pivoting

Different sources* all introduce LU with partial pivoting with multiplying matrix $A$ by permutation matrices $P_i$ and matrices used in classical LU decomposition $L_i$ such that $$ L_nP_n \cdots L_1 ...
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What's the fastest way to solve a system of equations a million times such that the coefficient matrix is same but the constant matrix is different?

I need an efficient way to solve $Ax=C$ a million times such that coefficient matrix $A$ is always the same but the constant matrix $C$ is always different for each of the million problems. To solve ...
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For any matrix $A \in M_{n\times n}(\mathbb {R})$, there exists a permutation matrix $P$ such that $PA=LU$.

For any matrix $A \in M_{n\times n}(\mathbb {R})$, there exists a permutation matrix $P$ such that $PA=LU$.Here P is defined as a matrix resulting from any number of row interchanges in the $I_{n\...
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Why $L^{-1}U^T=D$ in LU decomposition?

I learned that given a matrix $A$, we can apply LU decomposition to get $A=LU$, where $L$ is lower triangular and $U$ is upper triangular. Further, if $A$ is symmetric (or Hermitian for complex $A$), ...
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Determine if LU decomposition is possible on a matrix?

I am trying to understand how you determine if LU decomposition is possible on a given matrix. I believe the way to calculate this is to check if the leading-matrices have non-zero determinants. I ...
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How to determine if result of "LU decomposition" is rank revealing factorisation (RRF)

I want to do some LU decompositions with dofferent kind of pivoting and at the end determine if the results of "LU decomposition" is rank revealing factorisation (RRF). which metrics should ...
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Determining a basis for the column space of A from its LU factorisation?

There are two ways of finding a basis for the column space of A from A = LU. The pivot columns in U correspond to the linearly independent columns of A, which in turn form a basis for the column ...
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Do the rows used in row operations during LU factorisation matter?

A method I have seen for finding the LU factorisation of a matrix is that U is the row echelon form of A. The row operations we perform on A to get to U must involve replacing $R_i$ by $R_i - kR_j$ ...
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Solving System of linear equation using LU decomposition

I am working on a simultaneous linear equation problem using LU decomposition and I'm unsure if this is the correct approach/answer to solve a system of simultaneous equations using LU decomposition. ...
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Solving System of linear equation containing decimals

I am working on a simultaneous linear equation problem using LU decomposition. $5x1 + 6x2 + 2.3x3 + 6x4 = 4$ $9x1 + 2x2 + 3.5x3 + 7x4 = 5$ $3.5x1 + 6x2 + 2x3 + 3x4 = 6.7$ $1.5x1 + 2x2 + 1.5x3 + 6x4 = ...
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Is my LU Factorization Incorrect?

I have the following matrix I am trying to decompose into it's respective $L$ and $U$ parts for $A = LU$. So I have $$\begin{bmatrix} 1 && 4 && 3 \\ 0 && -10 && -5 \\ 0 ...
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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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LU- decomposition

Find an LU-decomposition of the coefficient matrix and solve the system $$\begin{pmatrix} 1 & 4 & 3\\ -1 & -1 & 3 \\ 2 & 9 & 8 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \...
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Prove that $\exists S$ a product of permutation matrices, such that the principal sub-matrix $B_n$ of $B=SA$ is invertible.

Let A be an invertible matrix that has no LU decomposition (L lower triangular with unit diagonal, U upper triangular). We will prove by induction on the order n of A that there exists an invertible ...
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A question about the product of commutators in an article of Vaserstein

The question comes from Lemma 13. It is stated as follows. Let $A$ be an associative ring with $1$, and $n\geq2$ an integer. Assume that either $n\geq3$ or $n=2$, and $1$ is the sum of two units in $...
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Necessary and suffcient condition for LR-decompositions

Let $ A=\left(a_{i j}\right)_{i, j=1, \ldots, n} \in \mathbb{R}^{n \times n} $ be an invertible matrix. Show that an $ L R $-decomposition of $ A $ exists if and only if $ \operatorname{det}\left(A^{[...
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LU Decomposition without Gaussian Elimination

I'm creating a program to compute the LU factorization of a matrix, and I was wondering if there was a way to compute an LU factorization without using Gaussian Elimination. I'm mainly worried about ...
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Block LU factorization with more than two blocks?

If I have a symmetric, positive definite block matrix there exists the following LU decomposition: $$\left[\matrix{A && B^\intercal \\ B && C}\right]=\left[\matrix{A^{\frac{1}{2}} &...
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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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Do you have to start with the first row during LU decomposition?

I guess I've never run into this pretty simple problem, but I'm doing a Cholesky Factorization of the matrix shown below, and ran into some weird ambiguity in how I usually do $LU$ decomp: $$ A = \...
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Efficient LU decomposition of matrix after updating diagonal

I am computing LU decomposition of $(kD + A)$ where $D$ is diagonal matrix with {$d_{1}$, $d_{2}$, ... , $d_{n}$}, $A$ is a real symmetric positive-definite matrix, $k$ is a number that changes on ...
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Is there any situation where the LDU decomposition is the same as the eigenvalue decomposition?

I was just wondering if there are any situation where the LDU decomposition is the same as eigenvalue decomposition (diagonalization)? The only way this can be possible if L and U are inverse so ...
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DU decomposition

My professor said that we can't express a square matrix as a product of upper triangular matrix and lower triangular although it can be expressed as a product of lower triangle matrix and an upper ...
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Example of regular tridiagonal matrix $A$ with given properties

I am looking for a regular tridiagonal matrix $A$ such that at the LU-decomposition with partial column pivoting the matrices $L$ and $U$ are also tridiagonal, but with total pivoting the matrices $L$ ...
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Calculate the LU-decomposition $PA=LU$ : How do we calulate $L$?

Calculate the LU-decomposition $PA=LU$ for the matrix $$A=\begin{pmatrix}3 & 1 & -3 & 2 \\ -2 & 1 & 0 & 0 \\ 2 & -2 & 4 & 1 \\ 0 & -1 & -1 & 3\end{...
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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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How to switch rows in matrix L when decomposing matrix A into PA = LU?

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 ...
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Diagonal entries of $U$ in $LU$ factorisation of positive definite matrix

Let $A\in M(n,\mathbb R)$ be a symmetric positive definite matrix. Let $L$ be a lower triangular matrix with real entries, all whose diagonal entries are $1$ and $LA$ is upper triangular. Then, is it ...
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Finding the LU decomposition of a matrix using the shortcut method

I was asked to find the LU decomposition of $$\begin{bmatrix}5&4\\-2&-3\\\end{bmatrix}$$ I know that the shortcut method means finding the upper and using the multiplier to find the lower. In ...
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Calculate number of eigenvalues in interval $[-2, 3>$ of matrix $A$ using Sylvesters law of inertia and $LDL^T$ decomposition.

I have a new one, and I am not sure about a few things. I hoped you might help me in understanding them. For matrix $$ A=\left[ \begin{matrix} 4 & 4 & 0 \\ 4 & 6 & 2 \\ 0 & 2 & ...
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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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proving that lu decomposition is not unique on singular matrix.

How to prove that the following isn't true (using 3 by 3 matrix): Given A is a square and a singular matrix (which means non invertible), if LU decomposition is possible without the use of ...
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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{...
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