All Questions
Tagged with matrix-decomposition lu-decomposition
76
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Decompositions of symplectic matrices over the integers
Given a symplectic matrix $S \in \text{Sp}(2n,\mathbb Z)$ whereby $S^T\Omega S=\Omega$ with
$$\Omega=\left(\begin{matrix}0&I_n\\-I_n&0\end{matrix}\right)$$
what known decompositions exist such ...
- 429
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2
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32
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Counting LU decompositions of 2 by 2 matrices over a finite field of 5 elements.
Let $G=GL(2, \mathbb{F}_5)$, i.e., group of invertible $2\times 2$ matrices over a finite field with $5$ elements. Let $S$ be the subset of those elements of $G$ that can be written in the form $LU$, ...
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Is the LU decomposition just Gauss-Jordan elimination?
I am watching Gilbert Strang's neat lecture on the LU decomposition, which is taught just after Gaussian elimination. $LU$ for a matrix $A$ was found doing $EA=U$ and finally $A=E^{-1}U$.
Seems to me, ...
- 129
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Missing the point of LU factorization / decomposition
Gaussian Elimination
The system of linear equations $Ax = b$ may be solved by using Gaussian Elimination (GE) arriving to a Row Echelon Form R of the augmented matrix $[A b]$, and then using back-...
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34
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Decomposing a matrix that has duplicate columns using PA=LU factorization
I am given the following matrix.
$$ A = \begin{bmatrix} 3 & 3 & 9 & 6 \\ 4 & 4 &4 &4 \\ 1 & 1 & 5 & 5 \\ 2 & 2 & 4 & 6\end{bmatrix} $$
As you can notice,...
- 37
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Ways of showing that the LU decomposition of a square matrix almost always exists?
We know that (in the measure-theoretic sense) almost all square matrices $M$ over $\mathbb R$ admit an LU decomposition: $M = LDU$. We are using the Lebesgue measure, and we may treat for each fixed $...
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153
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Do positive semidefinite matrix have LU decomposition? [closed]
Suppose we have a real matrix $A=R R^t$ where $R$ is triangular. Since there is no restriction on $R$ we can say that $A$ is positive semidefinite. Can I affirm that $A$ has LU decomposition (L lower ...
- 21
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358
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MATLAB Code Help. Using Crout's Method, solve the system of linear equation $Mz=f$, where $M=\begin{pmatrix}I &A\\A^T&0\end{pmatrix}$
Using Crout's Method, solve the system of linear equation $Mz=f$, where $$M=\begin{pmatrix}I &A\\A^T&0\end{pmatrix}$$
I have implemented algorithm of Crout's method. But I don't have any idea ...
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214
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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 $A$), ...
- 103
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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 ...
- 147
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343
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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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398
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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. ...
- 147
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LU- decomposition
Find an LU-decomposition of the coefficient matrix and solve the system $$\begin{pmatrix}
1 & 4 & 3\\ -1 & -1 & 3 \\ 2 & 9 & 8 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \...
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A question about the product of commutators in an article of Vaserstein
The question comes from Lemma 13. It is stated as follows.
Let $A$ be an associative ring with $1$, and $n\geq2$ an integer. Assume that either $n\geq3$ or $n=2$, and $1$ is the sum of two units in $...
- 727
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Block LU factorization with more than two blocks?
If I have a symmetric, positive definite block matrix there exists the following LU decomposition:
$$\left[\matrix{A && B^\intercal \\ B && C}\right]=\left[\matrix{A^{\frac{1}{2}} &...
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