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

# Questions tagged [lu-decomposition]

134 questions

**27**

votes

**2**answers

### LU Decomposition vs. Cholesky Decomposition

What is the difference between LU Decomposition and Cholesky Decomposition about using these methods to solving linear equation systems?
Could you explain the difference with a simple example?
Also could you explain the differences between these…

mertyildiran

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**10**

votes

**1**answer

### Decompose invertible matrix $A$ as $A = LPU$. (Artin, Chapter 2, Exercise M.11)

Decompose matrix $A$ as $A = LPU$, where $A \in Gl_n( \mathbb{R}^n)$, $L$ is lower triangular, $U$ is upper triangular with diagonal elements of $1$, and $P$ is a permutation matrix.
It is fairly easy to decompose any invertible such $A$ as $PA =…

gnometorule

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**9**

votes

**5**answers

### Is the $L$ in $LU$ factorization unique?

I was doing an $LU$ factorization problem
\begin{bmatrix}
2 & 3 & 2 \\
4 & 13 & 9 \\
-6 & 5 &4
\end{bmatrix}
and I was going to multiply the second row by .$5$ and subtract the result from row $1$, then do something similar to…

thecat

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**8**

votes

**1**answer

### Cholesky decomposition when deleting one row and one and column.

I've thought about this problem for days but could not find a good answer.
Given Cholesky decomposition of a symmetric positive semidefinite matrix $A = LL^T$. Now, suppose that we delete the $i$-th row and the $i$-th column of $A$ to obtain $A'$…

user97656

**8**

votes

**0**answers

### Can I go from the LU factorization of a symmetric matrix to its Cholesky factorization, without starting over?

I mistakenly computed the LU factorization and then realized that the question is asking for a Cholesky factorization, i.e., finding a lower triangular matrix L such that the symmetric matrix A has factorization $LL^T$.
Can I modify this…

User001

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**7**

votes

**1**answer

### 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 $A$ is ill conditioned ?

Pedro Alvarès

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**6**

votes

**1**answer

### What is the computation time of LU-, Cholesky and QR-decomposition?

I found these information about computation-time of following decompositions:
Cholesky: (1/3)*n^3 + O(n^2) --> So computation-time is O(n^3)
LU: 2*(n^3/3) --> So computation-time is O(n^3) also (not sure)
QR: (2/3)*n^3 + n^2 + (1/3)*n- 2 --> So…

ZelelB

- 305
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**6**

votes

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### Proof of uniqueness of LU factorization

The first question: what is the proof that LU factorization of matrix is unique? Or am I mistaken?
The second question is, how can theentries of L below the main diagonal be obtained from the matrix $A$ and $A_1$ that results from the row echelon…

Tashima Sasaki

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**5**

votes

**1**answer

### 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}
\begin{align} Ax = b\end{align}
Where
n is large
G: real…

Walden95

- 55
- 5

**5**

votes

**2**answers

### Complexity/Operation count for the forward and backward substitution in the LU decomposition?

If I have a linear system of equations $Ax=b$ where $A \in \mathbb{R} ^{n\times n}, x \in \mathbb{R} ^{n}, b \in \mathbb{R} ^{n} $ this system can be solved for $x$ via an LU decomposition: $$A = LU$$ where $U \in \mathbb{R} ^{n\times n}$ is upper…

silver96

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**4**

votes

**1**answer

### 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:
My thinking is that $(LD^{1/2})(D^{1/2}U)$ is…

Museful

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**4**

votes

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### Condition number and $LU$ decomposition

Consider $A: n \times n$ non-singular and the factors $L$ and $U$ of $A$ obtained with partial pivoting strategy, such as: $PA = LU$.
Proof that $$\kappa_{\infty}(A) \geq \dfrac{||A||_{\infty}}{\min_{j}|u_{jj}|}.$$
The condition number…

Samuel Francisco

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**3**

votes

**1**answer

### $PA = LU$ 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 following matrices:
(The i,j in $E_{ij}$ denotes the element in matrix A…

Kakarot_7

- 141
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**3**

votes

**3**answers

### Why is $LU$ preferred over $A^{-1}$ to solve matrix equations?

I understand the whole $LU$-decomposition vs Gaussian elimination argument. The fact that you can isolate the computationally expensive elimination step and re-use the $L$ and $U$ matrices for $Ax=b$ style equations with different $b$:s makes sense…

Erik

- 53
- 3

**3**

votes

**2**answers

### LU-decomposition of A

I have:
$A=\begin{bmatrix}
2 & -1 & 2 & 3 & 4
\\
4 & -2 & 7 & 7 & 6
\\
2 & -1 & 20 & 9 & -8
\end{bmatrix}$
and I'm asked to LU-decomposition A, then solve $Ax=0$.
What I did:
With these steps:
$1)$ $R2=R2-2\times R1$
$2)$ $R3=R3-R1$
I…

RedRose

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