## Description

It is possible to compute matrix factorizations with Octave, including the $LU$, $QR$, and singular value decompositions of a matrix. Suppose

(1)## Sage Cell

#### Code

```
A=[1 3 7; 3 -3 1; 4 1 4];
[L U] = lu(A)
L*U
```

## Options

#### Option

The $QR$ factorization of a matrix can be computed with the command `qr`.

#### Code

```
A=[1 3 7; 3 -3 1; 4 1 4];
[Q R] = qr(A)
Q*R
```

#### Option

The singular value decomposition factors any $m \times n$ matrix $A$ into $A = Q_1 \Sigma Q_2^{-1}$, where $Q_1$ is an $m \times m$ orthogonal matrix whose columns are the eigenvectors of $AA^T$, $Q_2$ is an $n \times n$ orthogonal matrix whose columns are the eigenvectors of $A^TA$ and $\Sigma$ is an $m \times n$ matrix whose diagonal entries are the square roots of the eigenvalues of $AA^T$ ($A^TA$ has the same eigenvalues as $AA^T$).

#### Code

```
A=[1 3 7 3; 3 -3 -5 1; 4 1 -7 4];
[U,S,V] = svd(A,0)
U*S*V'
```

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Author: T. Judson

Date: 10 Dec 2018 16:07

Submitted by: Tom Judson