# Matrix

## Matrix

### Definition

A Matrix is typically used as a *linear transformation* to map vectors to vectors. That is: Y = MX where X is a Vector and M is a Matrix applying any or all transformations (scale, rotate, translate).

There are a few special matrices:

*zero matrix* is the Matrix with all zero entries.

0 | 0 | 0 |
---|---|---|

0 |
0 |
0 |

0 |
0 |
0 |

The *Identity Matrix* is the matrix with 1 on the diagonal entries and 0 for all other entries.

1 | 0 | 0 |
---|---|---|

0 |
1 |
0 |

0 |
0 |
1 |

A Matrix is *invertible* if there is a matrix *M ^{-1}* where

*MM*.

^{-1}= M^{-1}= IThe *transpose* of a matrix *M = [m _{ij}]* is

*M*. This causes the rows of

^{T}= [m_{ji}]*M*to become the columns of

*M*.

^{T}1 | 1 | 1 | 1 | 2 | 3 | |
---|---|---|---|---|---|---|

2 |
2 |
2 |
⇒ |
1 |
2 |
3 |

3 |
3 |
3 |
1 |
2 |
3 |

A Matrix is symmetric if *M* = *M ^{T}*.

X | A | B |
---|---|---|

A |
X |
C |

B |
C |
X |

Where X, A, B, and C equal numbers

jME includes two types of Matrix classes: Matrix3f and Matrix4f. Matrix3f is a 3x3 matrix and is the most commonly used (able to handle scaling and rotating), while Matrix4f is a 4x4 matrix that can also handle translation.

### Transformations

Multiplying a Vector with a Matrix allows the Vector to be transformed. Either rotating, scaling or translating that Vector.

#### Scaling

If a *diagonal Matrix*, defined by D = [d_{ij}] and d_{ij} = 0 for i != j, has all positive entries it is a *scaling matrix*. If d_{i} is greater than 1 then the resulting Vector will grow, while if d_{i} is less than 1 it will shrink.

#### Rotation

A *rotation matrix* requires that the transpose and inverse are the same matrix (R^{-1} = R^{T}). The *rotation matrix* R can then be calculated as: R = I + (sin(angle)) S + (1 - cos(angle)S^{2} where S is:

0 | u_{2} |
-u_{1} |
---|---|---|

-u |
0 |
u |

u |
-u |
0 |

#### Translation

Translation requires a 4x4 matrix, where the Vector (x,y,z) is mapped to (x,y,z,1) for multiplication. The *Translation Matrix* is then defined as:

M | T |
---|---|

S |
1 |

where M is the 3x3 matrix (containing any rotation/scale information), T is the translation Vector and S^{T} is the transpose Vector of T. 1 is just a constant.