= Matrix
:author:
:revnumber:
:revdate: 2020/07/15



== Matrix

See link:{link-javadoc}/com/jme3/math/Matrix3f.html[Javadoc of Matrix3f] and link:{link-javadoc}/com/jme3/math/Matrix4f.html[Javadoc of Matrix4f]


=== 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.
[cols="3", options="header"]
|===

a|0
a|0
a|0

a|0
a|0
a|0

a|0
a|0
a|0

|===

The _Identity Matrix_ is the matrix with 1 on the diagonal entries and 0 for all other entries.
[cols="3", options="header"]
|===

a|1
a|0
a|0

a|0
a|1
a|0

a|0
a|0
a|1

|===

A Matrix is _invertible_ if there is a matrix _M^-1^_ where _MM^-1^ = M^-1^ = I_.

The _transpose_ of a matrix _M = [m~ij~]_ is _M^T^ = [m~ji~]_. This causes the rows of _M_ to become the columns of _M^T^_.
[cols="7", options="header"]
|===

a|1
a|1
a|1
<a|
a|1
a|2
a|3

a|2
a|2
a|2
a| ⇒
a|1
a|2
a|3

a|3
a|3
a|3
<a|
a|1
a|2
a|3

|===

A Matrix is symmetric if _M_ = _M^T^_.
[cols="3", options="header"]
|===

a|X
a|A
a|B

a|A
a|X
a|C

a|B
a|C
a|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 xref:tutorials:intermediate/terminology.adoc#vectors[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:
[cols="3", options="header"]
|===

a|0
a|u~2~
a|-u~1~

a|-u~2~
a|0
a|u~0~

a|u~1~
a|-u~0~
a|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:
[cols="2", options="header"]
|===

a|M
a|T

a|S^T^
a|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.


=== jME Class

Both Matrix3f and Matrix4f store their values as floats and are publicly available as (m00, m01, m02, …, mNN) where N is either 2 or 3.

Most methods are straight forward, and I will leave documentation to the Javadoc.