Matrix Details

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Matrix Details

A matrix is a rectangular array that can describe a transformation from one coordinate system to another. Transformations that matrices can describe include translation (shifting), scaling, and rotation. Functions dealing with matrices begin with “mat”. A 3 × 3 or 4 × 4 matrix has 9 or 16 elements, respectively.

This section contains detailed information on matrices.

Contents

• Matrix Details
• Contents
• Arrangement
  • A Matrix Transforms Between Coordinate Systems
  • Why a 4 × 4 Matrix?
• Transforming Points
  • Scaling
  • Translation
  • Rotation
  • Matrix Multiplication
  • Projective Transformations
  • Matrix Inversions
    • Inverting a General NxN Matrix
• Rotation Example

Arrangement

In mathematical publications, matrices are often notated in column-major order, in which each element of the matrix is placed in columns as opposed to rows, as in the following example:

$\left[\begin{array}{cccc}\mathrm{matrix[0]}& \mathrm{matrix[4]}& \mathrm{matrix[8]}& \mathrm{matrix[12]}\\ \mathrm{matrix[1]}& \mathrm{matrix[5]}& \mathrm{matrix[9]}& \mathrm{matrix[13]}\\ \mathrm{matrix[2]}& \mathrm{matrix[6]}& \mathrm{matrix[10]}& \mathrm{matrix[14]}\\ \mathrm{matrix[3]}& \mathrm{matrix[7]}& \mathrm{matrix[11]}& \mathrm{matrix[15]}\end{array}\right]$

The numbers in brackets in the matrix above are the zero-based indices into the matrix arrays passed to MathUtil’s matrix methods.

For 3 × 3 matrices, the elements are arranged in the following order:

$\left[\begin{array}{ccc}\mathrm{matrix[0]}& \mathrm{matrix[3]}& \mathrm{matrix[6]}\\ \mathrm{matrix[1]}& \mathrm{matrix[4]}& \mathrm{matrix[7]}\\ \mathrm{matrix[2]}& \mathrm{matrix[5]}& \mathrm{matrix[8]}\end{array}\right]$

A Matrix Transforms Between Coordinate Systems

A transformed 3D coordinate system is made up of an X, Y, and Z axis, and a center of the coordinate system. These are four 3-element vectors that describe how the three axes and the center map to the new coordinate system in relation to the old coordinate system.

The following depiction of a 4 × 4 matrix illustrates the meaning of each of its elements. To keep things simple, this matrix’s transformation is one that keeps straight lines straight and parallel lines parallel.

$\left[\begin{array}{cccc}\mathrm{[0]\; X-axis\; X}& \mathrm{[4]\; Y-axis\; X}& \mathrm{[8]\; Z-axis\; X}& \mathrm{[12]\; Center\; X}\\ \mathrm{[1]\; X-axis\; Y}& \mathrm{[5]\; Y-axis\; Y}& \mathrm{[9]\; Z-axis\; Y}& \mathrm{[13]\; Center\; Y}\\ \mathrm{[2]\; X-axis\; Z}& \mathrm{[6]\; Y-axis\; Z}& \mathrm{[10]\; Z-axis\; Z}& \mathrm{[14]\; Center\; Z}\\ \mathrm{[3]\; 0}& \mathrm{[7]\; 0}& \mathrm{[11]\; 0}& \mathrm{[15]\; 1}\end{array}\right]$

The following is an example of a transformation matrix.